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 4* 
 
 ARITHMETIC, 
 
 
 
 1 -'ainnnuMt AJM-r-Vorks, Philadelphia 1 1 
 
 
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 VJ. PHILADELPHIA. H 
 
 
 

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 IN MEMORIAM 
 FLOR1AN CAJ0R1 
 
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Digitized by the Internet Archive 
 
 in 2007 with funding from 
 
 Microsoft Corporation 
 
 http://www.archive.org/details/adamsarithmOOadamrich 
 
REV 
 
 ISED AND IMPROVEDlijDITION. 
 
 ARITHMETIC: 
 
 IN TWO PARTS. 
 PART FIRST, 
 
 ADVANCED LESSONS IN MENTAL ARITHMETIC. 
 
 PART SECOND, 
 
 RULES AND EXAMPLES FOR PRACTICE IN 
 WRITTEN ARITHMETIC. 
 
 FOR CO'MMON AND RTGII SCHOOLS. 
 
 BY FREDERIC A. ADAMS, A.M., 
 
 FORMER PRINCIPAL OF DUMMER ACADEMY. 
 
 THIRTEENTH THOUSAND. 
 
 PHILADELPHIA: 
 THOMAS, COWPERTHWAIT & CO. 
 
 New York, Roe Lockwood &. Son : —Boston, Phillips, Sampson <fc Co ; B. B. Mussev 
 
 & Co. ; W. J. Reynolds <k Co. :— Baltimore, Cushing & Bailey :— Charleston, S. C, 
 
 McCarter <& Allen : — Louisville, Ky., Morton & Griswold; Beckwith & 
 
 Morton: — St. Louis, Fisher <fc Bennett; Wm. D. Skillman; Amos H. 
 
 Shultz: — Cincinnati, J. F. Desilver: — Nashville, Wm. T. Berry; 
 
 Chas.W. Smith :-Memphis,C. C.Cleaves :-Lexington,C. S. Bodley : 
 
 — Macon, Geo , J. M. Boardman:— Buffalo, T. & M. Butler. 
 
 1851. 
 
ifrs-7 
 
 Entered accoflding to Act of Congress, in the year 1848, 
 
 BY DANIEL BIXBY, 
 
 In the Clerk's Office of the District Court of the District of Massachusetts. 
 
 RECOMMENDATIONS. 
 
 From Mr. George B. Emerson, Boston. 
 I have carefully examined the plan of Mr. Adams's work on Mental Arithmetic, 
 and have given some attention to its execution ; and I am confident that it will prove 
 a very valuable addition to the means of instruction in Arithmetic. It is a suc-.| 
 cessful extension of the admirable method of Colburn's First Lessons, with such i 
 modifications as seemed to be required in a higher work on the same general model. 
 It occupies unappropriated ground ; and it deserves, and I think it will take, a high 
 place amongst the text-books. GEO. B. EMERSON. 
 
 From Mr. Thomas Shcrwin, Boston. 
 I have carefully examined, in manuscript, the work of Mr. Adams on Mental Arith- 
 metic, and am much pleased with it. His plan is good, and well executed. I would, 
 therefore, heartily recommend his book to Teachers and School Committees, as one 
 which will contribute very materially to the attainment of that very important, but 
 much-neglected branch of study, Intellectual Arithmetic. 
 
 T. SHERWIN, Principal of Boston English High School. 
 
 From Professor Chase, of Dartmouth College. 
 Mr. F. A. Adams. *•».*, ■ - ; Hanover, Oct. 12, 1846. 
 
 My Dear Sir: — I have ■eJcaJniaed, with some* c,a*e.; your treatise on Arithmetic, 
 and ajn much pleased with k. '.The practfce aiid halut of extending mental operations 
 to large numbers is of great utility. I have occasion, very frequently, to see the 
 inconvenience that yg^ng njen 'suffer /ran*, -ike; want of« such a habit. Not less valu- 
 able than the habit ©f. operating 'mentalty -upon la/ge^humbers, is the habit of per- 
 forming the more advanced operations" oi* arithmetic without the aid of the pencil. 
 
 I like very much, also, the manner in which you have treated several of the princi- 
 ples which you have developed ; as, for example, the subject of the common divisor, 
 the least common multiple, the roots, ratio, and proportion. These are but few of the 
 subjects, but I mention them as examples. 
 
 I think the book will do much to promote the proper method of teaching arithme- 
 tic, — by demonstration and explanation. I am, Dear Sir, very truly yours, &c. 
 
 S. CHASE. 
 
 From Mr. John Tatlock, Professor of Mathematics, and Mr. A. Hopkins, Professor of 
 JYutural Philosophy. 
 
 Williams College, Nov. GO, 1846. 
 I have examined a treatise on Arithmetic by F. A. Adams, and am much pleased 
 with it. I think it well adapted to teach the science and art of numbers, and at the 
 same time to teach the art of thinking. I am persuaded that a thorough training in 
 this Arithmetic would prepare students for the farther study of mathematics betier 
 than nine tenths are now prepared. 
 I should be glad if every student who enters college was master of this Arithmetic. 
 
 JOHN TATLOCK. 
 A. HOPKINS. 
 
 In School Committee, Roxburv, Feb. 17, 1847. 
 The undersigned, members of the Committee on Text-Books, recommend that the 
 Arithmetic prepared by Frederic A. Adams, be used in the grammar schools, by a!l 
 the classes which now use Leonard's Arithmetic ; and that hereafter no other Arkh- 
 me»ic be used in the public schools, than Adams's together with Colburn's First 
 Lessons. SAM'L. H. WALLEY, Jr. DAN'L. LEACH, 
 
 B. E. CUTTING. THEO. PARKER. 
 
 GEO. PUTNAM, 
 In School Committee, Feb. 17, 1847. — The above recommendation of the Cem- 
 mittee on Text-Books, was this day adopted. JOSHUA SEAVER, Secr'y. 
 
 STEREOTYPED AT THE BOSTON TYPE AND 8TEREOTYPE FOUNDRY. 
 
 Cajori 
 
PREFACE 
 
 The study of Arithmetic in the schools of this 'country received 
 its best impulse, unquestionably, in the publication of Colburn's 
 First Lessons. The use of this book, and of others made on a sim- 
 ilar plan, has done much towards placing this branch of study on its 
 proper ground. In all our best schools, in the elementary stages of 
 this study, the logical mode has taken the place of the merely for- 
 mal; reason is the guide, instead of rule. 
 
 The want of a higher work on Mental Arithmetic has long been 
 felt by teachers ; and by business men, who have been compelled, 
 after having received all the training of the school-room, to adopt 
 practical modes of calculation of their own, to meet the exigencies 
 of their daily business. It is to meet these wants that the following 
 work has been prepared. The design of it is, — 
 
 To accustom the pupil to perform, with ease and readiness, mental 
 calculations upon somewhat large numbers ; — 
 
 To present these operations in their natural form, freed from the 
 inverted and mechanical methods which belong of necessity to 
 operations in Written Arithmetic ; — 
 
 To train the student to such a power of apprehending the rela- 
 tions of numbers, as shall give him an insight into the grounds of 
 the. rules of Arithmetic; and, consequently, shall release him from 
 dependence on those rules; — * 
 
 To prepare the members of our schools, when they shall have left 
 school, and engaged in the active pursuits of life, to solve mentally, 
 and with ease and delight, a large share of those questions, of busi- 
 
 
4 PREFACE. 
 
 ness or curiosity, for which a process of ciphering is usually thought 
 indispensable. 
 
 The "Advanced Lessons" presuppose, of course, the knowledge 
 of some more elementary book. To study this work successfully, 
 therefore, the pupil must be acquainted with Colburn's First Les- 
 sons, or some other work occupying essentially the same ground. 
 
 In all the mental calculations in large sums, it will be found a 
 uniform characteristic of this work, to begin with the highest order 
 of numbers in the sum, — hundreds before tens, tens before units. 
 In this way, the numbers are presented in the same order in which 
 they are presented in the common usage of our language. In most 
 of the operations of Written Arithmetic, however, the smallest 
 number is taken first; and thus a method is pursued, the reverse 
 ot what the genius of our language would naturally suggest. 
 Another advantage of taking the highest numbers first, in Mental 
 Arithmetic, is, that we thus obtain a large approximation to the 
 final answer, at the first step. When the first step, however, as 
 in written addition or multiplication, furnishes only the units of 
 the answer, leaving the hundreds or thousands still unknown, only 
 a minute fraction of the answer is at first obtained. It is too plain 
 to require proof, that that method will be most interesting and 
 gratifying to the mind, which secures the largest portion of the 
 answer at the first step. Another advantage of the method here 
 used, is found in the fact, that we naturally make the higher 
 order the standard; and the lower order takes its value in the 
 mind from a comparison with the higher, as a certain part of it. 
 Thus 150 is apprehended by the mind, as one hundred and half 
 a hundred. This is not, indeed, the method of acquiring the 
 idea of large numbers, but the method of combining them after 
 the idea has been acquired ; consequently, it is the legitimate 
 method of instruction, just as soon as the pupil is qualified to 
 enter on the study of such combinations. If, now, we obtain the 
 number of the highest order first, we have a standard, under 
 which all the succeeding orders naturally fall, and from a com- 
 parison with which they successively take their value. If we 
 begin with units, however, and work upward through the higher 
 
PREFACE. S> 
 
 orders, we obtain no standard ; we must hold the successive num- 
 bers in suspense, until the last term shall furnish the nucleus for 
 the group, — the standard under which all the lower orders shall 
 take their rank. 
 
 It is on the basis of these facts, which are only indications of 
 the laws of the mind, that, throughout the mental part of this 
 Arithmetic, the author has, in all operations, taken the highest 
 order of numbers first. The increased interest which the per- 
 severing use of this method will awaken in the minds of pupils, 
 will be, to teachers, a better commendation of its correctness, 
 than any more extended mental analysis. 
 
 Other characteristic features of the Advanced Lessons are, the 
 extended Multiplication Table ; the use of the complement in 
 addition; the analytical treatment of fractions, vulgar and decimal ; 
 the careful separation of the three topics, linear, square, and solid 
 measure ; the construction of the square and the cube ; . and the 
 mode of treating proportion. 
 
 The Second Part contains examples in Written Arithmetic on 
 all the most important rules. They are designed to be sufficiently 
 numerous to lead the student to ready and accurate practice in 
 ciphering. In this Part the author has aimed to interest the 
 scholar by furnishing him with natural and reasonable questions, 
 and to aid both teacher and scholar by arranging them progres- 
 sively. 
 
 The rules and explanations will, probably, be found sufficient, 
 after a thorough mastery of the First Part. It is not necessary 
 that the pupil complete the First Part before beginning the 
 Second. He may carry on both Parts at the same time; but, 
 under each particular head, the mental part should be thoroughly 
 mastered before the written examples are begun. 
 
 The answers to the questions in the Second Part are given In 
 a separate work. This course his seemed to the author, on the 
 whole, the best, notwithstanding some incidental disadvantages 
 that may arise from it. It will enable the teacher to oversee a 
 much larger amount of work in Arithmetic than he could other- 
 wise attend to. 
 
 1* 
 
6 PREFACE. 
 
 The Key will be bound up with the Arithmetic, for the use 
 of teachers ; and such copies will be lettered " Teacher's Copy." 
 
 The present contains a considerable number of examples more 
 than the Third Edition, but no change in the numbering of the 
 sections or of the examples, to occasion inconvenience to the 
 teacher. 
 
 To aid in awakening a higher interest and zeal in this branch 
 of study, the author will offer a few suggestions. 
 
 Let the Key be used as little as the teacher's necessities will 
 permit. 
 
 Let original questions be proposed by the teacher in connection 
 with every Section. 
 
 Each member of the class should be encouraged to propose 
 original questions to be solved by the class. 
 
 It will often be "useful, especially in a review, to alter some 
 one figure in the conditions of each question. This often pro- 
 duces a happy excitement, and gives quite a new zest to the 
 study. 
 
REVISED EDITION 
 
 A recent revision of the work has led to a few alter- 
 ations in the First Part, designed to render the pupil's 
 course more strictly progressive. A thorough acquaint- 
 ance with the author's First Book in Arithmetic, em- 
 bracing the " lessons for practice in rapid calculation," 
 will prepare the pupil to enter successfully on the study 
 of this book. It is indispensable, however, to the rapid 
 progress of the pupils, that they commence at the begin- 
 ning, making the thorough mastery of each successive 
 lesson the basis of the studies that follow. 
 
 Orange, New Jersey, 
 September 1, 1850. 
 
CONTENTS 
 
 PART FIRST. 
 
 Section. Page. 
 
 Explanations, 11 
 
 I. Multiplication of Tens and Units, 13 
 
 II. Multiplication of Tens and Units. Complement, 16 
 
 III. Practical Questions, 19 
 
 IV. Division, 21 
 
 V. Time. Linear Measure, 28 
 
 VI. Federal Money. Sterling Money. Dry Measure. 
 Avoirdupois Weight. Troy Weight. Apothecaries' 
 Weight. Cloth Measure. Wine Measure. Beer 
 
 Measure. Measure of the Circle, 38 
 
 VII. Prime Numbers, 50 
 
 VIII. Multiplication and Division of Fractions. To find the 
 
 Divisors of Numbers, 59 
 
 IX. Multiplication of Fractions by Fractions. Division of 
 Fractions by Fractions. To Multiply or Divide Whole 
 Numbers by Fractions. Addition of Fractions. To find 
 
 a Common Denominator, G5 
 
 X. The Least Common Multiple, 73 
 
 XI. Practical Questions, 77 
 
 XII. Decimal Fractions. Addition and Subtraction of Deci- 
 mals. Multiplication of Decimals. Division of Deci- 
 mals, 80 
 
 XIII. Reduction of Vulgar Fractions to Decimals, 80 
 
 XIV. Interest. Banking. Discount. Loss and Gain. Per 
 
 Centage, 93 
 
 XV. Square Measure, ■ 101 . 
 
 XVI. Construction of the Square. Practical Questions, 106 
 
 XVII. Practical Questions in Square Measure, 114 
 
 XVIII. Analysis of Problems, 119 
 
 XIX. Solid Measure. Construction of the Cube, 122 
 
 XX. Ratio. Proportion. Comparison of Similar Surfaces. 
 
 Comparison of Similar Solids, 120 
 
 Notes to Part First, 143 
 
CONTENTS. 
 
 PART SECOND 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 X. 
 
 XI. 
 
 XII. 
 
 XIII. 
 
 XIV. 
 
 XV. 
 
 XVI. 
 
 XVII. 
 
 XVIII. 
 
 XIX. 
 
 XX. 
 
 XXI. 
 
 XXII. 
 
 XXIII. 
 
 XXIV. 
 
 XXV. 
 
 XXVI. 
 
 XXVII. 
 
 XXVIII. 
 
 XXIX.- 
 
 XXX. 
 
 XXXI. 
 
 XXXII. 
 
 XXXIII. 
 
 Numeration of Whole Numbers. Numeration of De- 
 cimals, ...» 147 
 
 Addition, 150 
 
 Subtraction, 152 
 
 Multiplication, 155 
 
 Division, 158 
 
 Reduction, 163 
 
 Reduction, 165 
 
 Compound Addition, 167 
 
 Compound Subtraction, '. 170 
 
 Compound Multiplication, 172 
 
 Compound Division, 173 
 
 Miscellaneous Examples, 174 
 
 Divisibility of Numbers, 175 
 
 Reduction of Fractions, 176 
 
 Change of Numbers and Fractions to Higher Terms, 179 
 
 Multiplication and Division of Fractions, 180 
 
 Multiplication and Division of Fractions, 182 
 
 Addition and Subtraction of Fractions, 184 
 
 Reduction of Denominate Fractions, 186 
 
 Change of Denominate Integers to Fractions, 188 
 
 Practical Examples, 189 
 
 Decimal Fractions. Addition and Subtraction. Mul- 
 tiplication and Division, 190 
 
 Reduction of Vulgar Fractions to Decimals. Repeat- 
 ing and Circulating Decimals, 192 
 
 Reduction of Denominate Integers to Decimals, 194 
 
 To find the Integral Value of Denominate Decimals, 195 
 
 Practical Examples, 196 
 
 Practical Questions in Vulgar and Decimal Fractions, 198 
 Reduction of Currencies. English Currency. Fed- 
 eral Money to Sterling. Canada Currency. New- 
 England Currency. New York Currency. Penn- 
 sylvania Currency, 200 
 
 Interest, , 203 
 
 Partial Payments. Annual Interest, 206 
 
 Discount, 210 
 
 Banking, 211 
 
 Loss and Gain. Per Centage, 212 
 
 Alligation, 216 
 
10 
 
 CONTENTS. 
 
 XXXIV. 
 
 XXXV. 
 
 XXXVI. 
 
 XXXVII. 
 
 XXXVIII. 
 
 XXXIX. 
 
 XL. 
 
 XLI. 
 
 XLII. 
 
 XLIII. 
 
 XLIV. 
 
 XLV. 
 
 XLVI. 
 
 Equation of Payments, • 220 
 
 Square Measure, • • • • 222 
 
 Duodecimals, 224 
 
 Extraction of the Square Root, 226 
 
 Extraction of the Cube Root, 230 
 
 Proportion. Practical Questions. Partnership, . . . 233 
 Arithmetical Progression. Descent of Falling 
 
 Bodies, 243 
 
 Geometrical Progression, 248 
 
 Mensuration of Surfaces, 250 
 
 Mensuration of Solids, 252 
 
 Miscellaneous Theorems and Questions. Specific 
 Gravity. Mechanical Powers. The Lever. The 
 Wheel and Axle. The Screw. Strength of 
 Beams to resist Fracture. Stiffness of Beams to 
 
 resist Flexure, 254 
 
 Business Forms and Instruments. Promissory 
 Notes ; — on Demand, with Interest ; on Time, 
 with Interest ; on Time, without Interest; payable 
 by Instalments, with Periodical Interest. Remarks 
 on Promissory Notes. Receipts. A General 
 Form ; for Money paid by another Person ; for 
 Money received for Another ; in Part of a Bond ; 
 for Interest due on a Bond ; on Account ; of Pa- 
 pers. Order at Sight. Order on Time. Award 
 by Referees. Letter of Credit for Goods. Power 
 
 of Attorney, 266 
 
 On the Standard of Weights and Measures. The 
 English System ; adopted by the Government of 
 the United States. French Decimal System. 
 French Long Measure. French Square Measure. 
 
 French Decimal Weight, 271 
 
 Miscellaneous Examples, 276 
 
EXPLANATIONS 
 
 1 The sign = indicates equality; as, 7 times 3 = 21. 
 
 2. The sign -|- indicates addition; as, 15 -f- 7 = 22. 
 
 3. The sign — , placed between two numbers, indicates 
 that the latter number is to be taken from the former; as, 
 9—4 = 5. 
 
 The larger number is called the minuend ; the smaller, the 
 subtrahend. 
 
 4. The sign X indicates multiplication; as, 6X7 = 42. 
 
 The two numbers are called factors ; the number multi- 
 plied is called the multiplicand; the number by which it is 
 multiplied, the multiplier. 
 
 5. The sign -£- indicates that the number placed before it 
 is to be divided by the number after it; as, 15-^-5 = 3. 
 
 The number to be divided is called the dividend; the 
 number by which it is divided is called the divisor. 
 
 6. When a number is multiplied by itself, the product is 
 called the second power of that number, or the square of it ; 
 as, 2X2 = 4, which is the second power, or the square of 2; 
 so 9 is the square of 3; 25, the square of 5. 
 
 7. When a number is multiplied by itself, so as to be 
 taken 3 times as a factor, the product is called the third 
 power or the cube of the number ; thus, 8 is the cube of 2, 
 for it is formed by multiplying 2X2X2; 27, or 3X3X3, is 
 the cube or third power of 3; 125, or 5X5X5, is the third 
 power of>6. The number thus used as a factor, is called the 
 root of the power; thus, 3 is the square root of 9, and the 
 cube root of 27 : 5 is the square root of 25. 
 
12 
 
 EXPLANATIONS. 
 
 The number of the power may be expressed by a small 
 figure; thus, 2 3 is the 3d power of 2; 3 2 is the 2d power of 
 3; 5 3 is the 3d power of 5. 
 
 An angle is formed when two lines meet, run- 
 ning in different directions. 
 
 A triangle is a figure bounded by three straight 
 lines. It is called a triangle, because it has three 
 angles. An equilateral triangle has all its sides 
 equal. 
 
 A rigid angle is formed when one line meets 
 _ another, making the angles on both sides equal. 
 
 A square is a four-sided figure, the sides of which 
 are all equal, and the angles of which are right angles. 
 The diagonal divides it into two equal parts. 
 
 A rectangle is a four-sided figure, the oppo- 
 site sides of which are equal, and the angles of 
 which are right angles. The diagonal divides 
 it into two equal parts. 
 
 A parallelogram is a four-sided figure, the 
 opposite sides of which are equal and parallel. 
 The diagonal divides it into two equal parts. 
 
 A circle is a figure bounded by a curved line, 
 called the circumference, every part of which is 
 equally distant from the centre. 
 
 A straight line from the centre to the circumference is 
 called the radius. 
 
 The diameter is a line drawn from side to side of the cir- 
 cle, through the centre. It follows that the diameter is equal 
 to twice the radius. 
 
 Any portion of the circumference, considered by itself, is 
 called an arc. 
 
 A sector of a circle is a portion of it bounded by two radii 
 and the arc between them. 
 
 A sphere is a solid bounded by a curved surface every part 
 of which is equally distant from the centre of the solid. 
 
 O 
 
MENTAL ARITHMETIC. 
 
 PART FIRST 
 
 SECTION I. 
 
 MULTIPLICATION OF TENS AND UNITS. 
 
 1. A man drove six oxen to market, and sold three 
 of them for 50 dollars apiece. What did they come 
 to? 
 
 Three times 50 are 150. Ans. 150 dollars. 
 
 He sold the remaining three for 52 dollars apiece. 
 What did they come to? 
 
 Three times 50 are 150, and three times 2 are 6, 
 which added to 150 makes 156. Ans. 156 dollars. 
 
 What did they all come to ? 
 
 Twice 100 is 200, and twice 50 is 100, which 
 added to 200 makes 300, and 6 added to 300 makes 
 306. Ans. 306 dollars. 
 
 2. A merchant bought 45 barrels of flour for 6 dol- 
 lars a barrel. What did it come to ? 
 
 6 times 40 are 240, and 6 times 5 are 30 ; 30 added 
 to 240 makes 270. Ans. 270 dollars. 
 
 He bought 75 barrels more at 5 dollars a barrel. 
 What did it come to? 
 
 5 timers 70 are 350 ; 5 times 5 are 25, which added 
 to 350 makes 375. Ans. 375 dollars. 
 
 What did all the flour come to ? 
 2 
 
14 
 
 MENTAL ARITHMETIC. 
 
 ;300 aud 200 are 5.00; 70 and 70 are 140, which 
 added to 500 makes 640, and 5 are 645. Aus. 645 
 dollars. 
 
 3. What will 87 barrels of flour come to at 6 dol- 
 lars a barrel ? 
 
 6 times 80 are 480; and 6 times 7 are 42, which 
 added to 480 makes 522. Ans. 522 dollars. 
 
 4. What are 7 times 68 ? What are 8 times 72 ? 
 What are 9 times 84 ? What are 4 times 96 ? 
 
 8 times 64 ? 5 times 72 ? 7 times 83 ? 5 times 79 ? 
 4 times 98 ? 3 times 81 ? 6 times 73? 6 times 86 ? 
 
 The preceding examples will show the importance 
 of being able readily to multiply tens by units. This 
 becomes easy after acquiring the Multiplication Table. 
 It may be connected with a review of the Multiplica- 
 tion Table in the following manner : 
 
 Twice 1 are how many? 
 Twice 2 are how many? 
 Twice 3? Twice 30? 
 Twice 5? Twice 50? 
 Twice 7 ? Twice 70 ? 
 Twice 9? Twice 90? 
 
 Twice 10 are how many? 
 Twice 20 are how many? 
 Twice 4? Twice 40? 
 Twice 6? Twice 60? 
 Twice 8? Twice 80? 
 Twice 10? Twice 100? 
 
 3 times 1 ? 
 3 times 3 ? 
 3 times 5? 
 3 times 7? 
 
 3 times 9 ? 
 
 4 times 1 ? 
 4 times 3 ? 
 4 times 5 ? 
 4 times 7 ? 
 
 4 times 9 ? 
 
 5 times 1 ? 
 5 times 3? 
 5 times 5 ? 
 5 times 7 ? 
 5 times 9 ? 
 
 3 times 10? 
 3 times 30 ? 
 3 times 50 ? 
 3 times 70 ! 
 
 3 times 90 ? 
 
 4 times 10? 
 4 times 30 1 
 4 times 50 ? 
 4 times 70 ? 
 
 4 times 90 ? 
 
 5 times 10 ? 
 5 times 30 ! 
 5 times 50? 
 5 times 70 ? 
 5 times 90 ? 
 
 3 times 2 ? 
 3 times 4 ? 
 3 times 6 ? 
 
 3 times 8 ? 
 :>, times 10 ? 
 
 4 times 2 ? 
 4 times 4? 
 4 times 6? 
 4 times 8 ? 
 
 4 times 10? 
 
 5 times 2 ? 
 5 times 4? 
 5 times 6? 
 5 times 8 ? 
 5 times 10? 
 
 3 times 20 ? 
 3 times 40 ? 
 3 times 60 ! 
 3 times 80 ( 
 
 3 times 100? 
 
 4 times 20 1 
 4 times 40 ? 
 4 times 60 ! 
 4 times 80 ? 
 
 4 times 100 ? 
 
 5 times 20 ? 
 5 times 40 'i 
 5 times 60 ? 
 5 times 80 ? 
 5 times 100? 
 
MULTIPLICATION OF TENS AND UNITS. 
 
 15 
 
 times 
 
 1? 
 
 6 times 
 
 10? 
 
 6 times 2 ? 
 
 6 times 20 ? 
 
 6 times 
 
 3? 
 
 6 times 
 
 30? 
 
 6 times 4 ? 
 
 •6 times 40? 
 
 6 times 
 
 5? 
 
 6 times 
 
 50? 
 
 6 times 6 ? 
 
 6 times 60 ? 
 
 6 times 
 
 7? 
 
 6 times 
 
 70? 
 
 6 times 8 ? 
 
 6 times 80 ? 
 
 6 times 
 
 9? 
 
 6 times 
 
 90? 
 
 6 times 10? 
 
 G times 100 ? 
 
 7 times 
 
 1? 
 
 7 times 
 
 10? 
 
 7 times 2? 
 
 7 times 20 ? 
 
 7 times 
 
 3? 
 
 7 times 
 
 30? 
 
 7 times 4 ? 
 
 7 times 40 ? 
 
 7 times 
 
 5? 
 
 7 times 
 
 50? 
 
 7 times 6? 
 
 7 times 60 ? 
 
 7 times 
 
 7? 
 
 7 times 
 
 70? 
 
 7 times 8 ? 
 
 7 times 80? 
 
 7 times 
 
 9? 
 
 7 times 
 
 90? 
 
 7 times 10? 
 
 7 times 100? 
 
 8 times 
 
 1? 
 
 8 times 
 
 10? 
 
 8 times 2 ? 
 
 8 times 20 ? 
 
 8 times 
 
 3? 
 
 8 times 
 
 30? 
 
 8 times 4 ? 
 
 8 times 40 ? 
 
 8 times 
 
 5? 
 
 8 times 
 
 50? 
 
 8 times 6 ? 
 
 8 times 60 ? 
 
 8 times 
 
 7? 
 
 8 times 
 
 70? 
 
 8 times 8 ? 
 
 8 times 80 ? 
 
 8 times 
 
 9? 
 
 8 times 
 
 90? 
 
 8 times 10 ? 
 
 8 times 100? 
 
 9 times 
 
 1? 
 
 9 times 
 
 10? 
 
 9 times 2 ? 
 
 9 times 20 ? 
 
 9 times 
 
 3? 
 
 9 times 
 
 30? 
 
 9 times 4 ? 
 
 9 times 40 ? 
 
 9 times 
 
 5? 
 
 9 times 
 
 50? 
 
 9 times 6? 
 
 9 times GO I 
 
 9 times 
 
 7? 
 
 9 times 
 
 70? 
 
 9 times 8 ? 
 
 9 times 80 ? 
 
 9 times 
 
 9? 
 
 9 times 
 
 90? 
 
 9 times 10 ? 
 
 9 times 100? 
 
 10 times 
 
 1? 
 
 10 times 
 
 10? 
 
 10 times 2 ? 
 
 10 times 20? 
 
 10 times 
 
 3? 
 
 10 times 
 
 30? 
 
 10 times 4 ? 
 
 10 times 40? 
 
 10 times 
 
 5? 
 
 10 times 
 
 50? 
 
 10 times 6? 
 
 10 times 60? 
 
 10 times 
 
 7? 
 
 10 times 
 
 70? 
 
 10 times 8 ? 
 
 10 times 80 ? 
 
 10 times 
 
 9? 
 
 10 times 
 
 90? 
 
 10 times 10? 
 
 10 times 100 ? 
 
 1 1 times 
 
 1? 
 
 11 times 
 
 10? 
 
 11 times 2? 
 
 It times 20? 
 
 11 times 
 
 3? 
 
 11 times 
 
 30? 
 
 11 times 4? 
 
 11 times 40? 
 
 11 times 
 
 5? 
 
 11 times 
 
 50? 
 
 11 times 6? 
 
 11 times 60? 
 
 11 times 
 
 7? 
 
 11 times 
 
 70? 
 
 11 times 8? 
 
 1 1 times 80 ? 
 
 11 times 
 
 9? 
 
 11 times 
 
 90? 
 
 11 times 10? 
 
 11 times 100? 
 
 11 times 11? 
 
 11 times 
 
 110? 
 
 11 times 12? 
 
 11 times 120? 
 
 12 times 
 
 1? 
 
 12 times 
 
 10? 
 
 12 times 2? 
 
 12 times 20 ? 
 
 12 times 
 
 3? 
 
 12 times 
 
 30? 
 
 12 times 4 ? 
 
 12 times 40? 
 
 12 times 
 
 5? 
 
 12 times 
 
 50? 
 
 12 times 6 ? 
 
 12 times 60? 
 
 12 times 
 
 7? 
 
 12 times 
 
 70? 
 
 12 times 8? 
 
 12 times 80? 
 
 12 times 
 
 9? 
 
 12 times 
 
 90? 
 
 12 times 10? 
 
 12 times 100? 
 
 12 times ] 
 
 LI? 
 
 12 times 110? 
 
 12 times 12? 
 
 12 times 120 ? 
 
16 MENTAL ARITHMETIC. 
 
 A number which contains another number a certain 
 number of times, is a multiple of that number. 
 Thus 6 is a multiple of 2 ; 15 of 3 ; 28 of 7.* 
 
 Name all the multiples of 2, from 2 to 60. 
 
 Name the multiples of 20, from 20 to 600. 
 
 What are the multiples of 3 up to 75 ? Of 30 up to 
 750? 
 
 What are the multiples of 4 up to 80 ? Of 40 up to 
 800? 
 
 What are the multiples of 5 up to 100? Of 50 up 
 to 1000? Of 6 to 72? Of 60 to 720? Of 7 to 84? Of 
 70 to 840 ? Of 8 to 96 ? Of 80 to 960 ? Of 9 to 108 ? 
 Of 90 to 1080 ? Of 10 to 120 ? Of 100 to 1200 ? 
 
 SECTION II. 
 
 MULTIPLICATION OF TENS AND UNITS.— COMPLEMENT. 
 
 1. What will 17 tons of hay come to at 8 dollars a 
 ton? 
 
 Ans. 8 times 10 are 80, and 8 times 7 are 56 ; 56 
 added to 80 makes 136. 136 dollars. 
 
 2. What will 37 pounds of sugar come to at 9 cents 
 a pound ? J 
 
 3. A man drove 87 sheep to market, and sold them 
 for 6 dollars apiece. What did they come to ? 
 
 4. A man travelled on foot eight days ; he travelled 
 29 miles each day. How many miles did he travel in 
 all? i, 
 
 In each of the above examples, the second product, 
 when added to the first, makes a sum exceeding the 
 next even hundred : thus, in the 1st ex., 80 + 56 ; in 
 
 * See Note 1 , at the end of Part First 
 
MULTIPLICATION OF TENS AND UNITS. 17 
 
 the 2d, 270 + 63; in the 3d, 480 + 42; in the 4th, 
 160 + 72. 
 
 In order to perform snch examples with ease, quick- 
 ness, and without mistake, each step in the process 
 should be made the subject of distinct practice. To 
 illustrate these steps by the first example, 80 + 56, the 
 first thing to be done is to think of the number which 
 must be added to 80 to make 100, namely, 20 ; the 
 next is to take this 20 from the 56, and what remains 
 (36) will belong to the next hundred. 
 
 The number which, in such cases, must be added to 
 a given number to make up an even hundred, may 
 be called the complement of that number. Thus the 
 complement of 80 is 20 ; of 60, 40 ; of 90, 10 ; of 56, 
 
 44. 
 
 What 
 
 is the 
 
 compl 
 
 ement of 10 
 
 ? 30? 
 
 50? ' 
 
 ro? 
 
 # 
 
 What is the complement 
 
 of 
 
 
 
 
 10? 
 
 20? 
 
 30? 
 
 40? 
 
 50? 
 
 60? 
 
 70? 
 
 80? 
 
 90? 
 
 11 \ 
 
 21? 
 
 HI? 
 
 41? 
 
 51? 
 
 61? 
 
 71? 
 
 81? 
 
 91? 
 
 12? 
 
 22? 
 
 32? 
 
 42? 
 
 52? 
 
 62? 
 
 72? 
 
 82? 
 
 92? 
 
 13? 
 
 23? 
 
 33? 
 
 43? 
 
 53? 
 
 63? 
 
 73? 
 
 83? 
 
 93? 
 
 14? 
 
 24? 
 
 34? 
 
 44? 
 
 54? 
 
 64? 
 
 74? 
 
 84? 
 
 94? 
 
 15? 
 
 25? 
 
 35? 
 
 45? 
 
 55? 
 
 65? 
 
 75? 
 
 85? 
 
 95? 
 
 16? 
 
 26? 
 
 36? 
 
 46? 
 
 561 
 
 66? 
 
 76? 
 
 86? 
 
 96? 
 
 17? 
 
 27? 
 
 37? 
 
 47? 
 
 57? 
 
 67? 
 
 77? 
 
 87? 
 
 97? 
 
 18? 
 
 28? 
 
 38? 
 
 48? 
 
 58? 
 
 68? 
 
 78? 
 
 88? 
 
 98? 
 
 19? 
 
 29? 
 
 39? 
 
 49? 
 
 59? 
 
 69? 
 
 79? 
 
 89? 
 
 99? 
 
 How many are 40 + 76? 80 + 34? 70 + 91? 90 + 17? 
 25 + 83? 36 + 71? 45 + 82? 56 + 73? 43 + 82? 95 + 36? 
 37 + 84? 45 + 76? 88 + 37? 94 + 17? 76 + 87? 
 
 f How many are How many are $ 
 
 12X2, 3, 4, 5, 6, 7, 8, 9, 10? 16X2, 3, 4, 5, 6, 7, 8, 9, 10? 
 
 13X2, 3, 4, 5, 6, 7, 8, 9, 10 ? 17X2, 3, 4, 5, 6, 7, 8, 9, 10 ? 
 
 14X2, 3, 4, 5, 6, 7, 8, 9, 10 ? 18x2, 3, 4, 5, 6, 7, 8, 9, 10 ? 
 
 15X2, 3, 4, 5, 6, 7, 8, 9, 10? 19X2, 3, 4, 5, 6, 7, 8, 9, 10? 
 
 See Note 2. t Note 3. 
 
 2* R 
 
18 
 
 MENTAL ARITHMETIC. 
 
 How many are 
 
 20X2,3,4,5,6,7,8 
 
 4 
 
 How many are 
 
 21X2 
 22X2 
 23X2 
 24X2 
 25X2 
 26X2 
 27X2 
 28X2 
 29X2 
 30X2 
 31X2 
 32X2 
 33X2 
 34X2 
 35X2 
 36X2 
 37X2 
 38X2 
 39X2 
 40X2 
 41X2 
 42X2 
 43X2 
 44X2 
 45X2 
 46X2 
 47X2 
 48X2 
 49X2 
 
 9, 10?)< 50X2 
 9,10? 51X2 
 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9, 10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9, K)? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 9,10? 
 
 52X2 
 53X2 
 54X2 
 55X2 
 56X2 
 57X2 
 58X2 
 59X2 
 60X2 
 61X2 
 62X2 
 63X2 
 64X2 
 65X2 
 66X2 
 67X2 
 68X2 
 69X2 
 70X2 
 71X2 
 72X2 
 73X2 
 74X2 
 75X2 
 76X2 
 77X2 
 78X2 
 79X2 
 
 , 3, 4, 5, 6 
 
 ,7 
 
 »8 
 
 A 
 
 ,3,4,5,6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9< 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7' 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7, 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 ~, 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7 
 
 8 
 
 9, 
 
 ■ 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7, 
 
 8, 
 
 9„ 
 
 , 3, 4, 5, 6, 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7, 
 
 8, 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7 
 
 6 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7, 
 
 e, 
 
 9, 
 
 ,3,4,5,6, 
 
 7 , 
 
 s, 
 
 9, 
 
 , 3,4, 5, 6, 
 
 7, 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 7 ; 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6 
 
 7 
 
 8 
 
 9, 
 
 , 3, 4, 5, 6, 
 
 ", 
 
 8, 
 
 9, 
 
 To multiply any number less than 10 by 11, repeat 
 the figure expressing the number ; as, 3 times 11 is 33, 
 4X11 = 44. 
 
 To multiply by 11 any number of two figures. 
 Think of the first figure, then of the sum of the two 
 
PRACTICAL QUESTIONS. 19 
 
 figures, then of the last figure. These three figures 
 will express the answer. Thus, 11X23; the first, 2; 
 the sum of the two, 5; the last, 3. A?is. 253. 11X 
 24 = 264, 11X32 = 352, 11X43 = 473. 
 
 Remember, if the sum of the two is as much as 10, 
 you must increase the first figure by 1. 
 
 How many are 11X26? 11X28? 11X29? 11X41? 
 11X43? 11X45? 11X61? 11X62? 11X64? 11X71? 
 11X73? 11X81? 11X94? 11X75? 11X86? 11X89? 
 11X82? 11X84? 
 
 SECTION III. 
 
 PRACTICAL QUESTIONS. 
 
 1. If a railroad car travels 23 miles in one hour, 
 how far will it travel in 9 hours ? 
 
 2. If a horse travels 38 miles in one day, how far 
 will he travel in 6 days ? 
 
 3. If a man earns 14 dollars a month, how much 
 will he earn in 7 months? 
 
 4. If a man spends 6 cents a day for ardent spirit, 
 how much will that amount to in 10 days ? How 
 much in 30 days ? How much in 300 days ? How 
 much in 60 days? How much in 5 days? How 
 much in 365 days ? 
 
 5. If a man earns 10 cents in an hour, and works 12 
 hours in a day, how much will he earn in a week, there 
 being 6 working days in a week ? How much in 10 
 weeks ? How much in 50 weeks ? 
 
 6. If a scholar in school is idle 18 minutes in the 
 forenoon, and 18 minutes in the afternoon, how much 
 time will he lose in a week, if there are 6 forenoons 
 and 4 afternoons of school time in a week ? 
 
 7. If a town is 6 miles long, and 5 miles broad, how 
 
26 MENTAL ARITHMETIC. 
 
 many square miles does it contain? If there are 40 
 inhabitants on every square mile, how many inhab- 
 itants does the town contain? 40 times 30. 4 times 
 
 30 are 120. 40 times 30 are 10 times as many. If 
 one in 12 of the inhabitants were able-bodied men, 
 how many able-bodied men would there be ? If one 
 in 6 are able-bodied men, how many such are there? 
 
 8. What will 146 yards of broadcloth come to at 
 
 5 dollars a yard? 
 
 9. What will 86 yards of broadcloth come to at 
 
 6 dollars and a half a }-ard ? . 
 
 10. What will 710 barrels of rlour come to at 6 dol- 
 lars a barrel ? At 5 dollars a barrel ? At 5 dollars and 
 a half a barrel ? 
 
 11. What will 33 gallons of molasses come to at 
 
 31 cents a gallon ? At 34 cents a gallon ? At 40 cents 
 a gallon ? 
 
 12. What will 38 pounds of coffee come to at 
 14 cents a pound? £At 16 cents a pound ? U ^ 
 
 13. If there are 12 windows in the front of a fiouse, 
 10 in the rear, and 6 in each end, each window contain- 
 ing 12 lights ; how many lights are there in the front of 
 the house? How many in the rear? How many in 
 both ends? How many in all? If each light cost 10 
 cents, how much would they all cost ? If the setting of 
 each light cost 5 cents, how much would the setting of 
 them all cost? 
 
 14. There are 8 houses in a row: each house has 8 
 windows in front, 7 in the rear, and 5 in each end, each 
 window containing 12 lights ; how many lights are 
 there in each house? How many in ajl the houses? 
 What would be the cost of them all at "If cents for each 
 light ? What would be the cost of setting them all at 3 
 cents for each light? At the above price, what would be 
 the cost of the glass and the setting of it for one of the 
 houses? What would be the cost of the glass and the 
 setting of it for two of the houses ? For three of the 
 houses ? For four of the houses ? For five of the houses? 
 
DIVISION. 21 
 
 15. What are 8£ tons of hay worth, at 13 dollars a 
 ton? 
 
 16. If one acre of ground produce 65 bushels of 
 corn, how much would grow on 9 acres ? 
 
 \1. If an acre of ground produce 228 bushels of 
 potatoes, how many bushels would grow on 5 acres ? 
 
 18. If standing wood is worth 2 dollars a cord, what 
 is the value of the wood on 7 acres, each of which 
 furnishes 18 cords? 
 
 19. If there are 200 families in a town, and each 
 family consumes 12 cords of wood annually, how 
 many cords are used in the town each year? 
 
 What is the whole value of the wood, at 3£ dollars 
 a cord ? How much money will be saved in the town 
 if each family burns 2 cords less than before ? 
 
 SECTION IV. 
 
 DIVISION. 
 
 1. What is one half of 20? Of 40? Of 60? Of 
 80? Of 100? Of 120? Of 140? Of 160? 
 
 2. What is one half of 22 ? Of 42 ? Of 62 ? Of 82 1 
 Of 102 ? Of 112 ? Of 122 ? Of 142 ? Of 162 ? Of 182 ? 
 
 3. What is one half of 44 ? 64 ? S6 ? 48 ? 66 ? 28 ? 
 84? 68? 46? 24? 26? 62? 
 
 1 4. What is one half of 70 ? Divide it into 60 and 10. 
 
 What is one half of 90? Divide it into 80 and 10. 
 
 What is one half of 50 ? Of 30? Of 110? Of 130? 
 Of 150? 
 
 5. What is one half of 32 ? Of 54 ? Divide it into 
 50 and 4. 
 
 What is one half of 76 ? Of 74 ? Of 78 ? Of 96 ? 
 Of9Sr Of 92? Of 94? Of 72? Of 76 ? Of 58? Of 56? 
 
22 MENTAL ARITHMETIC. 
 
 6. What is one half of 43? One half of 40 is 20. 
 One half of 3 is H ; this added to 20 makes 21*. 
 
 What is one half of 47 ? Of 49 ? Of 63 ? Of 65 ? 
 Of 67 ? Of 69 ? Of 83 ? Of 85 ? Of 87 ? Of 89 ? 
 
 7. What is one half of 33 ? Divide into 30 and 3. 
 What is one half of 35 ? 37 ? 39 ? 51 ? 53 ? 55 ? 
 
 57} 59? Of 71? Of 73? Of 75 ? Of 77 ? Of 79 ? 
 Of 91? Of 93? Of 95? Of 97? Of 99 ? 
 
 8. What is one half of 367? Divide into 300, 
 60, and 7. 
 
 What is one half of 674 ? Of 895 ? Of 724 ? Of 
 632? Of 945? Of 424? Of 688? 01546? Of 392? 
 
 We can now find a very quick way of multiplying 
 any number by 5. Take one half the number : mut 
 tiply that by 10.* We will take the numbers in ques- 
 tion 2, and multiply them by 5 in this way. 
 
 Multiply 22 by 5. Half of 22 is 11, and ten times 
 11 is 110. 
 
 Multiply 42 by 5. Half of 42 is 21 ; ten times that 
 is 210. 
 
 Multiply 62 by 5. Half of 62 is 31 ; 310. 
 
 Multiply 82 by 5. Half is 41 ; 410. 
 
 Multiply 102 by 5. Half is 51 ; 510. 
 
 Multiply 112 by 5. Half is 56; 560. 
 
 Multiply 122 by 5. Half is 61 ; 610. 
 
 Multiply 142 by 5. Half is 71 ; 710. 
 
 Multiply 162 by 5. Half is 81 ; 810. 
 
 Multiply 182 by 5. Half is 91 ; 910. 
 
 9. Multiply by 5, in this way, the numbers in ques- 
 tion 3; 44; 64; 86; 48; 66;-28; 84; 68; 46; 24; 
 26; 62. 
 
 You can, if you wish, perform these examples by 
 both methods, and thus prove the work correct. 
 
 Multiply 862 by 5. Half is 431; 4310.— Multiply 
 672 by 5. Half is 336 ; 3360. 
 
 10. Multiply 6S6 by 5; 748 by 5; 932 by 5; 896 
 by 5 ; 1262 by 5. 
 
division. 23 
 
 If the number to be multiplied is an odd number, 
 so that half of it will show the fraction £, this, when 
 you multiply by 10, will become 5 ; for teu halves 
 are 5. 
 
 Multiply 781 by 5. Half is 3904- ; 3905, Ans. 
 
 11. Multiply 963 by 5. Half is 481J; 4815.— 
 Multiply 845 by 5; 3S1 by 5; 953 by 5; 845 by 5; 
 637 by 5; 429 by 5. 
 
 12. What is one fourth of 40? One fourth of 80? 
 One fourth of 120? One fourth of 12 is 3 ; a fourth 
 of 120 is 10 times as much ; 30. — What is one fourth 
 of 160? One fourth of 200? Of 240? Of 280? Of 
 320? Of 360? Of 400. 
 
 13. What is one fourth of 60 ? Take half of it ; 
 then half of that half. Half of 60 is 30 ; half of 30 
 is 15. — What is one fourth of 100? One fourth of 
 140? One fourth of 180? One fourth of 220? One 
 fourth of 260 ? One fourth of 300 ? 
 
 14. Another way of finding one fourth of the num- 
 bers in the last example, is as follows : 
 
 What is one fourth of 60 ? Divide 60 into 40 and 
 20. One fourth of 40 is 10 ; one fourth of 20 is 5 ; 15. 
 
 What is one fourth of 100? Divide into 80 + 20. 
 
 What is one fourth of 140? Divide into 120 + 20. 
 
 What is one fourth of 180? Divide into 160 + 20, 
 &c. 
 
 15. What is. one fourth of 30? Of 50? Of 70? 
 Find the best way of answering these, for yourself. 
 What is one fourth of 90? Of 110? Of 130? Of 
 
 150? Of 170? Of 190? Of 210? Of 230? Of 250. 
 
 16. What is one fourth of 76 ? Divide the number 
 into 40 and 36. — What is one* fourth of 96 ? Divide 
 the number into 80 and 16. 
 
 What is one fourth of 52 ? Of 64 ? Of 84 ? 
 
 17. What is one fourth of 368 ? There are several 
 ways of dividing this number ; first, into 200 + 100 
 + 6(\+ 8 ; a second way would be, into 200 + 160 + 8 j 
 
24 MENTAL ARITHMETIC. 
 
 another way, into 320 -|- 48. This is shorter than 
 either of the former. A better division still is into 
 360 + 8. 
 
 What is one fourth of 496? Into what different 
 sets of numbers, each divisible by 4, can you divide 
 this ? What is one fourth of 964 ? Of 336 ? Of 836 ? 
 596? 472? 1324? 1728? 2236? 
 
 18. What is one tenth of 10? Of 20? Of 30? 
 Of 40? Of 50? Of 60? 70? 80? 90? 100? 110? 
 120? 130? 140? 150? 
 
 19. What is one tenth of 5? Ans. 5 tenths of 1, or 
 yV, equal to £. 
 
 What, then, is one tenth of 15? Of 25? 35? 45? 
 55} 65? 75} 85? 95? 14? 17? 36? 47? 52? 
 91? 43? 28? 65? 86? 47? 
 
 20. What is one fifth of 25? 40? 45? 50? 55} 
 60? 65? 70? Divide 70 into 50 and 20. — Of 75} 
 Divide into 50 and 25. — Of 80? Divide into 50 
 and 30. — Of 85? Of 90? Of 95? Of 100? 
 
 21. What is one fifth of 64? Of 82? Of 91? Of 
 67? Of 73? Of 59? Of 63? Of 72? Of 78? Of 
 83? Of 87? 
 
 22. What is one fifth of 140? Of 385? Of 260? 
 Of 480? Of 390? Of 580? Of 470? Of 865? Of 
 395? 
 
 23. The following is a short way of dividing a 
 number by 5. Take one tenth of the number, and 
 double it. That, of course, gives 2 tenths, which is 
 equal to 1 fifth. Take the numbers in the last ex- 
 ample, and divide by 5 in this way. One fifth of 140 ; 
 one tenth is 14; double that is 28. One fifth of 385; 
 one tenth is 38 and 5 tenths ; twice that is 77. One 
 fifth of 260 ; one tenth is 26 ; twice 26 is 52. One 
 fifth of 480 ; one tenth is 48 ; twice that is 96. What 
 is one fifth of 390? Of 580? 470? 865? 395? 
 
 24. The following is a short way of multiplying a 
 number by 25. Take one fourth of the number; 
 
DIVISION. 25 
 
 multiply that by 100. This will give 100 fourths, 
 which are equal to 25 whole ones. 
 
 Multiply 40 by 25. One fourth is 10 ; one hundred 
 times that are 1000. — Multiply 60 by 25. One fourth 
 is 15 ; 1500. 
 
 Multiply 80 by 25. One fourth is 20 ; 2000. 
 
 Multiply 120 by 25. A fourth is 30; 3000. 
 
 Multiply 112 by 25. A fourth is 28 ; 2800. 
 
 Multiply 116 by 25. A fourth is 29; 2900. 
 
 25. Multiply 22 by 25. One fourth is 5 and a half; 
 100 times this are 5 hundred and half a hundred ;• 550. 
 
 Multiply 26 by 25. One fourth is 6Jj 650. 
 
 Multiply 28 by 25. One fourth is 7; 700. 
 
 Multiply 30 by 25; 32 by 25; 34 by 25; 36 by 25; 
 40 by 25. 
 
 Multiply 42 by 25; 44 by 25; 46 by 25; 48 by 
 25; 50 by 25. 
 
 26. Multiply 13 by 25. One fourth is 3 and one 
 fourth ; one hundred times this is 300 and one fourth 
 of a hundred or 25 ; 325. 
 
 Multiply 15 by 25. One fourth is 3 and three 
 fourths ; one hundred times this are 3 hundred and 
 3 fourths of a hundred or 75; 375. 
 
 Multiply 17 by 25; 19 by 25; 21 by 25; 23 by 25; 
 27 by 25 ; 29 by 25'; 31 by 25 ; 33 by 25 ; 35 by 25. 
 
 27. Multiply 116 by 25. One fourth is 29; 2900. 
 Multiply 117 by 25. One fourth is 29±; 2925. 
 Multiply 121 by 25; 87 by 25; 156 by 25; 960 by 25. 
 28.* What is one third of 60? Of 90? Of 120? 
 
 Of 15? Of 150? Of 45? Of 450? 
 
 What is one third of 18? Of 180? Of 21? 210? 
 Of 36? Of 360? Of 30? Of 390? 
 
 What is one third of 72 ? Divide into 60 and 12. 
 
 What is one third of 54 ? Of 85 ? Of 98 ? 
 
 What is one sixth of 60 ? Of 80 ? Divide into 60 
 and 20. — Of 74? Of 84? Of 96? Of 100? 
 
 * Note 4. 
 
20 MENTAL ARITHMETIC. 
 
 What is one sixth of 12? Of 120? Of 130? Of 
 140 >; Of 144? 
 
 What is one sixth of 18 ?5 Of 180? Of 200? Of 
 210? Of 220? 
 
 What is one sixth of 384 ? Of 492 ? Divide into 
 480 and 12. — Of 5551 Divide into 540 and 15.-^ 
 Of 620? Of 726? Of 947? 
 
 29. What are the two factors of 18? Of 180? 
 What are the two factors of 27 ? Of 270 ? Of 22 ? 
 
 Of 220? Of 35? Of 350? Of 54? Of 540? Of 
 45? Of 450? Of 21? Of 210? Of 28? Of 280? 
 Of 42? Of 420? 
 
 30. What two numbers, multiplied together, will 
 produce 24? 
 
 What other two factors will produce 24? What 
 other two? 
 
 What two factors will produce 240? What other 
 two? What others? 
 
 What two factors will produce 30? What others? 
 
 What two factors will produce 300? What others? 
 
 What two factors will produce 18? What others? 
 
 What two will produce 180? What others? 
 
 Name all the pairs of factors that will produce 36 ; 
 360; 48; 480; 60; 600; 64; 640; 72; 720. 
 
 31. What is one ninth of 27? A man divided 270 
 dollars equally among 9 persons. How much did he 
 give to each? 
 
 32. What is£pne fourth of 48 ? If 480 dollars are 
 divided into 4 equal shares, what will each share be ? 
 
 What is one eighth of 480? One sixth of 480? 
 One twelfth of 480? 
 
 33. What is one seventh of 63 ? If a ship sails, at 
 a uniform rate, 630 miles in a week, how many miles 
 does she sail in a day ? 
 
 What is one ninth of 630? What is one sixth of 
 630 ? What is one third of 630 ? 
 
 34. What is one fifth of 25? If 250 trees are 
 

 DIVISION. Xf 27 
 
 - / I I — • 
 
 placed in 5 equal rows, how many will there be in 
 each row? 
 
 If placed in 50 equal rows, how many would there 
 be in each row? 
 
 35. What is one fourth of 36? In a circle there 
 are 360 degrees. How many are there in one fourth of 
 a circle ? How many in one eighth of a circle ? How 
 many in one sixteenth of a circle? 
 
 36. What is one eighth of 56 ? If 560 trees were 
 planted in 8 equal rows, how many would there be in 
 each row? If planted in 16 rows, how many would 
 there be in each row ? 
 
 37. What is one eleventh of 551 If you place 550 
 trees in 11 equal rows, how many will there be in a 
 row ? If you place them in 50 rows, how many will 
 there be in each row ? If you place them in 25 rows, 
 how many will there be in each row ? 
 
 38. What is one twelfth of 96 ? If a man spends 
 960 dollars in a year, how much will be his average 
 expense for each month? 
 
 39. What is one tenth of 40? Of 400? Of 4000? 
 What is one fourth of 40? 400? Of 4000? Of 
 
 80? Of 800? Of 8000? One fourth of 12 ? Of 120? 
 Of 1200? Of 12,000? 
 
 40. What is one fifteenth of 60? Of 600? Of 
 6000? 
 
 What is one thirtieth of 60? Of 600? Of 6000? 
 Of 1200? Of 12,000? 
 
 41. What is one fifth of 92? What is one third of 
 51? One fourth of 65? One fifth of 78? One sixth 
 of 96? One seventh of 100? Divide into 70 and 30. 
 — What is one ninth of 117? Ans. One ninth of 90 
 is 10 ; one ninth of 27 is 3 ; 10 and 3 are 13. 
 
 42. What is one third of 49? One sixth of 84? 
 One fifth of 79 ? One eighth of 100 ? One seventh of 
 91? One sixth of 79? One fourth of 76? Of 92? 
 Of 57? Of 60? Of 52? Of 65 ? Of 70? 
 
28 
 
 MENTAL ARITHMETIC. 
 
 43. What is one fourth of 480? What is one fifth 
 of 155? Divide into 150 and 5. — What is one fourth 
 of 920? What is one fifth of 15,765? This number 
 may be divided into 15,000, 750, and 15; or 15,000, 
 500, 250, and 15. 
 
 44.* What is one sixth of 4836? One eighth of 
 336? Divide into 320 and 16. — What is one seventh 
 of 574? One third of 684? One "sixth of 43,248? 
 One ninth of 72, 108 ? Of 64,827 ? One fifth of 5275 ? 
 One fourth of 92,648? 
 
 45. What is one third of 6156? Of 8436? 
 
 46. What is one fourth of 6428? Of 9648?! .\ I t 
 
 47. What is one fifth of 7655? Of 12,535? 
 
 48. What is one sixth of 13,218? Of 1944? 
 
 49. What is one seventh of 10,542? Of 14,280? 
 
 50. What is one eighth of 1632? Of 2560? 
 
 SECTION V. 
 
 TABLE OF TIME. 
 
 60 seconds, (sec.) . make . 1 minute, . marked . m. 
 
 60 minutes, 1 hour, h. 
 
 24 hours, 1 day, d. 
 
 7 days, 1 week, w. 
 
 4 weeks, 1 month, mo. 
 
 52 weeks, 1 day, 6 hours, . 1 year, y. 
 
 365 days, 6 hours, 1 year, y. 
 
 12 calendar months, 1 year, y. 
 
 In common reckoning, 4 weeks are called a month ; 
 but this is merely for convenience in doing business. 
 
 * Note 5. 
 
time. 29 
 
 The number of days in a calendar month is 30 or 31 ; 
 except February, which has 28 days, and in leap year 
 29. The 6 hours over and above the 365 days in a 
 year, will in 4 years amount to a whole day ; it is then 
 added to February, making 29 days, and that year is 
 called leap year. The number of days in the other 
 months may be seen in the line below. The months 
 connected by a tie drawn over the words have 31 
 days ; those connected by a tie underneath have 30. 
 
 Jan. Feb. March. April. May. June. July. August. Sept. Oct. Nov. Dec. 
 28. 29. s f * ' 
 
 You observe that, beginning with January, every 
 alternate month has 31 days, till you come to July and 
 August. Here there are two months together that 
 have 31, and then the alternation goes on as before to 
 the end of the year. 
 
 The leap year may be easily known from the fact 
 that the number of the year is exactly divisible by 4. 
 Thus 1844 was leap year ; the number can be divided 
 by 4. 
 
 What years in the present century have been leap 
 years? What years will be leap years from now to 
 the close of the century? 
 
 1. In one minute there are 60 seconds ; in one hour 
 there are 60 minutes. How many seconds are there 
 in one hour? How many in 10 hours? How many 
 in 20 hours? How many in one day? How many 
 in seven days, or one week ? How many in ten days ? 
 In 100 days? In 300 days? In 350 days? In 365 
 days? 
 
 2. If you save 30 minutes from idleness each day, 
 how many hours will you save in a week? How 
 many in 5 weeks ? How many in 50 weeks ? How 
 many in 52 weeks? 
 
 3. If you read 40 pages each day, how many 
 
 3 * 
 
30 MENTAL ARITHMETIC. 
 
 pages will you read in one week ? How many in 10 
 weeks ? How many in 52 weeks ? 
 
 4. If a printer sets 4 pages of type in a day, in how 
 many days will he set the type for a book of 500 
 pages? What will his wages come to, at $1.50 a 
 day? 
 
 5. If there are 300 members in the Legislature of 
 Massachusetts, and each member receives 2 dollars a 
 day during the session, what does the pay of all the 
 members come to for one day ? What does the pay of 
 the Legislature amount to for one week ? For 10 
 weeks? 
 
 6. The number of members in Congress is about 
 275. At 8 dollars a day, what is 'the amount of their 
 pay each day ? What would be the amount of their 
 pay for 10 days ? For 100 days ? 
 
 7. How many days are there in the 3 months of 
 spring ? How many days in the 3 months of sum- 
 mer ? jHow many days in autumn ? 
 
 8. How many days in the winter of leap year? 
 How many days were there in the winter of 1844 ? 
 How many days in the winter of 1845? 
 
 9. If January comes in on Monday^ on what day 
 of the week will February come in ? * 
 
 If March comes in on Wednesday, on what day of 
 the week will April come in ? 
 
 If August comes in on Saturday, on what day of 
 the week will September come in ? 
 
 10. If April comes in on Sunday, on what day of 
 the week will it go out ? 
 
 If June comes in on Tuesday, on what day will it 
 go out ? m . 
 
 If September comes in on Saturday, on what day 
 will it go out ? 
 
 11. If January comes in on Friday, how many Sun- 
 days will there be in that month ? 
 
REDUCTION OF LINEAR MEASURE. 31 
 
 If it comes in on Thursday, how many Sundays 
 will there be in that month ? 
 
 If June comes in on Friday, how many Sundays 
 will there be in that month ? If it comes in on Sat- 
 urday, how many ? 
 
 If February comes in on Saturday, and that year is 
 leap year, hoW many Saturdays will there be in the 
 month ? If it is not leap year, how many ? 
 
 In 1845, February came in on Saturday. How many 
 Saturdays were there in that month ? 
 
 In 1844, February came in on Thursday. How many 
 Thursdays were there in that month ? 
 
 12. If January comes in on Monday, on what day 
 of the week will March come in, if it is leap year ? 
 On what day, if it is not leap year ? 
 
 13. If June comes in on Wednesday, what day of 
 the week will the 1st of August be? The 9th? The 
 12th? The 15th? 
 
 TABLE OF LINEAR MEASURE. 
 
 12 inches, (in.) . make . 1 foot,. . . marked ... ft. 
 
 3 'feet, 1 yard, yd. 
 
 5f yards, 16£ feet, 1 rod, rd. 
 
 40 rods, . 1 furlong, . ; fur. 
 
 8 furlongs = 320 rods, . 1 mile, m. 
 
 3 miles, 1 league, 1. 
 
 694- miles, 1 degree of latitude, . . deg. 
 
 For lengths less than an inch, the inch is divided 
 into fourths, eighths, tenths, or twelfths. 
 
 I. How many inches in 2 feet? In 4 feet? In 5 
 feet ? In 7 feet ? In 10 feet ? In 12 feet ? How 
 many inches in 4 yards ? In 1 rod ? In 3 rods ? How 
 many feet in 1 furlong ? In 2 furlongs ? In 4 fur- 
 longs ? In 1 mile ? 
 
32 MENTAL ARITHMETIC. 
 
 2. How many miles in 46 leagues ? In 132 leagues ? 
 How many miles in 2 degrees of latitude ? In 3 de- 
 grees? In 4£ degrees ? In 6 degrees? 
 
 In estimating the miles in any number of degrees 
 of latitude, it is most convenient to call a degree 70 
 miles ; and then, if we wish to be accurate, we may 
 subtract from the answer half as many miles as there 
 are degrees. In this way, the distance of places from 
 each other may be determined on a map. The degrees 
 of latitude on the margin may be used as a scale of 
 miles. If the distance of two places from each other 
 is equal to 6£ degrees of latitude, how many miles are 
 they apart ? 
 
 3. How many yards in 10 rods ? In 20 rods ? In 
 30 rods ? In 1 furlong ? In 8 furlongs, or 1 mile ? 
 
 4. In measuring land or a road with a chain 4 rods 
 long, how many times must the chain be applied to 
 the ground in measuring one mile ? How many times 
 in measuring the road from Boston to Salem, 15 miles? 
 How many in measuring from Boston to Providence, 
 40 miles? 
 
 5. If a man walks 3 miles in an hour, how many 
 minutes will he be in walking 1 mile ? How many 
 minutes in walking 1 fourth of a mile? How many 
 rods will he walk in 1 minute ? Ans. 16. 
 
 How many seconds will he be, then, in walking 1 
 rod? 16 will go into 60, 3 times and 12 over. He 
 will be, then, a little less than 4 seconds in walking 1 
 rod. 
 
 Let us now suppose he is precisely 4 seconds in 
 walking 1 rod ; how many rods would he walk in a 
 minute ? How many in 10 minutes ? How many 
 in 60 minutes? How many miles ? 
 
 6. If a man in walking takes 6 steps to a rod, how 
 many steps will he take in walking a mile ? How 
 many in walking 10 miles ? How many in walking 
 40 miles ? 
 

 REDUCTION OF LINEAR MEASURE, 33 
 
 7. If a man in walking takes 6 steps to a rod, and 
 takes 2 steps in a second, how many seconds will he 
 be in walking one rod ? How many seconds in walk- 
 ing 10 rods ? 20 rods ? If a man walks 20 rods in 
 one minute, how many minutes will it take him to 
 walk a mile ? 20 are contained in 320 just as many 
 times as 2 are contained in 32. 
 
 8. If a man walks 20 rods in one minute, how long 
 will it take him to walk 4 miles ? 
 
 9. If a railroad train goes 30 miles in an hour, how 
 far does it go in one minute ? How many rods in one 
 second ? 
 
 Ans. 30 miles in 60 minutes is 1 mile in 2 minutes ; 
 half a mile in one minute ; quarter of a mile in half 
 a minute ; that is 80 rods in 30 seconds ; that is 8 rods 
 in 3 seconds ; and in 1 second, one third of 8 rods, or 
 2 rods and two thirds. 
 
 10. How many rods in 14 miles ? 
 
 In 1 rod there are 16£ feet. In 1 mile there are 320 
 rods. How many feet are there in a mile ? There 
 are various ways of finding the answer to this ques- 
 tion ; some of them will be suggested, and the pupil 
 left to take his choice. 
 
 First, how many feet are there in 300 rods ? 
 
 This is not difficult ; for in 3 rods there are three 
 times 16^ feet, and in 300 rods there are 100 times as 
 many. 3 times 15 feet are 45 feet ; 3 times l£ feet 
 are 4|, which added to 45 make 49£ feet in 3 rods. 
 Now, 100 times 49 are 4900, and 100 halves are 50 ; 
 4950 feet in 300 rods. In 20 rods there are ten times 
 as many feet as in 2 rods ; in 2 rods there are twice 
 16£ or 33; in 20 rods, therefore, there are 330 feet; 
 300 added to 4900 make 5200, and 30 added to 50 
 make 80 ; there are, then, 5280 feet in a mile. 
 
 Another method would be to multiply 320 first by 
 8, and that product by 2, for 8 and 2 are the factors 
 of 16 ;. then, as there was £ a foot in each rod left out, 
 
 C 
 
34 MENTAL ARITHMETIC. 
 
 there must be added half as many feet as there are 
 rods, or half of 320. 
 
 Another method would be to multiply 320 by 10, 
 then by 6, and add the products, and lastly by J-, and 
 add that to the other products. 
 
 The pupil can try each of these ways, and see if he 
 obtains the same answer. 
 
 Let us now see if our answer is correct. If there 
 are 5280 feet in a mile, how many are there in half a 
 mile ? One half of 5200 is 2600 ; one half of 80 is 40 ; 
 there are, then, 2640 feet in half a mile. How many 
 in 1 fourth of a mile? One half of 2640 feet, which 
 is 1320. Now, 1 fourth of a mile is 80 rods. If, then, 
 there are 1320 feet in 80 rods, how many will there 
 be in 8 rods ? One tenth as many. One tenth oY 1320 
 is 132. Now, how many are there in one rod ? One 
 eighth of 132 ; dividing 132 into 80 and 52 ; one eighth 
 of 80 is 10, and one eighth of 52 is 6 J or 4-, which added 
 to 10 make 164. We have now come down from 5280, 
 and arrived by successive divisions to 164, the number 
 from which we started at first. The answer is thus 
 proved to be correct. 
 
 11. How many feet are there in 2 rods? In 20 
 rods ? In 200 rods ? 
 
 12. How many feet are there in 3 rods? In 30 
 rods ? In 300 rods ? In 8 rods ? In 80 rods, or a 
 quarter of a mile ? 
 
 13. How many rods are there in 2 miles? In 4 
 miles? In 8 miles? In 20 miles? In 30 miles? In 
 50 miles ? 
 
 14. How many rods are there in half a mile ? In 
 three fourths of a mile ? In 1 mile and a half? In 1 
 mile and 3 furlongs ? In 2 miles and 5 furlongs ? In 
 4 miles and 7 furlongs ? 
 
 15. How many yards are there in 2 rods ? In 20 
 rods ? In 3 rods ? In 30 rods ? In 300 rods ? 
 
REDUCTION OF LINEAR MEASURE. 35 
 
 How many yards are there in 3 rods and 4 feet ? 
 How many yards are there in 17 rods and 11 feet? 
 
 16. How many inches are there in 7 feet ? In 9 
 feet? In 6 feet? In 8 feet and 6 inches? In 11 feet 
 9 inches ? 
 
 17. How many inches are there in 1 rod, or 16£ feet ? 
 How many inches in 2 rods ? In 3 rods ? In 4 rods ? 
 
 18. A house is 46 feet and 5 inches in length. 
 How many inches long is it? 
 
 A creeping vine grows on an average 3 inches a 
 day. How many days will it take to grow from the 
 ground to the top of a house that is 25 feet high ? 
 
 19. A stage-horse travels 13 miles and 20 rods each 
 day. How far will he travel in 60 days ? How far 
 in 120? 
 
 20. If a horse travels 16 miles in a day, how many 
 miles will he travel in 5 days? In 10 days? In 15 
 days ? In 20 days ? In 200 days ? In 300 days ? 
 
 21. What is the weight of iron used in one mile of 
 railroad, allowing 50 pounds for a yard of rail ? 
 
 If one yard of rail weighs 50 pounds, the two rails 
 for one yard of the road would weigh 100 pounds: from 
 this may be obtained the weight for one rod ; for 10 
 rods; for 100 rods; for 300 rods; for 320 rods. 
 
 22. If a yard of iron rail weighs 75 pounds, what 
 would be the weight of the two rails of a single track 
 for one rod ? For 2 rods ? For 3 rods ? For 300 
 rods ? For 20 rods ? For 200 rods ? 
 
 What would be the cost of the rail for one rod, at 2 
 cents a pound? At 3 cents a pound? At 4 cents a 
 pound ? 
 
 23. If the cost of the iron for a single track of railroad 
 is 6000 dollars a mile, and the cost of the land and the 
 labor of construction equals that of the iron, what 
 would be the cost of 15 miles of railroad ? Of 24 miles ? 
 
 24. What would be the cost of constructing 1 mile of 
 common road at 225 dollars a rod ? 
 
36 MENTAL ARITHMETIC. 
 
 25. What would be the cost of building 80 rods of 
 common wall at 54 cents a rod ? 
 
 26. If a horse travels 10 miles in an hour, how long 
 is he in travelling 1 mile ? How long in travelling £ 
 of a mile, or 80 rods? How many seconds is he in 
 travelling 8 rods ? How long in travelling 1 rod ? 
 
 27. A body falling through the air, falls in the first 
 second 16£ feet, and in each succeeding second it falls 
 twice 16^ feet farther than in the preceding second. 
 How far would a stone fall in 2 seconds ? 
 
 28. How far would it fall in the third second? 
 How far would it fall in 3 seconds? 
 
 29. How far would it fall in the fourth second? 
 How far would it fall in 4 seconds? 
 
 30. Sound moves through the air at the rate of 
 1090 # feet in a second. How many feet will it move 
 in 3 seconds ? How many feet in 4 seconds ? How 
 many feet in 5 seconds? 
 
 As sound is found thus to pass 5450 feet in 5 sec- 
 onds, and as there are 5280 feet in a mile, we see that 
 in 5 seconds sound moves 170 feet more than a mile. 
 Now, as 165 feet is just 10 rods, we say, without 
 much error, that sound moves 1 mile and 10 rods in 
 5 seconds. This is accurate enough for all common 
 purposes, and you will do well to fix it in your mem- 
 ory, and make your calculations from it. 
 
 31. How many rods will sound move in 1 second? 
 One fifth of 320+10 rods = 66 rods. 
 
 32. How many rods in 2 seconds? How many 
 rods in 4 seconds? 
 
 Thus, if you watch the stroke of an axe used by 
 some one at a distance, and observe that the sound 
 comes to you one second later than you see the stroke, 
 you may know that the distance is 66 rods. If the 
 sound of a bell comes to you two seconds after the 
 
 * Professor Pierce on Sounds. 
 
REDUCTION OF LINEAR MEASURE. 37 
 
 stroke is given, you must be distant from the bell 132 
 rods. In these cases, no allowance is made for the 
 transmission of light. You are supposed to see the 
 motion as soon as it occurs. This is not strictly the 
 fact ; but the time is so exceedingly small, that it need 
 not be taken into the account. 
 
 33. In a still night, a church bell is sometimes 
 heard at the distance of 12 miles. How many sec- 
 onds, or nearly how many, after the stroke, would 
 the sound be heard at that distance? 
 
 34. If the report accompanying a flash of lightning 
 is heard 4 seconds after the flash is seen, how far from 
 the hearer was the discharge ? How far, if the time 
 between the flash and the report is 6 seconds? How 
 far, if the time is 8 seconds? How far, if the time is 
 10 seconds ? How far, if the time is 15 seconds ? 
 
 35. The report of a cannon has, in some instances, 
 been heard at the distance of 100 miles. Allowing that 
 the sound moves one mile in 5 seconds, in how many 
 seconds after the discharge would the report be heard at 
 the distance of 100 miles? 
 
 36. By means of a magnetic telegraph, it is possi- 
 ble to communicate intelligence instantly from New 
 Orleans to Boston, a distance of 1500 miles. If this 
 intelligence could be communicated by sound passing 
 through the air, how long would it be in travelling 
 that distance, allowing 5 seconds to a mile ? 
 
 A ball discharged from a gun moves at first with a 
 greater speed than sound, but it moves slower and 
 slower, and before it is spent the report overtakes it, 
 and passes by it ; for sound moves always at the same 
 rate. 
 
 37. If a cannon ball moves a mile in 8 seconds, 
 how long would it be in moving 3 miles ? How long 
 in moving one fourth of a mile ? How long in moving 
 one eighth of a mile ? How long in moving If miles ? 
 
 4 
 
38 MENTAL ARITHMETIC. 
 
 SECTION VI. 
 
 TABLE OF FEDERAL MONEY 
 
 10 mills . . . make ... 1 cent, . . . marked . . . ct. 
 
 10 cents, 1 dime, d. 
 
 10 dimes, 1 dollar, D. 
 
 10 dollars, 1 eagle, E. 
 
 This is established by law as the currency of 
 the United States. 
 
 The general mark for Federal Money is $ ; as, 
 $5.14, five dollars fourteen cents. A period must 
 always be placed between dollars and cents. 
 
 1. How many mills in 2 cents? In 10 cents? In 
 12 cents? In 5£ cents? In 12^ cents? In 36 cents? 
 In 1 dollar? 
 
 2. How many cents in 5 dimes ? In 1 1 dimes ? In 
 16 dimes? In U dollars? In 17 i dollars? In 12£ 
 dollars ? 
 
 3. How many dimes in 7 dollars ? In 13i dollars ? 
 In 3 eagles ? In 56 dollars ? In 100 dollars ? 
 
 4. How many cents in 35 mills? In 180 mills? In 
 600 mills? How many dimes in 80 cents? In 210 
 cents? In 740 cents? 
 
 5. How many dollars in 350 cents? In 325 cents? 
 In 700 cents? In 850 cents? In 1400 cents? In 
 1675 cents? In 925 cents? 
 
 TABLE OF STERLING MONEY. 
 
 4 farthings, (qr.) make 1 penny, . . marked'. . d. 
 
 12 pence, 1 shilling, s. 
 
 20 shillings, 1 pound, £. 
 
 This is the currency of Great Britain. 
 
REDUCTION. 39 
 
 1. How many farthings are there in 3 pence? In 
 7 pence? In 8 pence? In 10 pence? In 11 pence? 
 
 2. How many pence in 2 shillings ? In 12 shillings ? 
 In 15 shillings? In 18 shillings? In 16 shillings? 
 
 3. How many shillings in 4 pounds ? In 7 pounds ? 
 In 18 pounds ? In 36 pounds ? In 84 pounds ? 
 
 4. How many farthings in 1 shilling and 6 pence ? 
 In 2 shillings and 6 pence? In 15 shillings and 4 
 pence ? 
 
 How many pence in 10 shillings? In 20 shillings? 
 In 2 pounds ? In 4 pounds ? In 12 pounds ? 
 
 5. How many farthings in 1 pound ? In 5 pounds ? 
 In 8 pounds ? In 1 pound 2 shillings ? 
 
 6. How many pence in 45 farthings? In 128 far- 
 things? In 464 farthings? In 1296 farthings? In 
 648 farthings? 
 
 7. How many shillings in 80 pence ? In 67 pence ? 
 In 372 pence ? In 649 pence ? In 840 pence ? 
 
 8. How many pounds in 267 shillings? In 845 
 shillings? In 432 shillings? In 640 shillings? In 
 4000 shillings? 
 
 9. How many pounds in 890 pence? In 16,000 
 farthings? In 720 pence? In 1200 pence? In 456 
 pence ? 
 
 10. How many pence in 5 pounds 4 shillings ? In 
 7 pounds 8 shillings ? In 12 pounds 3 shillings ? 
 
 How many farthings in 4 shillings 6 pence? In 9 
 shillings ? How many farthings in 6 pounds 3 shil- 
 lings 8 pence ? 
 
 11. A man set out on a journey with £4 8s. 6d. in 
 his pocket. Before spending any thing, he received 
 in payment of a debt £2 3 s. 8d. How much had he 
 then ? When he arrived home, he had spent £1 4 s. 6d. 
 How much had he then? 
 
 These denominations, you must bear in mind, have 
 not the same value in English currency that they 
 have iii the United States. 
 
40 
 
 MENTAL ARITHMETIC. 
 
 In our country, they have different values in the 
 different States, but in none of them so high a value 
 as in England. In the New-England States, a shil- 
 ling is equal to 16 cents and two thirds, and 6 shillings 
 make a dollar. In New York, 12 and a half cents are 
 a shilling, and 8 shillings a dollar. In other States, 
 the values are still different. But these denominations 
 are gradually giving way to those of the Federal cur- 
 rency. They are now used only in naming prices. 
 Accounts are not kept in them, and all that is impor- 
 tant in them may be learned by practice without fur- 
 ther notice here. 
 
 In the Sterling currency, used in England, a pound 
 is equal to 4 dollars, 44 cents, and 4 mills ; 10 shillings, 
 therefore, or half a pound, are 2 dollars, 22 cents, 2 
 mills ; and 1 shilling is one tenth part of that, or 22 
 cents 2 mills. An English sixpence is, therefore, 11 
 cents 1 mill. The following table will be useful in 
 exchanging English money to our own. 
 
 1 -pound (£ ) is $ 4.44 4 
 
 10 shillings, or half a pound, 2.22 2 
 
 1 shilling, 22 2 
 
 6 pence, or half a shilling, Ill 
 
 4 shillings 6 pence, 1.00 
 
 1 guinea, 21 shillings, 4.66 6 
 
 The actual value of the English money is a little 
 higher than is here stated, but this is sufficiently accu- 
 rate for a general table. 
 
 1. What is the value, in dollars and cents, of 2£ ? 
 3£? 4£? 5£? l£6s.? 2£8s.? 3s. 6d.? 5s. 9d.? 
 
REDUCTION. 
 
 TABLE OF DRY MEASURE. 
 
 41 
 
 2 pints (pt.) . . make . . 1 quart, . . . marked . . qt. 
 
 4 quarts, . , 1 gallon, gal. 
 
 8 quarts, 1 peck, pk. 
 
 4 pecks, 1 bushel, bu. 
 
 8 bushels, 1 quarter, qr. 
 
 3(5 bushels, 1 chaldron, ch. 
 
 These denominations are used for measuring grain, 
 fruit, and coal. The pint, quart, and gallon, are larger 
 than the same denominations in wine measure, and 
 less than those of beer measure. 
 
 1. How many pints in 1 peck ? In 3 pecks ? In 1 
 bushel ? In 3 bushels ? In 4 bushels ? 
 
 2. How many quarts are there in 1 bushel ? In 4 
 bushels ? How many pecks in 7 quarters ? In 2 
 chaldrons ? 
 
 3. If a horse eat 4 quarts of oats each day, how 
 many bushels will he eat in 10 weeks ? How many 
 bushels in 50 weeks ? In 52 weeks ? 
 
 What will they cost at 50 cents a bushel ? 
 
 4. In 80 quarts how many pecks? How many 
 bushels ? 
 
 In 644 quarts how many pecks ? How many 
 bushels? 
 
 In 7840 quarts how many pecks? How many 
 bushels? 
 
 5. In 100 pints how many pecks? How many 
 bushels ? 
 
 In 620 pints how many pecks? How many 
 bushels ? 
 
 4*' 
 
42 MENTAL ARITHMETIC. 
 
 TABLE OF AVOIRDUPOIS WEIGHT. 
 
 16 drams (dr.) . . make . . 1 ounce, . . marked . . oz. 
 
 16 ounces, 1 pound, lb. 
 
 25 pounds, 1 quarter, (net wt.) . . qr. 
 
 28 pounds,. 1 quarter, (gross wt.) . qr. 
 
 4 quarters, 1 hundred weight, . . . cwt. 
 
 20 hundred weight, .... 1 ton, T. 
 
 These denominations are used in weighing hay, 
 grain, meat, flour, and all the most common articles 
 bought and sold by weight. On account of the waste 
 in handling such articles, their shrinking in drying, 
 and worthless admixtures sometimes found in them, 
 112 pounds are sometimes allowed for one hundred 
 weight ; this makes 28 pounds one quarter, and is 
 called gross weight. In all the following questions 
 of avoirdupois weight, understand gross weight, un- 
 less net weight is expressed. 
 
 1. How many drams in 3 oz. ? In 5 oz. ? In 8 oz. ? 
 In 11 oz. ? How many oz. in 12 lbs.? In 15 lbs.? 
 In 20 lbs. ? In 32 lbs. ? In 45 lbs. ? 
 
 2. How many lbs. in 4 cwt. net weight ? In 4 cwt. 
 gross ? In 6 cwt. net weight ? In 6 cwt. gross ? In 
 5 cwt. 2 qrs. net? In 5 cwt. 2 qrs. gross ? In 7 cwt. 
 3 qrs. net weight ? In 7 cwt. 3 qrs. gross ? 
 
 3. How many lbs. in a ton net weight ? In a ton 
 gross ? How many lbs. in 5 tons 3 cwt. net ? In 5 
 tons 3 cwt. gross ? 
 
 4. There are 2 loads of hay whose net weight is as 
 follows : the first, 25 cwt. 3 qrs. 17 lbs. ; the second, 
 
 17 cwt. 2 qrs. 21 lbs. What is the weight, of both ? 
 
 5. A man set out for market with a load of hay 
 weighing 36 cwt. 2 qrs. 15 lbs. net weight. He lest 
 
REDUCTION. 43 
 
 a part of it ; the remainder weighed 25 cwt. 1 qr. 8 
 lbs. How much did he lose ? 
 
 6. If there are 196 lbs. in a barrel of flour, how 
 many lbs. net weight are there in 10 barrels ? 
 
 196 lbs. are 7 quarters gross. How many cwt. gross 
 are there in 10 barrels of flour ? 
 
 7. How many pounds are there in 100 oz. ? In 
 650 oz. ? 
 
 8. A barrel of flour weighs 7 quarters gross. How 
 many tons gross are there in 100 barrels of flour? 
 
 9. What will be the expense of transporting by 
 railroad 100 barrels of flour, 100 miles, at the rate 
 of 3 dollars a ton ? 
 
 What will be the expense of transporting a single 
 barrel ? 
 
 100 barrels are 700 qrs. gross weight. 400 qrs. = 
 100 cwt. = 5 tons; 300 qrs. = 75 cwt. = 3 fons 15 
 cwt. ; this, added to 5 tons, makes 8 tons 15 cwt. 
 
 10. The freight of goods by wagon is about 20 dol- 
 lars a ton gross for 100 miles. .At this rate, what will 
 be the cost of carrying a barrel of flour 100 miles ? 
 
 TABLE OF TROY WEIGHT. 
 
 24 grains (gr.) . make . 1 pennyweight, marked dwt. 
 
 20 pennyweights, 1 ounce, oz. 
 
 12 ounces, 1 pound, lb. 
 
 This is used for weighing gold and silver. The 
 pound Troy is nearly one fifth less than the pound 
 Avoirdupois. 
 
 1. How many grains in 6 pennyweights? In 8 
 pennyweights ? In 12 pennyweights ? In 1 oz. ? In 
 / 2 oz. ? In 4 oz. ? In 6 oz. ? 
 
44 MENTAL ARITHMETIC. 
 
 2. How many pennyweights in 8 oz. ? In 11 oz. ? 
 In 1 lb. ? In 3 lbs. ? In 8 lbs. ? In 5 lbs. ? In 1 lb. 
 3 oz.? In 2 lbs. 5 oz. ? 
 
 3. How many oz. in 120 dwt. ? In 480 dwt. ? In 
 060 grs. ? How many lbs. in 100 oz. ? In 860 dwt. ? 
 In 1200 dwt. ? 
 
 TABLE OF APOTHECARIES' WEIGHT. | 
 
 20 grs make 1 scruple, . . . marked ... 9 
 
 3 scruples, 1 dram, 5 
 
 8 drams, 1 ounce, . . . . , § 
 
 12 ounces, 1 pound, lb 
 
 This table is used only by apothecaries in mixing 
 medicines. The pound and ounce are the same as in 
 Troy weight. 
 
 TABLE OF CLOTH MEASURE. 
 
 2£ inches (in.) . make . 1 nail, . . . marked . . . na. 
 4 nails, 1 quarter, qr. 
 
 4 quarters, 1 yard, yd. 
 
 3 quarters, . 1 ell Flemish, Fl. e. 
 
 5 quarters, 1 ell English, E. e. 
 
 6 quarters, 1 ell French, Fr. e. 
 
 1. How many inches in 1 q* ? In 1 yd. ? In 3 
 yds. ? In I ell Eng. ? In 1 ell Fr. ? In 1 ell Fl. ? 
 
 2. How many inches in 4 yds. ? In 7 yds. ? In 12 
 yds.? In 10 yds.? In 20 yds.? In 6 yds. 3 qrs.? 
 
 ? 
 
 In 4 yds. 1 qr. ? 
 
REDUCTION. 45 
 
 TABLE OF WINE MEASURE. 
 
 4 gills (gi.) . . make . . 1 pint, . . .marked. . . pt. 
 
 2 pints, 1 quart, qt. 
 
 4 quarts, 1 gallon, gal. 
 
 3H gallons, 1 barrel, bl. 
 
 63 gallons, 1 hogshead, hhd. 
 
 2 hogsheads, 1 pipe, -p. 
 
 2 pipes, 1 tun, T. 
 
 This table is used for measuring wine, spirits, cider, 
 and water. 
 
 1. How many gills in 1 quart ? In 1 gal. ? In 4 
 gals. ? In 6 gals. ? In 10 gals. ? In 13 gals. ? In 
 15 gals. ? 
 
 2. How many pints in 1 gal. ? In 4 gals. ? In 20 
 gals. ? How many qts. in 1 barrel ? In 1 hogshead ? 
 
 3. How many gallons in 5 barrels? In 8 barrels? 
 How many gals, in half a barrel ? In one fourth of a 
 barrel ? 
 
 4. In 100 gals, how many barrels? In 300 gals, 
 how many bis. ? 
 
 5. At 14 cents a gallon, what is 1 qt. of vinegar 
 worth ? 3 qts. ? 6 qts. ? 10 qts. ? 15 qts. ? 21 qts. ? 
 30 qts. ? 
 
 6. What is 1 barrel of vinegar worth at 15 cts. a 
 gallon ? How much, if the price is 20 cts. a gal. ? 
 
 TABLE OF ALE OR BEER MEASURE. 
 
 (Used in measuring malt liquors and milk.) 
 
 2 pints" (pt.) . . make . . 1 quart, . . .marked. . . qt. 
 
 4 quarts, 1 gallon, gal. 
 
 36 gallons, 1 barrel, bl. 
 
 The beer gallon is a little more than one fifth larger 
 
46 MENTAL ARITHMETIC. 
 
 than the wine gallon. There are other measures of 
 beer besides those in the tables ; as, the firkin, of 9 
 gallons ; the kilderkin, 18 ; the hogshead, 54 • but 
 these are not much used in this country. A barrel of 
 wine contains not quite three fourths as much as a 
 barrel of beer. 
 
 1. In 1 bl. how many pints ? How many pints in 
 3 bis. ? How many gallons in 5 bis. ? In 12 bis. ? 
 In 15 bis. ? In 21 bis.? 
 
 2. In 100 gallons how many bis. ? How many 
 bis. in 400 gals. ? First consider how many bis. there 
 are in 360 gals. 
 
 MEASURE OF THE CIRCLE. 
 
 Every circle is supposed to have its circumference 
 divided into 360 equal parts, called degrees ; and each 
 degree into 60 parts, called minutes ; and each minute 
 into 60 pjarts, called seconds. Whether the circle is 
 great or small, it is still divided into 360 degrees. A 
 degree, therefore, is always the same fixed part of the 
 circumference of a circle, although its actual length is 
 longer or shorter, according as the circle is great or 
 small. The line passing from the centre to the cir- 
 cumference is called the radius of the circle. To 
 give you some idea of the length of a degree in circles 
 of different magnitudes, I will state that, on comparing 
 a degree in any circle with its radius, it has been 
 found to be about one fifty-eighth part of it. In other 
 words, 58 degrees on the circumference of a circle are 
 about equal to the radius. If a degree is 1 inch, the 
 radius of that circle is 58 inches. If the radius of a 
 carriage wheel is 29 inches, a degree on the rim of the 
 same wheel will be half an inch. 
 
 If we take for illustration one of the largest-sized 
 water-wheels, 29 feet in diameter, a degree on its rim 
 would measure only 3 inches. 
 
MEASURE OF THE CIRCLE. 47 
 
 You may enlarge the circle in your mind, till you 
 suppose it extending over a plain, with a radius of 58 
 rods. A degree on such a circle will measure 1 rod. 
 If the radius is 58 miles, a degree will measure 1 mile. 
 Now, the circle round the earth is so great in extent 
 that a degree measures 69£ miles. This may aid you 
 in forming a conception of the vast magnitude of the 
 earth. 
 
 Each of these degrees is divided into 60 minutes, or 
 geographical miles. A geographical mile, therefore, is 
 about one sixth greater than a common mile. The 
 Table of Circular Measure is as follows : 
 
 60 seconds ( /7 ). . . make. . . 1 minute, . . marked . '. 
 
 60 minutes, (or geog. miles,) 1 degree, °. 
 
 360 degrees a circle. 
 
 The term miles, instead of minutes, can be used 
 only in reference to the great circle of the earth. 
 
 As the earth turns round on its axis once in 24 
 hours, every place upon it passes in that time through 
 the 360 degrees of its circle ; and on the equator, 
 which is the great circle, each of these degrees, we 
 have seen, is 69£ miles. 
 
 How swiftly, then, does a body lying on the equator 
 move in consequence of the daily revolution of the 
 earth ? 
 
 In 24 hours, it passes through 360 degrees ; in one 
 hour, then, it will pass through one twenty-fourth 
 part as many, which is 15 degrees. If it pass through 
 15 degrees in one hour, how many minutes will it be 
 in passing through 1 degree ? One fifteenth of 60 
 minutes is 4 minutes. If it pass through a degree in 
 4 minutes, what part of a degree will it pass through 
 in 1 minute ? One fourth of a degree, or 15 geo- 
 graphical miles. If it pass through 15 geographical 
 miles in 1 minute, in how many seconds will it pass 
 
48 MENTAL ARITHMETIC. 
 
 through one geographical mile ? In 4 seconds ; and 
 in 1 second it will pass through one fourth of a geo- 
 graphical mile. 
 
 Now, a geographical mile on the equator is, as we 
 have seen, longer than a common mile. We will here 
 suppose it no longer, but of the same length, and it 
 appears that an object on the equator moves, as the 
 vast earth whirls round on its axis, one quarter of a 
 mile every second of time. Reflect now, that, while 
 the surface of the earth moves with such amazing 
 speed, so vast is its size, that it occupies an entire day 
 and night in turning once round. 
 
 If, as above stated, the earth turns from west to east 
 at the rate of 15 degrees in an hour, we can, by 
 knowing the time of day in any place, ascertain what 
 time it is at a place any particular number of degrees 
 east or west of it. It is noon at any place when the 
 meridian of that place passes under the sun. 
 
 1. When it is noon at Boston, what time is it at a 
 place 15 degrees west of Boston ? At a place 15 de- 
 grees east of Boston ? 
 
 2. When it is 12 o'clock at Boston, what time is it 
 at a place 1 degree west of Boston ? At a place 1 
 degree east of Boston ? At a place 2 degrees west of 
 Boston ? At a place 2 degrees east of Boston ? 3 
 degrees east ? 3 degrees west ? 4 degrees east ? 4 
 degrees west ? 5 degrees east ? 5 degrees west ? 
 
 3. Indianapolis is 15 degrees west of Boston. 
 When it is noon at Boston, what time is it at Indian- 
 apolis ? When it is sunset at Boston, where will the 
 sun be at Indianapolis ? , 
 
 4. Niagara Falls are 8 degrees west of Boston. 
 When it is noon at Boston, what time is it at Niagara 
 Falls ? When it is 4 o'clock at Niagara Falls, what 
 time is it at Boston ? 
 
 5. Washington city is 6 degrees west of Boston. 
 If you set your watch with the sun at Boston, and 
 
LONGITUDE AND TIME. 49 
 
 then carry it to Washington, your watch keeping ac- 
 curate time all the while, when you arrive at Wash- 
 ington, will it be too fast or too slow? and how 
 much ? 
 
 6. Two travellers met at a public house. When 
 one of them said to the other, " Friend, are you travel- 
 ling east or west?" "I am direct from home," said 
 the other, " where niy watch agrees exactly with the 
 sun, but here I find it is 10 minutes too fast. Now, if 
 you can tell which way I am travelling, you are wel- 
 come to know." 
 
 Had he travelled east, or west ? and how far ? 
 
 7. Boston is 71 degrees west of London. When it 
 is noon at Boston, what time is it in London ? 
 
 8. The English convicts are transported to Botany 
 Bay, 150 degrees east of London. When it is noon at 
 London, what time is it in Botany Bay ? 
 
 9. English traders are settled on Columbia River, 
 120 degrees west from London. What time is it there 
 when it is noon in London ? 
 
 10. If a man is on the equator, which way must he 
 travel, and how many geographical miles, to have the 
 day 4 minutes longer than 24 hours ? How far, to 
 have the day 2 minutes longer ? How far, to have it 
 1 minute longer? How far must he travel to have 
 the day 1 minute and a half longer? Which way 
 must he travel, and how far, to have the day 1 minute 
 shorter ? 2 minutes shorter ? 5 minutes shorter ? 
 
 11. Suppose two birds start from the same place on 
 the equator, and fly, one east and the other west, at 
 the rate of 60 geographical miles an hour, and at the 
 end of the hour it is just sunset to the bird flying east ; 
 how high is the sun then at the place where the other 
 bird is ? 
 
 How high was the sun at the place of their starting, 
 when they set out ? 
 
 12. A shipmaster sails from New York for Europe, 
 
 5 D 
 
50 MENTAL ARITHMETIC. 
 
 and for three days it is so cloudy that he cannot see 
 the sun. On the fourth day, he takes an observation 
 of the sun at noon j and by his chronometer, which 
 gives the New York time, it is half past eleven. How 
 many degrees east from New York has he sailed ? 
 
 In what longitude is he then, if New York is 74° 
 V west from Greenwich ? 
 
 SECTION VII. 
 
 PRIME NUMBERS. 
 
 Numbers may be divided into two great classes. 
 The first class comprises such numbers as cannot be 
 formed by the multiplication of any two or more num- 
 bers together ; as, 1, 2, 3, 5, 11, 17. These are called 
 prime numbers. The other class may be formed by 
 multiplying two or more numbers together, as 4, which 
 is formed by multiplying 2 by 2 ; 6, which is equal to 
 2X3; 10, which is equal to 2X5, &c. These are 
 called composite numbers. These may always be 
 formed by multiplying two or more prime numbers 
 together. Thus all numbers are either prime, or are 
 formed by the multiplication of prime numbers to- 
 gether. 
 
 In separating numbers into their factors, care should 
 be taken that the factors be all prime. Thus, in re- 
 solving 30 into its factors, we may say it is formed by 
 multiplying 5 by 6 ; but this is not sufficient, for 6 is 
 not prime ; it is formed of the factors 2 and 3. The 
 prime factors of 30, therefore, are 2, 3, and 5. We 
 may say that 30 is formed of the factors 3 and 10; but 
 here, again, the analysis is not complete, for 10 is not 
 prime ; it is composed of the factors 2 and 5. Thus 
 
PRIME AND COMPOSITE NUMBERS. 
 
 51 
 
 we are brought to the same three factors as before, 
 namely, 2, 3, and 5. 
 
 The following table of numbers from 1 to 100 will 
 show what of them are prime, and what are the prime 
 factors of those which are composite. This table 
 should be carefully studied and made perfectly familiar. 
 The analysis of composite numbers into their prime 
 factors lies at the foundation of some of the most im- 
 portant operations in numbers, and affords an insight 
 into some of the most intricate rules of arithmetic. 
 
 1, prime. 
 
 27 = 3X3X3. 
 
 2, prime. 
 
 28 = 2X2X7. 
 
 3, prime. 
 
 29, prime. 
 
 4 = 2X2. 
 
 30 = 2X3X5. 
 
 5, prime. 
 
 31, prime. 
 
 6 = 3X2. 
 
 32 = 2X2X2X2X2. 
 
 7, prime. 
 
 33 = 3X11. 
 
 8 = 2X2X2. 
 
 34 = 2X17. 
 
 9 = 3X3. 
 
 35 = 5X7. 
 
 10 = 2X5. 
 
 36 = 2X2X3X3. 
 
 11, prime. 
 
 37, prime. 
 
 12 = 2X2X3. 
 
 38 = 2X19. 
 
 13, prime. 
 
 39 = 3X13. 
 
 14 = 2X7. 
 
 40 = 2X2X2X5. 
 
 15 = 3X5. 
 
 41, prime. 
 
 16 = 2X2X2X2. 
 
 42 = 2X3X7. 
 
 17, prime. 
 
 43, prime. 
 
 18 = 2X3X3. 
 
 44 = 2X2X11. 
 
 19, prime. 
 
 45 = 5X3X3. 
 
 20 = 2X2X5. 
 
 46 = 2X23. 
 
 21 = 3X7. 
 
 47, prime. 
 
 22 = 2X11. 
 
 48 = 2X2X2X2X3. 
 
 23, prime. 
 
 49 = 7X7. 
 
 24 = 2X2X2X3. 
 
 50 = 2x5X5. 
 
 25 = 5X5. 
 
 51 = 3x17- 
 
 y 26 = 2X13. 
 
 52 = 2x2X13. 
 
52 
 
 MENTAL ARITHMETIC. 
 
 53, prime. 
 
 54 = 2X3X3X3. 
 
 55 = 5X11. 
 
 56 = 2X2X2X7. 
 
 57 = 3X19. 
 58=2X29. 
 59, prime. 
 
 60 = 2X2X3X5. 
 61, prime. 
 
 62 = 2X31. 
 
 63 = 3X3X7. 
 
 64 = 2X2X2X2X2X2. 
 
 65 = 5X13. 
 
 66 = 2X3X11. 
 67. prime. 
 
 68 = 2X2X17. 
 
 69 = 3X23. 
 
 70 = 2X5X7. 
 71, prime. 
 
 72 = 2X2X2X3X3. 
 73, prime. 
 
 74 = 2X37. 
 
 75 = 3X5X5. 
 
 76 = 2X2X19. 
 
 77 = 7X11. 
 
 78 = 2X3X13. 
 79, prime. 
 
 80 = 2X2X2X2X5. 
 
 81 = 3X3X3X3. 
 
 82 = 2X41. 
 83, prime. 
 
 84 = 2X2X3X7. 
 
 85 = 5X17. 
 
 86 = 2X43. 
 
 87 = 3X29. 
 
 88 = 2X2X2X11. 
 89, prime. 
 
 90 = 2X3X3X5. 
 
 91 = 7X13. 
 
 92 = 2X2X23. 
 
 93 = 3X31. 
 
 94 = 2X47. 
 
 95 = 5X19. 
 
 96 = 2X2X2X2X2X3. 
 97, prime. 
 
 98 = 2X7X7. 
 
 99 = 3X3X11. 
 100 = 2X2X5X5. 
 
 On examining this table, several things may be ob- 
 served. 
 
 n 1. All the even numbers are composite; for they 
 •are all divisible by 2. So it appears in the table, with 
 the exception of the number 2, which is regarded as 
 prime because it is divisible only by itself. 
 
 2. Several of the numbers given above are powers 
 of their prime factors. Thus 4 is the 2d power of 2, 
 8 the third power of 2, 16 the 4th power of 2, 32 the 
 5th, 64 the 6th. 9, 27, and 81, are the 2d, 3d, and 
 4th powers of 3. 25 is the 2d power of 5, 49 the 2d 
 power of 7. 
 
PROPERTIES OF PRIME AND COMPOSITE NUMBERS. 53 
 
 3. If you double the number of times a factor is 
 taken, you obtain the square of the number they at 
 first made. Thus 4 is obtained by taking 2 twice as 
 a factor. If you take it twice as many times, that is, 
 4 times, as a factor, you obtain 16, which is the square 
 of 4. 
 
 9 is obtained by taking 3 twice as a factor. If you 
 double the number of times it is taken, thus, 3X3X3 
 X3, you obtain the square of 9. 
 
 8 is obtained by taking 2 three times as a factor. 
 If you take it 6 times, you obtain 64, the square of 8. 
 
 So universally, if you double the number of times 
 a factor is taken to produce a certain number, you ob- 
 tain, not twice that number, but the square of it. 
 
 I will make a single remark here about the prime 
 numbers, and then call your attention to the compos- 
 ite numbers. 
 
 Since the prime numbers are not formed by multi- 
 plying any two or more numbers together, they can- 
 not be divided by any number. You will observe, 
 however, that any number whatever may be divided 
 by itself, and may also be divided by 1 ; but 1 is a 
 unit, and not a number ; and by dividing a number by 
 itself, or by 1, you obtain no new number. Dividing 
 the number by itself, you obtain 1 ; and dividing by 1, 
 you obtain the number itself. Such an operation, 
 therefore, brings out nothing new. It is only another 
 way of expressing what was just as plain before. In 
 the same way, we may sometimes regard a number 
 as produced by multiplying itself into 1 ; thus, 7 = 7 
 XI; but this is not multiplication, but only an ex- 
 pression in the form of multiplication. It produces 
 no new number, and is employed only for convenience, 
 in order to make the reasoning more plain. 
 
 Composite numbers can be divided by their factors. 
 Thus you can divide 10 either by 2 or by 5, and by 
 no other number. If you divide by 2, you obtain 5 
 5* 
 
54 MENTAL ARITHMETIC. 
 
 for the answer, or quotient ; if you divide by 5, you 
 obtain 2 for the answer. Dividing by a number, then, 
 is the same as erasing that number as a factor, and 
 will always give for the answer the other factor, or 
 factors. Thus, dividing 10 by 2 you may represent 
 thus, $X5, leaving the factor 5 for the answer; di- 
 viding 10 by 5, thus, 2X0, leaving 2 for the answer. 
 Divide 21 by 3, thus, £X7. Divide 12 by 3, thus, 
 4X3, or $X^X3. 
 
 It is plain, therefore, that if you express any num- 
 ber by its factors, you can\at once see what numbers 
 you can divide it by. You can divide it by each of 
 its prime factors, or by any combination of them, and 
 by no other number. Thus 6 = 2X3 you can divide 
 by 2 or by 3; 8 = 2X2X2 you can divide by 2, and 
 that quotient by 2, and that by 2 again; 30 = 2X3 
 X5 you can divide by 2, or 3, or 5, or by any two 
 of them combined. 
 
 Any composite number may be divided by any of 
 its prime factors, or by any combination of them. 
 
 By what numbers can you divide 15? 18? 20? 21? 
 26? 27? 36? 42? 46? 48? 49? 50? 
 
 Sometimes we have two numbers, and we wish to 
 know if there is any number that will divide them 
 both. This we can ascertain if we express each num- 
 ber by means of its prime factors, and then see if the 
 same factor is found in both. If so, they are both di- 
 visible by that number. Thus, if we wish to know 
 whether any number will divide both 9 and 15, we 
 express them thus, 3X3 and 3X5. Now, 3 appears 
 as a factor in both; they can both, therefore, be di- 
 vided by 3. This number, 3, is called the common 
 divisor, because it is a divisor common to several num- 
 bers. If we wish to know whether any number will 
 divide both 15 and 8, we express 15 by its factors, 
 5X3; and 8 by its factors, 2X2X2. Now, there is 
 no factor common to both ; no number, therefore, will 
 
PROPERTIES OF PRIME AND COMPOSITE NUMBERS. 55 
 
 divide them both ; in other words, they have no com- 
 mon divisor. Numbers which have no common di- 
 visor are said to be prime to each other. They may 
 be composite considered by themselves, as is the case 
 with 8 and 15 ; but if they have no common divisor, 
 they are said to be prime to each other. Numbers 
 which have a common divisor are said to be compos- 
 ite to each other. If there are more than two num- 
 bers, they must be treated in the same way. Each 
 must be written in the form of its prime factors; and 
 then, if any one number appears as a factor in them 
 all, they are divisible by that. 
 
 Is there any common divisor to 9, 14, and 27? 
 Written in the form of their factors, they stand thus, 
 2X3; 2X7; 3X3X3. They have, therefore, no 
 common divisor; for, though 3 or 9 will divide both 
 the first and the third number, it will not divide the 
 second ; and neither 2 nor 7, which are the factors of 
 the second number, appear in the first or third. 9, 14, 
 and 27, are, therefore, prime to each other. 
 
 What is the common divisor of 15 and 27? Of 14 
 and 22? Of 21 and 49? Of 35 and 28? Of 6 and 21? 
 
 Let us now take the following question : What is 
 the common divisor of 18 and 30? By inspecting 
 their factors, 2X3X3, and 2X3X5, we find that 2X3, 
 or 6, is common to both ; 6 is therefore the greatest 
 common divisor. 
 
 What is the greatest common divisor of 18 and 27 ? 
 Of 4, 8, and 36? Of 15 and 45? Of 27 and 45? 
 Of 40, 64, and 16? Of 44 and 24? Of 75 and 15? 
 Of 80 and 100? Of 60 and 24? Of 35, 21, and 49? 
 Of 15 and 50? 
 
 We have seen that a composite number can be di- 
 vided only by its factors, and that prime numbers can- 
 not be divided at all. It is frequently necessary, how- 
 ever, to attempt the division of prime numbers, and to-, 
 divide composite numbers by some number different 
 
56 MENTAL ARITHMETIC. 
 
 from their factors. For example, we may wish to 
 divide 9 by 4, or to obtain one fourth of 9. Now, 4 
 is not a factor of 9, and the actual division of 9 by 4 
 is, strictly speaking, impossible. We proceed in this 
 way. We divide 8 by 4, and obtain 2 for the answer, 
 and we have a remainder of 1, which we have not 
 divided. To show that we design this to be divided 
 by 4, we write the 4 under it, with a line between, 
 thus, i. In this way we indicate plainly enough what 
 the answer is, although we have no one figure that 
 will express it. 
 
 Again, let us divide 15 by 4. We divide 12 by 4, 
 obtaining for the quotient 3, and we have 3 remaining. 
 This we cannot divide by 4 so as to express the quo- 
 tient by a single figure. We therefore indicate the 
 division, without performing it, by writing the divisor, 
 4, under the dividend, 3, thus, f. When we divide a 
 number by 4, we obtain one fourth of it. The ex- 
 pression |, therefore, signifies one fourth of three. 
 But one fourth of three is the same quantity as three 
 fourths of one. 
 
 Thus 1 1 the perpendicular stroke 
 
 cuts oif, . . on the left, one fourth 
 
 part of 1 -, the three whole lines. 
 
 By looking at what is thus cut off, you see that it is 
 just equal to the three fourths of one whole line, on 
 the right hand of the perpendicular mark. We may, 
 therefore, call £ one fourth of three, or three fourths 
 of one; or, simply, three fourths, meaning threa 
 fourths of one; and, as this is the shorter expres- 
 sion, it is the one usually employed. 
 
 Questions. 
 
 1. What will be the remainder in dividing 18 by 5? 
 How will you express this part of the division, which ^ 
 
FRACTIONS. 57 
 
 it is impossible to perform? In what different ways 
 can yon read the expression? 
 
 2. How will you draw lines on the board, or slate, 
 so as to show that one fifth of three is equal to three 
 fifths of one? 
 
 3. What will be the remainder in dividing 23 by 6 ? 
 How will you express the part of the division which 
 is not performed? In what different ways can you 
 read the expression? 
 
 4. How will you draw lines, so as to show that one 
 sixth of five is equal to five sixths of one ? 
 
 5. In all the above cases, which is the larger, the 
 dividend, or the divisor? 
 
 6. In all the above expressions, what is the value of 
 the quantity expressed, compared to one ? Is it greater, 
 or less, than one ? 
 
 In the same way as in the above cases, we may de- 
 note the division of any number by a number greater 
 than itself; by writing the divisor under the dividend; 
 as ) fj fj t£- These expressions are called Fractions. 
 A fraction, therefore, is an expression denoting the 
 division of a number by a number greater than itself; 
 and as the quantity signified by such an expression is 
 necessarily less than one, we may say, more briefly, 
 
 A fraction is an expression for a quantity less than 
 one. 
 
 It is expressed by means of two numbers, — the 
 smaller (the dividend) written above, the larger (the 
 divisor) written below, a horizontal line. 
 
 The number below the line is called the denom- 
 inator, because it shows into how many equal parts 
 the number, or unit, is divided ; the number above the 
 line is called the numerator, because it shows how 
 many units are divided, or how many parts of a di- 
 vided unit are taken. 
 
58 MENTAL. ARITHMETIC. 
 
 7. In the expressions f , f, -f T , is the quantity denoted 
 in each expression greater than 1, or less? and why? 
 
 8. In the expressions f , £, f, is the quantity denoted 
 in each case greater than 1, or less? and why? 
 
 It is sometimes convenient to express quantities 
 which are not less than 1, in the form of fractions; 
 as, f, f, &c. These are called Improper Fractions^ 
 while the others are called, in distinction from these, 
 Proper Fractions. An improper fraction may always 
 be reduced to a whole number and a proper fraction. 
 
 A whole number and a fraction taken together, is 
 called a Mixed Number. 
 
 9. Reduce to a mixed number, -V 3 -; j tS' ft- 
 
 10. Reduce to an improper fraction, 4|j 7£; 9|. 
 
 Questions. 
 
 What is meant by a common divisor? 
 
 What is meant by the greatest common divisor? 
 
 When are numbers prime to each other? 
 
 When are numbers composite to each other? 
 
 What is the process of dividing 13 by 4? 
 
 In dividing 16 by 5? In dividing 25 by 6? 
 
 What are Fractions ? 
 
 Explain what is signified by each of the numbers 
 in the fraction §. In \. In T 6 T . In tV* In f . In T £. 
 
 A man bought a barrel of flour, and gave away two 
 fifths of it. What fraction will express what he gave 
 away? What fraction will express what he kept? 
 
 A man bought a load of hay, and sold two elevenths 
 of it. What fraction will express what he sold? 
 What fraction will express what he kept? 
 
 What is a proper fraction ? Give an example. 
 
 What is an improper fraction? Give an example. 
 
 When is the value of a fraction just equal to 1 ? < 
 
FRACTIONS. 59 
 
 SECTION VIII. 
 
 MULTIPLICATION AND DIVISION OF FRACTIONS. 
 
 We have seen that a fraction is not a simple ex- 
 pression, but composed of two numbers; and its value 
 cannot be determined by one of these numbers alone, 
 but by both taken in connection. By looking at the 
 numerator, you cannot tell the value of the fraction, 
 unless you know what the denominator is. By look- 
 ing at the denominator, you cannot tell the value of 
 the fraction, unless you know what the numerator is. 
 
 Let us now observe the effect of altering one of the 
 terms of the fraction without altering the other. We 
 will take the fraction f. If we increase the numer- 
 ator by 1, making it f, we increase the value of the 
 fraction, for we take one fifth more than we had be- 
 fore. So, if we multiply the numerator by 2, making 
 it |, we double the value of the fraction ; and so of 
 any other numbers, if we multiply the numerator, we 
 multiply the value of the fraction. And, by the same 
 reasoning, if we divide the numerator by 2, we divide 
 the fraction by 2, for | is plainly one half as great as 
 f. So of all other numbers, by dividing the numer- 
 ator we divide the fraction. 
 
 Let us now observe the effect of altering the denom- 
 inator. If we increase the denominator of the fraction § 
 by 1, making it f, we have not increased the fraction, 
 but diminished it; for one sixth is less than one fifth, 
 and any number of sixths are less than the same num- 
 ber of fifths. We will multiply the denominator of the 
 fraction f by 2, making T 2 n . What effect has been 
 produced on the value of the fraction? One tenth 
 is half as great as one fifth, and two tenths are half as 
 great as two fifths. The fraction is, .therefore, half 
 
00 MENTAL ARITHMETIC. 
 
 as great as it was before ; that is, it has been divided 
 by 2. Multiplying the denominator, therefore, divides 
 the value of the fraction. 
 
 We will now divide the denominator. Take the 
 fraction f ; dividing the denominator by 2, we have f- . 
 Now, as this is twice as great as f , we have multiplied 
 the fraction, by dividing the denominator. 
 
 There are, then, two ways of multiplying a fraction. 
 We may multiply the numerator; or, if the multiplier 
 is a factor of the denominator, we may divide the de- 
 nominator. Thus, to multiply f by 2, we may multi- 
 ply the numerator, which gives f , or divide the denom- 
 inator, which gives f , equal to f . 
 
 To divide a fraction, we may either divide the 
 numerator, if the divisor is a factor of it, or we may 
 multiply the denominator. Thus, to divide f by 3, 
 we may divide the numerator, giving f , or we may 
 multiply the denominator, which gives / T , which is 
 equal to f . 
 
 We will now multiply both terms of the fraction by 
 the same number. Multiplying both terms of the 
 fraction § by 3, we have f. Here the denominator, 
 expressing the number of parts into which the unit is 
 divided, is three times as great as it was before ; con- 
 sequently each of the parts is only one third as great ; 
 but the numerator has also been multiplied by 3, so 
 that three times as many parts are taken, and this 
 makes the value of the fraction just equal to what it 
 was before. So we may multiply, by any number 
 whatever, both terms of the fraction f, and the value 
 will still be the same as before ; for example, £, f , T 8 j, 
 i£> If j eacn °f which is equal to §. We may, then, at 
 any time multiply both terms of a fraction by the same 
 number, without altering the value of the fraction. 
 By the same reasoning, we may divide both terms of 
 a fraction by the same number without altering its 
 value. Taking the examples above, we may divide ^ 
 
FRACTIONS. 61 
 
 the terms of f by 2, and we obtain f ; dividing the 
 terms of f by 3 gives us § , and so of the others, § is 
 the same fraction as $, f, t 8 ?, &c, but it is expressed in 
 lower terms, and therefore is more convenient. It is 
 easier to write ^ than it is to write Jf, though both 
 have the same value. 
 
 To reduce a fraction to its lowest terms, we divide 
 both the numerator and denominator by their greatest 
 common divisor. To find the greatest common divi- 
 sor, separate each term into its prime factors, and erase 
 those which are common to both. The remaining 
 factors will express the value of the fraction in its 
 lowest terms. 
 
 Treating the above fractions in this way, they ap- 
 pear thus, 
 4_2X2 6_2X2 8 _2X^X2 10__2X# 12_ gX2X# 
 
 6~^X3' 9~ 3X? 12~2X#X3' To~ 3X? 18~~ ft X 3 X 0' 
 leaving, in each case, f. 
 
 In how many ways can you obtain the answer to the 
 following questions ? |X2? fX3? fx4? t^X2? 
 
 In how many ways can you obtain the answer to the 
 following? fX3? |X2? fX4? t 2 tX6? t VX5? 
 AX3? 
 
 In how many ways can you obtain the answer to 
 the following? f-i-3? f-r-4? -ft-r-3?. if-^-2? 
 
 In how many ways can you obtain an answer to 
 the following? f-r-2? f -=-4? i-^3? t 9 <j-^2? 
 t§-^5? 
 
 Reduce to their lowest terms each of the following 
 fractions : f; J+; *; §§; f$; U] U] ¥t) ft 5 A ) 
 is i M J s 8 ^ ) If J it • 
 
 TO FIND THE DIVISORS OF NUMBERS. 
 
 Reduce the fraction {£§ to its lowest terms. 
 You will not see immediately that these two num- 
 6 
 
b2 MENTAL ARITHMETIC. 
 
 bers have any common divisor. To assist yon to 
 reduce fractions of this kind, something will here be 
 said about the way of finding the divisors of num- 
 bers. Let us first inquire what numbers can be divi- 
 ded by 2. 
 
 We have seen that all even numbers, and only those, 
 can be divided by 2. 
 
 What numbers can be divided by 4 ? 
 
 If you examine, you will find that all even tens are 
 divisible by 4 ; as, 20, 40, 60, &c. If, therefore, the 
 tens are even, and the units are divisible by 4, then the 
 whole is divisible by 4. But the only unit numbers 
 divisible by 4 are 4 and 8 ; therefore, if the tens are 
 even, and the unit number is 4 or 8, the whole is di- 
 visible by 4; as, 84, 88; 124, 128; 148, 364; &c. 
 
 Again; as 10, when divided by 4, leaves a remainder 
 of 2, any odd number of tens will do the same ; as, 30, 
 50, 70, 90; for every odd number of tens is an even 
 number of tens + 10. If, then, the number of tens is 
 odd, the units must be two less than 4 or 8, in order 
 to be divisible by 4; that is, if the tens are odd, and 
 the units 2 or 6, the whole is divisible by 4 ; as, 72, 
 96, 52, &c. 
 
 Are the following even numbers divisible by 4, or 
 only by 2? and why? 126; 82; 94; 92; 138; 156; 
 346; 548; 76; 58; 392. 
 
 What numbers can be divided by 8? 
 
 As 100 divided by 8 leaves a remainder of 4, (8X12 
 = 96,) it follows that 200 will be exactly divisible by 
 8, for the two remainders of 4 will make 8. If 200 
 is divisible by 8, it follows that all even hundreds are 
 divisible by 8 ; as, 400, 600, 1400, &c. 
 
 If, therefore, the hundreds are even, and the tens 
 and units are divisible by 8, the whole number will be 
 divisible by 8; as, 248, 672, 1456, &c. 
 
 Again ; if the hundreds are odd, and the tens and 
 units are 4 less than some multiple of 8, the whole 
 
FRACTIONS. 63 
 
 number will be divisible by 8 j for the odd hundred, 
 divided by 8, leaves a remainder of 4 ; and this, added 
 to the tens and units, will make an exact multiple of 8. 
 
 Are the following numbers divisible by 8, or by 4 ? 
 and why? 444; 944; 136; 1328; 712; 532; 816; 
 516; 384; 128; 1236. 
 
 What numbers are divisible by 5 ? All tens are di- 
 visible by 5 ; consequently, if the unit figure is 5 or 0, 
 the whole number is divisible by 5. 
 
 What numbers are divisible by 3 ? By examining 
 the multiples of 3, we shall find this singular fact, that 
 the sum of the figures which express any multiple of 3 
 is itself a multiple of 3. Take the multiples of 3 from 
 12 to 24; 12, 15, 18, 21, 24; by adding the figures 
 which express any one of these multiples, we find that 
 the sum is a multiple of 3. The figures of 12 added 
 are 1+2 = 3, of 15 are 1 + 5 = 6, of 18 are 1 + 8 = 
 9, of 21 are 2+ 1 = 3, of 24 are 2 + 4 = 6. The same 
 is true of all multiples of 3. 
 
 It will also be found, that, if you add the figures of 
 any number, and the sum is a multiple of 3, the whole 
 number is a multiple of 3. To know, then, if a num- 
 ber is a multiple of 3, add together the figures that 
 express the number, and if the sum is a multiple of 
 3, the whole number is a multiple of 3. 
 
 Are the following numbers divisible by 3 ? 471 ; 59 ; 
 115; 642; 624; 138; 234; 742; 894. 
 
 It follows from what has been said, that, if any 
 number is divisible by 3, any other number expressed 
 * by the same figures differently arranged will also be 
 divisible by 3 ; for the sum made by adding the fig- 
 ures will be the same in whatever order they are 
 taken. 
 
 Thus, if 936 is divisible by 3 ; 369, 396, 963, 639, 
 1 693, are each divisible by 3. 
 
 We will next inquire what numbers are divisible 
 » by 6. As 6 = 2X3, any number that is divisible by 
 
64 MENTAL ARITHMETIC. 
 
 2 and by 3 is divisible by 6. You have learned what 
 numbers are divisible by '3, and what by 2. If a 
 number combines both these conditions, it is divisible 
 by 6 ; that is, all numbers are divisible by 6, the sum 
 of whose figures is a multiple of 3, and whose last 
 figure is an even number. 
 
 What combinations of the figures 1, 2, 3, will give 
 numbers divisible by 6 ? and what by 3 only ? 
 
 Next let us inquire what numbers are divisible by 9. 
 
 If the figures which express any multiple of 9, as 
 18, 27, 36, 45, 54, be added together, the sum will be 
 a multiple of 9. 
 
 Also, if the figures of any number be added to- 
 gether, and the sum is a multiple of 9, the whole 
 number is divisible by 9. 
 
 Are the following numbers divisible by 9 ? and 
 why? 936; 972; 396; 423; 387; 527; 441; 416; 
 315; 756. 
 
 Any number divisible by 9 and by 2 is divisible by 
 9X2, or 18. Which of the above numbers are divis- 
 ible by 18 ? 
 
 Any number divisible by 9 and by 4 is divisible by 
 9X4, or 36. Which of the above numbers are divis- 
 ible by 36 ? ^ 
 
 Any number divisible by 9 and by 8 is divisible by 
 9X8, or 72. Is either of the above numbers divisible 
 by 72? 
 
 Any number divisible by 9 and by 5 is divisible by 
 9X5, or 45. Which of the above numbers is divisi- 
 ble by 45 ? 
 
 What are divisors of 124? Of 176? Of 252? Of 
 384? Of 153? Of 186? Of 207 ? Of 702? Of 
 4041? 
 
 We will now return to the fraction that was first 
 given. Reduce f|f to its lowest terms. 
 
 Reduce to lowest terms, T ££; g|f ; J$f. 
 
 Reduce to lowest terms, fff ; %{$; Iff. 
 
FRACTIONS. 65 
 
 SECTION IX. 
 
 MULTIPLICATION OF FRACTIONS BY FRACTIONS. 
 
 We have seen how we may multiply or divide a 
 fraction by a whole number. We will now inquire 
 how we can multiply or divide one fraction by another. 
 Let us multiply $ by f. First multiply f by 2, w r hich 
 gives f- for the -answer. But here we have multiplied 
 by 2, instead of the real multiplier, f. Now, 2 is 5 
 times greater than f; the product f, then, is 5 times 
 greater than it should be. It must therefore be di- 
 vided by 5. We divide f by 5 by multiplying the 
 denominator by 5, giving ^ for the answer. 
 
 In the same way multiply -| by T \; f X|; |-X£. 
 
 DIVISION OF FRACTIONS BY FRACTIONS. 
 
 Let us now divide f by f. First divide f by 3. 
 This we do by multiplying the denominator by 3, 
 giving for the answer T V Here, however, we have 
 divided by 3, instead of the true divisor, f . We have 
 used a divisor seven times too large. The quotient, 
 therefore, will be seven times too small; T \ must 
 therefore be multiplied by 7, making the answer ff. 
 In the same way perform the following: § — £; f-~$ ; 
 ?-H-; f-H; i-H. 
 
 The above analysis shows the grounds of the rules 
 usually given in arithmetics for the multiplication and 
 division of fractions. 
 
 For Multiplication, multiply the numerators to- 
 gether for a new numerator, and the denominators 
 for a new denominator. 
 
 For Division, invert the divisor, and proceed as 
 in multiplication. 
 
 Sometimes we wish to find the value of a eom- 
 6 * E 
 
66 MENTAL ARITHMETIC. 
 
 pound fraction, as § of f . In such cases, we may un- 
 derstand the sign of multiplication, X, to stand in the 
 place of the word of, and treat it as a case of multi- 
 plication; for, in the above example, it is plain that 
 one third off is T %] and two thirds is twice as much, 
 that is, T %. 
 
 What is f of | of f ? Multiplying as. we have done 
 above, we have for the answer t 2 <jV But this opera- 
 tion may be shortened. We see that 4 appears as a 
 factor both in the numerator and the denominator. 
 We may, then, cancel them both, which will have the 
 same effect as dividing both terms of the answer by 4. 
 Again ; 3 appears in both, the numerator and the de- 
 nominator; for in the denominator it is a factor of 9. 
 We may therefore cancel 3 in both terms. 
 
 2 $ 4 
 The question will then appear thus, 75X-X-5, sub- 
 
 stituting 3 in place of the 9. Multiplying together the 
 terms that now remain, we have § for the answer. 
 This is the same fraction as T 2 ^V If you separate the 
 terms of T %\ into their prime factors, and cancel what 
 are common to both, the remaining factors will give 
 the fraction f . 
 
 Multiply the fractions f Xf X T f Xf Writing the 
 terms that are composite in the form of their prime fac- 
 tors, and cancelling factors that are common in both, it 
 
 # $ %X7t $ ^ . ' . 
 
 will stand - — - — -x-3 — „X-^ X^, which gives T V 
 
 $X£X2 $X3^3X£ f & 
 
 Multiply fXfXf; ?iX«XJ. 
 
 Multiply §£XtV: t|X z 8 t ; HXfXf. 
 
 TO MULTIPLY OR DIVIDE WHOLE NUMBERS BY 
 FRACTIONS. 
 
 The above examples will show how to multiply or 
 divide a whole number by a fraction. 
 
FRACTIONS. 07 
 
 Multiply 7 by |. Multiplying 7 by 4 gives 28, 
 which is 5 times too great, because 4 is five tijnes 
 greater than f. We must therefore divide the answer 
 by 5, thus, - 2 ^. As this is more thau 1, we can re- 
 duce it to a whole number and a fraction. As f is 
 equal to 1, ^ will be equal to 5; - 2 5 8 -, therefore, is equal 
 to 5f. 
 
 In this way multiply 6 by f- ; 9 by f ; 8 by f . 
 
 This operation is in fact the same as multiplying a 
 fraction by a whole number, which has been treated 
 of already. 
 
 Let us next divide 7 by f . Dividing 7 by 3, we 
 have £. Here, however, we have divided by a number 
 4 times too great; for 3 is four times greater than £. 
 If the divisor is 4* times too great, the quotient will be 
 4 times too small; f, therefore, must be multiplied by 
 4, giving 2 u 8 - for the answer. 
 
 Divide 8 by f; 9-rfi ll«Hj *<B-Aj 
 
 To reduce an improper fraction, as \ 3 -, to a whole 
 number and a proper fraction, we have only to con- 
 sider how many whole ones the fraction is equal to, 
 and how much remains. Thus, -^ is equal to 3; 
 -V 3 -, therefore, is equal to 3£. 
 
 Reduce §; ->/-; f; -^-; tf; Vj «J U] ¥• 
 
 In like manner, if we have a whole number and a 
 fraction, we may always reduce it to an improper 
 fraction. 
 
 ADDITION AND SUBTRACTION OF FRACTIONS. 
 
 Suppose we wish to add together 3£, so that its 
 value shall be expressed in a single expression; we 
 must change 3 to halves, which will be f ; adding £ 
 to this, we have £ for the answer. 
 
 In order to unite separate numbers into one expres- 
 sion, they must be of the same kind. We cannot unite 
 '2 bushels and 3 pecks in one expression. It is still 2 
 
G8 MENTAL ARITHMETIC. 
 
 bushels and 3 pecks, and we can make nothing else of 
 it. w But if we change the bushels to pecks, making 8 
 pecks, we can then add the 3 pecks, and bring it all into 
 one expression, 11 pecks. So, to unite 5§, we must 
 change the 5 to thirds, making -V 5 -, and add the f, 
 making -y% This is called reducing a mixed number 
 to an improper fraction. 
 
 Reduce to an improper fraction, 7£; 8£; 4f ; 5£; 
 6j; 9£; 3§; 5?; 154; 16f; 13$; 20f; &1£ 
 
 Supposing we wish to add £ to £, we must change 
 the £ to fourths, making f ; adding these, we have £ 
 for the answer. > 
 
 Add 4 to T V J = & ; ft + T V = A, -4ws. 
 
 Add|to T 3 ¥ ; f+A; *+&; * + ,&; $ + «. 
 
 Let us now add § and f« This question, you per- 
 ceive, has a difficulty which the former ones had not; 
 for § is no number of fifths, and therefore we cannot 
 bring the fraction into fifths by any multiplication. 
 We want a number for the denominator which can be 
 divided both by 3 and by 5. Now, if you examine, 
 you will find no such number until you come to 15. 
 This is, of course, divisible by 3 and by 5, for these 
 are its factors. We will then take 15 for the de- 
 nominator. This we call the common denominator. 
 Taking, now, the fractions § and $, and changing the 
 denominator, 3, to 15, we see that we have made it 
 5 times as large as it was before; that is, we have 
 multiplied it by 5. We must therefore multiply the 
 numerator by 5, to preserve the value of the fraction. 
 The fraction §- then becomes T §, without altering its 
 value. Passing, now, to the second fraction, f, we see 
 that, in changing the denominator to 15, we have 
 multiplied it by 3; we must therefore multiply its 
 numerator by 3. This will make the fraction T f. 
 The two fractions will stand, then, t£ + t|, which 
 added together are ??=lrV 
 
FRACTIONS. 69 
 
 TO FIND A COMMON DENOMINATOR. 
 
 We can always obtain a common denominator, by 
 multiplying all the denominators together. Then, 
 for the numerators, consider, in the case of each frac- 
 tion, what its denominator has been multiplied by, in 
 order to change it to the common denominator, and 
 multiply the numerator by the same number. Thus 
 each fraction will have had its numerator and its de- 
 nominator multiplied by the same number, and so its 
 value will not be changed. 
 
 What is the value of £ + *? Of§ + j? Off + |? 
 Of*+|? Ofi + f? Off + f? Off + f? Off + f? 
 
 Supposing we wish to add the fractions £ and £; 
 we can proceed as above, and, with the common de- 
 nominator, 24, the fractions will be Jf+ff. But we 
 need not employ so large a denominator as 24. We 
 seek the smallest denominator that shall contain both 
 4 and 6 as a factor. If, now, we separate 4 and 6 
 into their prime factors, we shall find the factor 2 
 belonging both to 4 and to 6; thus, 2X2, 2X3. 
 Now, one of these may be cancelled, and we shall 
 still have 2X2 for the number 4, and 2X3 for the 
 number 6. Multiplying the factors which remain, 
 2X2X3, we have 12 for the smallest common de- 
 nominator. 
 
 From this we see, that, when both the denomina- 
 tors contain the same factor, we may reject it from 
 one of them, and multiply together the factors that 
 remain. 
 
 Add } to T 5 2-. Here 2X2 is common to both de- 
 nominators. Rejecting it in one, and multiplying, 
 we obtain 24 for the least common denominator. 
 
 Add T 5 s t0 uV Here 3x3 is common to both de- 
 nominators. Rejecting it in one, and multiplying what 
 remains, we have 54 for the least common denominator. 
 
70 MENTAL ARITHMETIC. 
 
 Add JJ to &. Add § to &. Add T \ to &. 
 
 When more fractions than two are to be added, it is 
 often most convenient to add two together first, and 
 then add a third to the sum of these, and so on. 
 
 Add § + F + I- First add % and £, which equal f. 
 Next, |'+t- » = «; and ft=& ; t2 + A = t! = 1 t V, 
 
 Add £+§+tV First add { and & ; then to the 
 sum of these add J. 
 
 Addf+A+t. Addf + f + A- Addf + f + f. 
 
 Add A + J. Add T \ + f AddrV + ^ + yV 
 
 From \± subtract T 5 Z . From % sub. T V. From T £ 
 sub. 3 5 j- 
 
 From T f sub. £. From ff sub. tV From §£ sub. T 9 v. 
 
 Miscella neons Examp les. 
 
 1. A man spends | of a dollar in a day. What part 
 of a dollar will he spend in 5 days? How much will 
 he spend in 9 days? How much in 11 days? 
 
 2. A man earns f of a dollar in a day. How much 
 will he earn in half a day ? How much in J of a day ? 
 How much in £ of a day ? 
 
 Here consider whether you can divide the numerator. 
 
 3. A man earns I of a dollar in a day. How much 
 can he earn in half a day ? How much in £ of a day ? 
 How much in l of a day? 
 
 Consider whether you can divide the numerator, 
 and, if you cannot, what you must do. 
 
 4. A vessel filled with water leaks so that f of its 
 contents will leak out in a week. At this rate, what 
 part will leak cut in a day? 
 
 What is } of | ? 
 
 5. If a team ploughs £ of an acre in 6 hours, how 
 much will it plough in one hour? How much in 3 
 hours ? 
 
FKACTIOxNS. 71 
 
 What is i of f ? What is £ of f ? 
 
 6. If a horse runs ^ of a mile in one minute, how 
 far will he run in f of a minute ? 
 
 How far will he run in f of a minute ? 
 What is £ of J ? What is f of i ? 
 
 7. A man has f- of a dollar, which he wishes to dis- 
 tribute equally among several persons, giving T % of a 
 dollar to each. How many can receive this sum ? and 
 what will be the remainder? 
 
 How many times is T V contained in 7 ? T % in 7 ? T \ in £■ ? 
 How many times is T V contained in 4 ? T 2 3 in 4 ? T 2 ^ in f- ? 
 How many times is 5 V contained in 6 ? ^ in 6 ? jfo in f- ? 
 
 8. A man gave ^ of a bushel of oats to some horses, 
 giving to each |- of a bushel. To how many did he 
 give it ? and what was the remainder ? 
 
 How many times will T V go in 5 ? In>f ? How 
 many times will T 2 F go in f ? 
 
 9. A man has £■ of a dollar. He gives £ of a dol- 
 lar to one person, and f of a dollar to a second. What 
 part of a dollar has he left ? 
 
 How many cents had he at first ? How many cents 
 did he give away ? How many cents had he left ? 
 
 10. If 13 pounds of figs cost | of a dollar, what is 
 that a pound? 
 
 11. If 5£ lbs. of figs cost T 5 2- of a dollar, what is 
 that a pound? Find first what one half pound will 
 cost. 
 
 12. If f of a cwt. of iron cost 4£ dollars, what will 
 a hundred weight, cost ? 
 
 13. If 34^ lbs. of tea cost llf dollars, what will 1 
 pound cost? 
 
 Here you find - 6 ^ 9 - pounds cost V °f a dollar. There- 
 fore 69 pounds must cost V of a dollar. 
 
 14. If § of a barrel of Hour cost 3| dollars, what is 
 that a barrel ? / 
 
 15. If wood is 5} dollars a cord, what will tb of & 
 cord cost ? What will 4£ cords cost ? 
 
72 MENTAL ARITHMETIC. 
 
 16. If 33£ gals, of molasses cost llf dollars, what 
 is that a gallon ? 
 
 17. If 31^ gals, of vinegar cost 4f dollars, what is 
 that a gallon ? 
 
 18. If a bottle of wine, containing l£ pints, cost | 
 of a dollar, what would a barrel of wine come to at 
 that rate ? 
 
 19. In a pile of wood there are 13^ cords. How 
 many loads, of £ of a cord each, are there in the pile? 
 
 20. How many times will 2£ go in 7V. In 9£? 
 In 11? 
 
 21. How many loaves, of 8| oz. of flour each, can 
 be made from 7 pounds of flour ? « 
 
 22. If a family consume 3± pounds of flour a day, 
 how long will a barrel of flour, that is, 196 pounds, 
 last them? 
 
 How long will it last if they consume 2£ lbs. a day ? 
 
 23. If a barrel of flour last a family 40 days, how 
 long will 14 pounds last them? 
 
 24. A garrison of 100 men is allowed 12 oz. of flour 
 a day to each man. How long will 10 barrels last 
 them ? 
 
 25. Two men hire a horse, a week, for 5 dollars. 
 One travels with him 30 miles, the other 45 miles. 
 What ought each to pay ? 
 
 26. Two men hire a pasture in common for $4.80. 
 One pastures his horse in it 1\ weeks ; the other pas- 
 tures his horse 9 weeks. What ought each to pay? 
 
 I 27. A boy bought 3 doz. of oranges for 37^- cents, 
 and sold them for l£ cents apiece. What did he gain ? 
 
 28. A man bought 7 yds. of cloth for 16 dollars, and 
 sold it for 3 dollars a yard ? What did he gain on 
 each yard ? 
 
 29. A man, worth 1690 dollars, left f of his prop- 
 erty to his wife. How much did she receive ? The 
 remainder he divided equally among 3 sons. What 
 did each one receive ? 
 
FRACTIONS. 73 
 
 30. A man bequeathed his estate of 14,000 dollars, 
 one third to his wife, and the remainder to be divided 
 equally among four sons. What did the wife, and what 
 did each son, receive ? 
 
 31. In an orchard one third of the trees bear apples, 
 two fifths as many bear plums, and the rest bear cher- 
 ries. What portion of the trees bear plums? What 
 portion bear cherries ? The number of cherry-trees is 
 40. What is the whole number of trees in the or- 
 chard ? 
 
 32. What is f of 549 ? What is f of 374 ? 
 
 33. What is i of 175i ? What is | of 198 ? 
 
 34. What is £ of i of 1640 ? What is f of 972? 
 
 35. If 2 barrels of flour cost ll£ dollars, what will 
 17 barrels cost ? What will 22£ barrels cost ? 
 
 36. If 2£ cords of wood cost 15 dollars, what will 
 68| cords cost ? What will 200 cords ? 
 
 37. If a horse eat 2£ tons of hay in 30 weeks, 
 what part of a ton will he eat in 1 week ? 
 
 38. What is the cost of 23£ yds. of cloth at £ of a 
 dollar a yard ? 
 
 39. What is the cost of 31| gallons of molasses, at 
 T % of a dollar a gallon ? 
 
 40. A grocer drew from a cask, containing 31^ gal- 
 lons, i of its contents. Now, how much did he draw 
 out ? How much remained ? 
 
 SECTION X. 
 
 THJE LEAST COMMON MULTIPLE. 
 
 Name the multiples of 2 up to 20. 
 Name the multiples of 5 up to 30. 
 Name the multiples of 7 up to 63. 
 Name the multiples of 3 up to 15 : also of 4 up to 20. 
 
74 MENTAL ARITHMETIC. 
 
 What one of the numbers just named is a multiple 
 both of 3 and of 4? 
 
 Any product which has two or more numbers as 
 factors of it is a common multiple of those numbers. 
 
 Name the multiples of 6 up to 24, and of 9 up to 
 27, and state what common multiple you find of 6 
 and 9. 
 
 Name the multiples of 8 up to 32, and of 12 to 36, 
 and state what common multiple you find of 8 and 12. 
 
 Name all the multiples of 6 and of 8 below the 
 number 50. How many common multiples have you 
 found of 6 and 8? 
 
 What is the least common multiple of 6 and 8 ? 
 
 Find a common multiple of 6 and 9. 
 
 Find a second common multiple of 6 and 9. 
 
 What is the least common multiple of 6 and 9 ? 
 
 What is the least common multiple of 8 and 10 ? 
 
 What is the least common multiple of 3, 4, and 6 ? 
 
 What is the least common multiple of 4, 6, and 8 ? 
 
 From the above exercise it will be seen that the least 
 common multiple of two or more numbers is the small- 
 est product that contains them all as factors. 
 
 The least common multiple will always be the first 
 one found in enumerating the series, as in the above 
 examples. 
 
 But this method is often tedious ; and we now pro- 
 ceed to exhibit another method, independent of any 
 such series. 
 
 Suppose, now, we wish to find the smallest common 
 multiple of 3 and 5. The number, it is clear, must 
 be a certain number of 3's, and also a certain number 
 of 5's. Now, by multiplying 3 and 5 together, we 
 evidently obtain such a number ; for it will be 3 times 
 5, and it will be 5 times 3. Multiplying the two num- 
 bers together, then, will always give their common 
 multiple. The next question is, Will this product of v 
 
FRACTIONS. 75 
 
 the two numbers be their least common multiple? 
 This will depend on the character of the numbers. 
 If the numbers are prime to each other, their product 
 will be their least common multiple. For example, in 
 the numbers 3 and 5, if we take any number of 5's less 
 than 3, as 2X5, the factor 3 has disappeared, and the 
 number is no longer a multiple of 3. If we take any 
 number of 3's less than 5, as 4X3, the factor 5 has 
 disappeared, and the number is no longer a multiple of 
 5. The product, therefore, of numbers prime to each 
 other, is their least common multiple. In the above 
 example, the numbers 3 and 5 were prime in them- 
 selves, and not merely prime to each other. To make 
 the principle more clear, we will take two numbers 
 that are not prime in themselves, but are only prime 
 to each other. 
 
 What is the least common multiple of 8 and 9? 
 Multiplying them together, we have 72. 72 is, then, 
 a common multiple of .8 and 9. The question is, Is 
 it their smallest common multiple? Writing the num- 
 bers with their factors, they are 2X2X2, and 3X3. 
 Now, if we erase one of the 2's, we have no longer 
 the factors of 8, and the product of the factors will 
 not be divisible by 8. In the same way, if we erase 
 one of the 3's, the product will not be divisible by 9. 
 
 If, then, the numbers are either prime, or prime to 
 each other, the product is their least common multiple. 
 
 Next, let us inquire, What is the least common mul- 
 tiple of 4 and 6? Their product is 24; but this is 
 evidently not their least common multiple, for 12 con- 
 tains both 4 and 6 as factors. To show why it is, 
 that, in this case, something less than the product of 
 the numbers is their least common multiple, we will 
 express each by its factors, thus, 2X2, 2X3. Now, it 
 is clear that any number of times which you take 
 2X2 as a factor will be a multiple of 2x2. If, then, 
 we throw out the 2 in the 2X3, and multiply by the 
 
76 MENTAL, ARITHMETIC. 
 
 remaining 3. the product will be a multiple of 2X2, 
 or 4. Looking, now, at the 2X3, or 6, it is evident 
 that any number of times which you may take that 
 as a factor will be a multiple of 2X3. But the 2 we 
 may take from the 2X2, throwing away that in the 
 2X3: this leaves us to multiply the 2X3 by 2; as 
 we before multiplied the 2X2 by 3, making 12 as the 
 least common multiple. The *ule, therefore, is: Re- 
 tain of each prime factor the highest power which* 
 appears in any of the given numbers; erase the rest, 
 and multiply together what then remain. 
 
 Find the least common multiple of 8, 24, and 36. 
 Expressed by the factors, they are 2X2X2; 2X2X2 
 X3; 2X2X3X3. Now, 2X2X2 is common to 8 and 
 24; it may be thrown out of the latter, leaving only 3. 
 Examining again, you observe that 2X2 is common 
 to 8 and 36; we throw this out of 36, leaving 3X3. 
 Finally, 3, we find, is common to 24 and 36; throw- 
 ing this out of 24, we find the numbers appear as fol- 
 lows: 2X2X2; 2X£X£x£; £X2X3X3. These 
 multiplied together give for the least common multi- 
 ple, 72. This conforms to the rule; for 2X2X2 is 
 the highest power of the factor 2, and 3X3 of the 
 factor 3. What is the least common multiple of 24, 
 60, and 100? These factors are 2X2X2X3; 2X2X3 
 X5; 2X2X5X5. We see that 2X2 is common to 
 them all; expunge it in the second and third number. 
 Next, 3 is common to the 1st and 2d ; expunge it in 
 the 2d. Lastly, 5 is common to the 2d and 3d; ex- 
 punge it in the 2d, and the numbers will stand, 2X2 
 X2X3; &X<*X$X$; 2X2X5X5. These multiplied 
 together give 600. 
 
 To multiply these most easily, first take 2X2X5 
 X5=10(); then the remaining factors, 2X3, mul- 
 tiplied by 100, give 600. 
 
 What is the least common multiple of 24, 40, 
 and 72? 
 
FRACTIONS. 77 
 
 What is the least common multiple of 18, 54, 81 ? 
 What is the least common multiple of 15, 4, 7? Of 
 15, 40, 27? Of 16, 14, 6? Of 60, 12, 18? 
 
 From the foregoing reasoning and examples, you 
 wiU perceive that the least common multiple of sev- 
 eral numbers is the product of all their prime factors, 
 each taken in the highest power in which it appears 
 in any of the numbers. 
 
 SECTION XI. 
 
 PRACTICAL QUESTIONS. 
 
 1. What part of a shilling is 1 penny? 2 pence? 
 3 pence ? 4 pence ? 5 pence ? 6 pence ? 7 pence ? 
 
 2. What part of a penny are 2 farthings? 3 far- 
 things? 4 farthings? 5 farthings? 6 farthings? 8 
 farthings ? 
 
 3. What part of a shilling is 1 farthing? 2 far- 
 things? 3 farthings? 
 
 What part of a shilling is 1 penny and 1 farthing ? 
 1 penny 2 farthings? 3d. 3qrs. ? 4d. 2qrs. ? 6d. 1 
 qr.? 9d. 2qrs.? 
 
 4. What part of a pound is 1 shilling? 2s.? 3s.? 
 5s.? Is. Id.? 2s. Id.? 4s. 3d.? "5s. 6d.? 7s. 9d.? 
 3s. 8d.? 
 
 5. What part of a pound is 1 farthing ? 2 qrs. ? 
 3 qrs.? 2d. 3 qrs.? 5d. 2 qrs.? Is. Id. 1 qr. ? 6 s. 
 7d. 3 qrs.? 
 
 6. What part of a pound avoirdupois is 2 oz. ? 
 3 oz.? 4oz.? 5 oz. ? 6 oz. ? 7 oz. ? 8 oz. ? 9 oz. ? 
 10 oz.? 
 
 7. What part of one ounce is one dram ? What part 
 
 7* 
 
78 MENTAL ARITHMETIC. 
 
 of one pound is one dram ? 2 drs. ? 3 drs. ? 1 oz. 1 
 dr. ? 1 oz. 2 drs. ? 2 oz. 4 drs. ? 3 oz. 6 drs. ? 8 oz. 
 3 drs.? 9 oz. 11 drs.? 
 
 8. What part of a pound is T V of an oz. ? T \ of 
 an oz.? 
 
 What part of a pound is f an oz. ? 2} oz. ? 3£ 
 oz.? 4£ oz. ? 
 
 9. What part of a pound Troy is 1 dwt. ? 5 dwt. ? 
 6 dwt. ? 9 dwt. ? 1 1 dwt. ? 10 dwt. ? 1 oz. 1 dwt. ? 
 
 3 oz. 4 dwt. ? 
 
 What part of an oz. Troy is 1 dwt. ? 3 dwt. 1 gr. ? 
 
 4 dwt. 6 gr. ? 7 dwt. 3 gr. ? 8 dwt. 9 grs. ? 10 dwt. ? 
 12 dwt.? 16 dwt. ? 
 
 10. What part of an ell English is 1 qr. of a 
 yard? 2 errs. ? 3 qrs. ? What part of a qr. is 1 
 nail? 3 nails? 
 
 11. What part of a yd. is 1 qr. 1 nail ? 2 qrs. 3 n. ? 
 3 qrs. 2 n.? What part of an ell English is 3 nails? 
 
 1 qr. 3 n. ? 4 qrs. In.? 
 
 12. What part of a yd. is 1 inch ? 4 inches ? 7 
 inches ? 9 inches ? What part of a yard is 1 qr. 2 in. ? 
 
 2 qrs. 3 in. ? 3 qrs. 1 in. ? 
 
 13. From a vessel containing 3 gallons of wine 3 
 gills leaked out. What part of a gallon leaked out ? 
 What part of a gallon remained? 
 
 14. From a barrel full of wine 7 quarts were drawn. 
 How many quarts remained? What part of the barrel 
 had been drawn out? What part of the barrel had 
 remained ? 
 
 15. If f of a barrel of beer be divided into 4 equal 
 parts, what part of a barrel will each of the parts be? 
 How many gallons will each part be ? 
 
 16. If one quart be taken from a barrel full of beer, 
 what part of a barrel will remain? If 3 pints be 
 taken out, what part will remain? If 7£ gallons be 
 taken out, what part of a barrel is taken out? What 
 part of a barrel remains ? 
 
FRACTIONS. 79 
 
 17. A man distributed 7£ gallons of milk among 5 
 persons. What part of a gallon did he give to each ? 
 
 18. If you have 3^-. gallons of milk, and distribute 
 it to some poor persons, giving § of a gallon to each, 
 how many persons will you give it to ? How much 
 will remain? 
 
 19. What part of 1 foot is H in. ? 24 in. ? 5£ in. ? 
 62- in. ? 8| in. ? 9± in. ? 10§ in. ? 1 1* in. ? 
 
 20. What part of a yard is 2 inches? 34 inches? 
 14 in.? 5^ in.? 6±in.? 17^ in.? 24^ in.?" 
 
 21. What part of a rod is £ a foot? 14 feet? 24 
 feet ? 4 ft. 3 in. ? 6 ft. 7 in. ? 10 ft. 5 in. ? 
 
 22. What part of 3 rods is J- a foot? 1 foot? 3£ 
 feet? What part of a furlong are 2£ rods? 5£ rods? 
 
 23. What fraction of a foot is f of a yard? f of a 
 yd. ? What fraction of a foot is T V of a rod? T 2 T of a 
 rod ? T 3 3 of a rod ? 
 
 24. A man measured the length of his barn with a 
 stick half a yard long, and found the barn 314 times 
 the length of his stick. How long was it? 
 
 25. A carpenter is cutting up a board, 17i feet in 
 length, into pieces 2£ feet long. How many pieces 
 will there be, and how long will be the piece that 
 remains? 
 
 26. A man measures a piece of fence with a pole 
 9£ feet long. The fence is 15J- times the length of 
 the pole. How many rods is it in length? 
 
 27. What part of a peck is ^ of a bushel ? 
 
 What part of a gallon are T V of a peck ? f of a peck ? 
 What part of a quart is ^V of a peck ? -g s of a peck ? 
 What part of a quart are T \ of a bushel ? A of a 
 bushel ? 
 
 28. What part of a peck is |of a bush. ? f of a 
 bush. ? | of a bush. ? f of a bush. ? f of a bush. ? 
 
 29. Two men bought a lot of standing wood in 
 company, for 11 dollars. One cut off 2 cords, the 
 
 'Other 1 cord. What ought each to pay ? 
 
80 MENTAL ARITHMETIC. 
 
 30. Two boys bought the chestnuts on a tree for 
 50 cents. One had 11 quarts, the other 6 quarts and 
 1 pint. What ought each to pay ? 
 
 31. Three men bought a piece of cloth for 24 dol- 
 lars. The first took 2£ yds., the second the same 
 quantity ; and on measuring the remainder, it was 
 found to be 3 yards. What ought each to pay ? 
 
 32. Two men hire a horse for a month for 12 dol- 
 lars. One travels 200 miles with the horse, the other 
 150. How much should each pay ? 
 
 SECTION XII. 
 
 DECIMAL FRACTIONS. 
 
 [See Numeration, Part II.] 
 
 In the calculations in common fractions, a great in- 
 convenience arises from their irregularity. There is 
 no law regulating the magnitude of either of the 
 terms. The denominator may be in any ratio what- 
 ever to the numerator. From seeing one you can 
 make no inference at all respecting the magnitude of 
 the other. In calculations of addition, it is often more 
 than half the work to bring the fractions into a com- 
 mon denomination. 
 
 Now. it is evident, that, if fractions could be writ- 
 ten in the same manner as whole numbers, that is, 
 increasing in a tenfold rate as you advance to the left, a 
 and decreasing in a tenfold rate as you advance to the 
 right, an immense gain would be made in the con- 
 venience of calculating them. Operations in fractions 
 would then be just as easy as operations in whole 
 "lumbers. Now, this advantage is gained in decimal 
 fractions. They are brought under the same law as v 
 
DECIMAL FRACTIONS. 81 
 
 whole numbers. Let us observe the manner in which 
 whole numbers are written. Take the number 222 ; 
 the right-hand figure signifies two units, the next two 
 tens, the next two hundreds ; just as if it were writ- 
 ten in this manner, 2X100 + 2X10 + 2; two multi- 
 plied by 100, plus two multiplied by ten, plus two; 
 making two hundred and twenty-two. But this cum- 
 bersome method of writing is unnecessary, because 
 the law of notation determines what number the 
 figures in each place shall be multiplied by. It must 
 not be forgotten that the figure 2, in the above exam- 
 ple, in no case signifies of itself more than two. It is 
 the place it occupies that gives it the higher value of 
 tens or hundreds. 
 
 Now, it would evidently be a great convenience if 
 we could reduce fractions to the same law, so that 
 they would, like whole numbers, decrease in a deci- 
 mal ratio, in advancing from the left to the right. To 
 show this regularity to the eye, we will write the fol- 
 lowing numbers: two multiplied by 1000, two multi- 
 plied by 100, two multiplied by 10, two units, two 
 divided by 10, two divided by 100, and two divided 
 by 1000. Written in full, they would stand thus : 
 2X1000 + 2X100 + 2X10 + 2 + A + T ^ + To 2 ^. 
 
 But we have seen that we may write the whole 
 numbers without the multipliers, thus, 2222, because 
 we know, from the place each figure occupies, what its 
 multiplier must be. Just so we can write fractions 
 without the denominators, provided we know, from 
 the place of the numerator, what the denominator 
 must be. Thus the whole of the above series may 
 be written as follows : 2222.222. A decimal, there- 
 fore, is the numerator of a fraction, whose denomina- 
 tor is never written, but is always understood to be 1 
 with as many ciphers as there are places in the 
 decimal. 
 
 ' You observe that, in writing the series given above, 
 
 T 
 
82 MENTAL, ARITHMETIC. 
 
 there is a period placed at the right hand of the whole 
 numbers, separating the unit figure from that of 
 tenths. The period must never be omitted when 
 there are fractions, for it enables you to determine the 
 value of each figure in the sum. Instead of reading 
 .22 two tenths and 2 hundredths, we may call it 22 
 hundredths, which is more convenient, and amounts to 
 the same ; for two tenths is equal to 20 hundredths ; so 
 .222 is two hundred and twenty-two thousandths. 
 So, in all cases, read the decimal numbers as whole 
 numbers, and for their denominator take 1 with as 
 many ciphers as there are places in the written deci- 
 mals. 
 
 In all your study of decimals, be careful not to con- 
 found the words which express fractions with the 
 similar words which express whole numbers : as tenths 
 with tens, hundredths with hundreds. The following 
 questions will aid you in fixing this distinction clearly 
 in mind : 
 
 1. How many tenths are equal to ten whole 
 ones? 
 
 2. How many tenths are equal to two and a half 
 whole ones ? 
 
 3. How many hundredths are equal to three and a 
 quarter whole ones? 
 
 4. How many hundredths are equal to one hundred 
 whole ones? 
 
 5. How many thousandths are equal to ten whole 
 ones ? 
 
 6. In fifteen whole ones how many tenths ? How 
 many hundredths ? 
 
 7. In seventy-five hundredths how many tenths ? 
 
 8. In three tenths how many hundredths? 
 
 9. hi six tenths how many thousandths? 
 
 Thus, you observe, fractions have been brought 
 under the same law that regulates the writing of 
 whole numbers. They may now be added, sub- 
 
ADDITION AND SUBTRACTION OF DECIMALS. 83 
 
 tracted, multiplied, and divided, like whole numbers. 
 But. in doing this, it is important to determine the 
 place of the period that separates the whole numbers 
 from the fractional part of the sum. Where must the 
 period be placed in the answer ? 
 
 ADDITION AND SUBTRACTION OF DECIMALS. 
 
 Let us first observe how important it is that the 
 rule in this case be entirely correct. If I have this 
 number, 32.5, to write, and by any mistake I should 
 write it 3.25, it would denote a quantity only one 
 tenth as great as it should be; or, if I should write 
 325. it would denote a quantity ten times greater than 
 it should be. Moving the period one place to the 
 right, makes the number ten times as great as it was 
 before ; for tens become hundreds, and hundreds, 
 thousands ; and each figure ten times as great as be- 
 fore. So r by moving the period one place to the left, 
 the number becomes just one tenth what it was be- 
 fore. Removing the period two places from its true 
 place, makes the number 100 times larger or smaller 
 than it should be, according as you remove it to the 
 right or the left. Hence you may see that, in order to 
 multiply a number that has decimals, by 10, you have 
 only to remove the period one place to the right ; to 
 multiply by 100, remove it two places ; and so on. 
 To divide by 10, remove the period one place to the 
 left : to divide by 100, remove it two places ; and so 
 on. From the above you may see the importance of 
 being perfectly accurate in fixing the place of the 
 decimal in the answer to any question. 
 
 We will begin with addition. Add 4.46 to 3.21. 
 
 Here you observe the two whole numbers make 7, 
 
 and 46 hundredths added to 21 hundredths make 67 
 
 • hundredths. The answer, then, must be 7.67, having 
 
84 MENTAL ARITHMETIC. 
 
 two (Jgcimal places. Add 6.8 to 5.23. The 3 hun- 
 dredths must evidently stand alone, since there is noth- 
 ing like it to add to it ; 2 tenths added to 8 tenths 
 make 10 tenths, or one whole one ; this we carry to 
 the 5, which gives us for the answer, 12.03. This 
 will serve to suggest the rule for placing the period in 
 the answer to questions in addition. The number of 
 decimal places in the answer must be as great as can 
 be found in any one of the numbers to be added. 
 
 The same rule holds in subtraction. Take for 
 illustration the numbers given in the second example 
 of addition. From 6.8 subtract 5.23. Now, as in 
 the minuend there are no hundredths, we must borrow 
 10 in this place, and we shall have a remainder of 7 
 hundredths ; adding 1 tenth to the subtrahend, to com- 
 pensate for the 10 hundredths added to the minuend, 
 we have, in the place of tenths, a remainder of 5 ; 
 finally, in the place of units we subtract 5 from 6 ; 
 the answer is 1.57. In performing this operation, you 
 may, if you please, call the 8 tenths 80 hundredths ; 
 then 23 hundredths from 80 hundredths leaves 57 hun- 
 dredths. By performing slowly and with care exam- 
 ples of your own selection, you will see the verifica- 
 tion of the rule given above, both for addition and 
 subtraction. 
 
 Add 2.4 to 3.8. Add .6 to 1.3. Add .4 to .3. 
 Add .37 to .25. Add 3.7 to .24. Add 1.08 to .05. 
 
 From 4.6 subtract 2.4. From 7.1 subtract 6.4. 
 From .18 subtract .13. From 4.5 subtract .6. 
 
 In these examples, each step should be explained by 
 the pupil as he performs it. 
 
 MULTIPLICATION OF DECIMALS. 
 
 The rule in multiplication we shall find to be differ- 
 ent from the above. 
 
MULTIPLICATION OF DECIMALS. 85 
 
 1. First, we will multiply 2.4 by 3. If we regard 
 the multiplicand as a whole number, the answer will 
 be 72. But by regarding the multiplicand as a whole 
 number, — as 24 instead of 2 and 4 tenths, — we re- 
 garded it as ten times greater than it really is. The 
 answer, therefore, is ten times too great. Instead of 
 72, it must be 7.2. 
 
 2. Multiply 6.2 by 3.4. By regarding both as 
 whole numbers, we obtain the answer 2108. Now, in 
 calling the multiplicand 62 instead of 6.2, we treated 
 it as 10 times greater than it is. The answer must 
 therefore be 10 times too great, even if the multiplier 
 were a whole number. We must therefore divide it 
 by 10, or write 210.8. But the multiplier also is 10 
 times too great ; the answer must therefore be divided 
 again by 10, in order to bring it right. Thus the an- 
 swer will stand 21.08. 
 
 3. Again ; multiply .62 by 3.4. Here we obtain 
 the same figures as before, 2108 ; but, by treating the 
 multiplicand as a whole number, we regarded it as 
 100 times too great ; the answer, therefore, must be 
 divided by 100, or written 21.08. But the multiplier, 
 calling it a whole number, was taken 10 times greater 
 than it is ; the answer must be again divided by 10, 
 and thus it will stand 2.108. 
 
 4. Once more ; multiply .62 by .34. The figures 
 of the answer are, as before, 2108 ; but, by regarding 
 both the factors as whole numbers, we take each 100 
 times greater than it is. We must therefore divide by 
 100 to correct the error in the multiplier, and again 
 by 100 to correct the error in the multiplicand. This 
 will remove the point four places to the left, and the 
 true answer will be .2108. By examining these ex- 
 amples, you will see that the pointing in each case 
 conforms to the following rule : 
 
 Point off as many figures for decimals in the answer 
 8 
 
86 MENTAL ARITHMETIC. 
 
 as there are decimal places in both the factors taken 
 together. 
 
 5. Multiply 2.7 by .3.-6. Multiply .6 by .7.-7. 
 Multiply 6. by .7.-8. Multiply .02 by .3.-9. Multi- 
 ply .02 by .03.-10. Multiply .01 by ".01. 
 
 DIVISION OF DECIMALS. 
 
 1. Divide 48 by 12. Ans. 4. 
 
 2. Divide 4.8 by 12. The figure expressing the 
 answer is 4, as in the first case ; but, observe, the divi- 
 dend is only one tenth as large as before ; the quo- 
 tient, therefore, is only one tenth as large. Instead of 
 4. it is .4. 
 
 3. Divide .48 by 12. The figure of the quotient 
 is still 4; but, as the dividend is only one hundredth 
 part as large as in the first example, the quotient will- 
 be only one hundredth part of 4, or 4 hundredths, 
 written thus, .04. 
 
 4. Again ; divide 48 by 1.2. The quotient is still 
 4; but we must investigate the question to see where 
 this 4 must stand. You observe that the divisor is 
 now only one tenth of 12. Now, if the divisor is 
 only one tenth as great as it was before, you must 
 consider how that will affect the quotient. You will 
 perceive, on reflection, that, as you diminish the divisor, 
 you increase the quotient. If you make the divisor 
 half as great, the quotient will be twice as great ; and 
 so proportionally of other numbers. Now, as, in this 
 instance, the divisor is one tenth as great as before, 
 the quotient must be ten times greater. The figure 
 4, then, which is the quotient figure, instead of stand- 
 ing in the place of units, as before, must stand in the 
 place of tens-; that is, it must be 40, the cipher 
 merely showing that the 4 stands in the piace of tens. 
 
DIVISION OF DECIMALS. 87 
 
 5. Once more; divide 48 by .12. Here, again, 
 yon have 4 for the quotient figure, for yon can have 
 no other ; but, on comparing this example with the 
 first, yon perceive the divisor is only one hundredth 
 part as great ; the quotient must therefore be one hun- 
 dred times greater ; that is, it is 400, the ciphers merely 
 removing the 4 into the place of hundreds. 
 
 On examining these examples carefully, you will 
 see that each answer is unquestionably correct. " But 
 by what rule," you ask, " are these examples 
 wrought ? " They are not wrought by rule, but by 
 reasoning on the numbers themselves ; and the more 
 you habituate yourself to reason in arithmetic, the 
 less need you will have to depend on rules. 
 
 With this suggestion I will now state a rule, which 
 you may at any time follow, when you have not time 
 to look into the reason of the operation. 
 
 There must be as many decimals in the quotient as 
 the decimals in the dividend exceed those in the divi- 
 sor. When there are fewer decimals in the dividend 
 than there are in the divisor, ciphers must be added so 
 as to make the number equal. 
 
 We will now review the foregoing examples, and 
 observe their conformity with the above rule. Exam- 
 ple 1 has no decimals in the divisor or the dividend, 
 therefore none in the quotient. Ex. 2, the dividend 
 has one decimal, the divisor none ; the quotient has 
 therefore one. Ex. 3, the dividend has two decimals, 
 the divisor none; the quotient has two. Ex. 4, the 
 dividend has none, the divisor one ; there must then 
 be a cipher added to the dividend, and then the quo- 
 tient will be in whole numbers. Ex. 5, the divi- 
 dend has none, the divisor two ; there must then be 
 two ciphers added, and then the quotient will be in 
 whole numbers. 
 
 6. Divide 45 by 15. Divide 4.5 by 15. Divide .45 
 by 15. Divide 45 by 1.5. Divide 45 by .15. 
 
88 MENTAL ARITHMETIC. 
 
 7. Divide 66 by 11; 6.6 by 11; .66 by 11; 66 by 
 1.1; 66 by .11. 
 
 In calculations of Federal money, cents and mills 
 are regarded as decimals ; the point, therefore, separa- 
 ting the whole numbers from the fractions must be 
 placed between the dollars and the cents. Thus, 24.00 
 is 24 dols. ; 2.40 is 2 dols. 40 cts. ; 0.24 is 24 cts. 
 
 8. A man divided $24.00 among 3 men. How 
 much did each receive? 
 
 9. A man divided $2.40 among 3 men. How 
 much did each receive? Divide 2.4 by 3. 
 
 10. A man divided $0.24 among 3 men. How 
 much did each receive? Divide $0.24 by 3. 
 
 11. A man divided 36 dollars among 4 persons. 
 How much did each receive? Divide 36 by 4. 
 
 12. A man divided $3.60 among 4 persons; How 
 much did each receive ? What is one fourth of $3.60 ? 
 
 13. A man divided $0.36 among 4 men. How 
 much did each receive? What is one fourth of .36? 
 
 SECTION XIII. 
 
 REDUCTION OF VULGAR FRACTIONS TO DECIMALS. 
 
 We have now seen that decimal fractions have this 
 great advantage over vulgar fractions, — that they con- 
 form to the same law of notation as whole numbers, 
 and may be added, subtracted, multiplied, and divided, 
 in the same manner, and with the same ease, as whole 
 numbers. It is desirable, therefore, to introduce them 
 in a great many cases instead of vulgar fractions. The 
 next question that arises, therefore, is, Can a vulgar 
 fraction be changed to a decimal having the same 
 value ; and how can it be done ? Take the fraction 
 
ANALYSIS OF DECIMALS. 89 
 
 £ ; we wish to reduce it to tenths, or, in other words, 
 to express it in tenths. Now, we can change any 
 number to tenths by multiplying it by 10. Thus, 3 
 is 30 tenths, 4 is 40 tenths. We will now take £, and 
 change the numerator, 1, to tenths, and it will stand 
 r 1.0; but the fraction was not one, but one half of one ; 
 1.0, therefore, is twice as great as it should be ; we 
 must divide it, therefore, by 2, that is, by the denomi- 
 nator, and it will be .5. To reduce a vulgar fraction, 
 then, to a decimal ; add a cipher to the numerator, and 
 divide by the denominator. If one cipher is not 
 enough to render the division complete, add more. 
 
 Reduce to a decimal \. Change the numerator 
 to tenths; it will be 1.0; but the quantity to be 
 reduced to tenths was not one, but one fifth of one ; 
 1.0, therefore, is 5 times greater than it should be ; 
 dividing by 5, the answer is .2. 
 
 Reduce to a decimal the fraction §, explaining each 
 step in the operation. 
 
 Reduce to a decimal the fraction f. 
 
 Reduce to a decimal the fraction f. 
 
 Reduce to a decimal the fraction £. 
 
 Reduce to a decimal the fraction f. 
 
 I will here direct your attention to a fact that it is 
 interesting to notice. If the denominator of the vul- 
 gar fraction is one of the factors of 10, that is, if it is 
 either 2 or 5, the decimal figure will be as many times 
 the other factor as there are units in the numerator of 
 the vulgar fraction. This will appear self-evident 
 i when we express the numbers by their factors. Thus, 
 in obtaining the decimal for £, we divide 10 by 2; but 
 10 is 2X5 ; therefore, in dividing by 2, we simply ex- 
 punge the factor we divide by, and leave the other ; 
 2)^X5. So in the fraction -£, we obtain the decimal 
 by dividing 10 by 5, which expunges the factor 5 ; 
 , 5)#X2. In reducing f, we divide 2X10 by 5, thus, 
 5)2X2X£, leaving twice the factor 2 : in f, 5)3X2 
 8* 
 
90 
 
 MENTAL ARITHMETIC. 
 
 X$, leaving 3 times the factor 2; in f, 5 ) 2X2X2X0, 
 leaving 4 times the factor 2. 
 
 2. We will now take the fraction £. Proceeding as 
 before, we wish to divide 10 by 4, thus, 2X2)2X5. 
 Here we see the division cannot be complete ; for the 
 divisor contains the factor 2 twice, while the dividend 
 has it only once. If, however, we had multiplied 
 the original numerator, 1, by 100, instead of 10, we 
 should have had 10 twice as a factor in the dividend, 
 and of course each factor of 10 twice ; 100 is 10X10, 
 and 10 is 2X5. It would have stood then thus, 2X2 ) 
 2X5X2X5. The division is now complete; for the 
 dividend contains the factor 2 as many times as the 
 divisor has it. Expunging these, we have remaining 
 the factor 5 taken twice, or .25. 
 
 This process, you may observe, conforms to the 
 rule, to add as many ciphers as may be necessary to 
 render the division complete. 
 
 3. Reduce the vulgar fraction f to a decimal. 30 
 is composed of the prime factors 3X2X5; it con- 
 tains 2 only once, and therefore it is not divisible by 
 2X2; 30 must therefore be multiplied by 10. This 
 will introduce another 2, and it will stand thus, 2X2) 
 3X2X5X2X5. By expunging the two 2's, and mul- 
 tiplying together the other factors, we have .75 for the 
 answer. 
 
 4. Reduce the fraction £ to a decimal. 10, ex- 
 pressed by its factors, is 2X5, and 8 is 2X2X2. We 
 must therefore multiply 2X5 by 10 till it shall contain 
 the factor 2 as many times as 8 contains the same fac- 
 tor ; that is, the numerator, 1, must be multiplied by 
 a thousand. It will then stand, 2X2X2)2X5X2X5 
 X2X5. Expunging the three 2's, there remains for 
 the answer, .125. 
 
 By examining the above examples, you may observe 
 this fact, — that if the denominator of the vulgar fraction, 
 contains one of the factors of 10, that is, 2 or 5, one 
 
ANALYSIS OF DECIMALS. 91 
 
 or more times as a factor, the decimal will contain the 
 other factor just as many times. Thus, J=.5j i or 
 ^ = .25, or .5X.5; i or sxl** =^-*25, or .5X.5X.5. 
 
 In the same way, i=.2] A or ^ = .04, or .2X.2: 
 t** or *xix5 = -008, or .2X.2X.2. In this way you 
 may determine that T V, when reduced to a decimal, 
 will contain 5 four times as a factor, because 16 con- 
 tains 2 four times as a factor. So ^V will contain 5 
 five times as a factor. 
 
 This is conveniently expressed by saying, whatever 
 power of one of the factors of 10 the denominator of 
 the vulgar fraction contains, the same power of the 
 other factor will .appear in the decimal. 
 
 5. Reduce £ to a decimal fraction. Preparing the 
 numbers as before, it will stand 3)2X5. You ob- 
 serve that 3 is different from either of the factors of 
 10. Now, as 10 has only the factors 2 and 5, it is not 
 divisible by 3 without a remainder. 
 
 If you add to the numerator ever so many ciphers, 
 you will only increase the number of times that 2 and 
 5 appear in it as its factors, and the number can never 
 become divisible by 3 without a remainder. The an- 
 swer becomes .333+. and this indefinitely, as far as 
 you may please to carry on the operation. On the 
 same principle, we shall find that it is not possible to 
 express accurately, in decimals, any vulgar fraction 
 whose denominator contains as a factor any thing dif- 
 ferent from the factors of 10; for this denominator 
 becomes, in the reduction, a divisor of 10 or some 
 power of 10 ; and if it has any thing in it as a factor 
 which is prime to the factors of 10, the complete di- 
 vision is impossible. Thus £ cannot be exactly ex- 
 pressed in decimals ; because, though one of its factors, 
 2, is a divisor of 10, the other, 3, is prime to 10. On 
 this principle the following questions may be examined. 
 
 Can | be accurately expressed in decimals ? Why ? 
 
 Can £ be accurately expressed in decimals? Why? 
 
92 MENTAL ARITHMETIC. 
 
 Can £ be accurately expressed in decimals? Why? 
 Can T V be accurately expressed in decimals ? Why ? 
 Can ^ ? T V? A? A? A? A* A? A? A? A? 
 A? A? A? A? 
 
 6. Name all the denominators, from 2 up to 20, of 
 such fractions as can be accurately expressed in deci- 
 mals. From 20 to 40. From 40 to 60. From 60 to 
 80. 
 
 7. Name all the denominators, from 2 to 20, of such 
 fractions as cannot be expressed accurately in decimals. 
 From 20 to 40. From 40 to 60. From 60 to 80. 
 
 8. What is the value of 4 shillings, expressed in the 
 decimal of a £ ? As 1 shilling is j& of a £, 4 s. is 
 sV. We can change 4 to tenths by adding a cipher ; 
 it will then be 40. 4, however, was not the number 
 we wished to reduce to tenths, but / ff 5 the answer, 
 40, is therefore 20 times too great ; dividing by 20, it 
 stands .2. 4 shillings, then, is 2 tenths of a £. 
 
 9. Now, reverse the operation. What is the value, 
 in shillings, of .2 of a £ ? Now, shillings are twen- 
 tieths. We can change any number to twentieths by 
 multiplying it by 20; as, 1 is 20 twentieths, 2 is 40 
 twentieths, &c. Multiplying the .2 by 20, we have 
 40; but observe the 2 was not two wholes, but two 
 tenths; the answer, 40, therefore, is ten times too 
 great. Dividing by 10, the answer is 4 shillings. 
 
 10. Reduce to the decimal of a £, 2 shillings. 5 
 shillings. 
 
 11. What is the value, in shillings, of .1 of a £ ? 
 Of .25 of a £ ? 
 
 12. Reduce to the decimal of a shilling, 3 pence. 3 
 pence are f? of a shilling. Reducing to hundredths, 
 to render the division complete, the answer is .25. 
 
 13. What is the value, in pence, of .25 of a shilling ? 
 
 14. Reduce 9 pence to the decimal of a shilling. 
 
 15. Reduce 1 peck to the decimal of a bushel. 
 
 16. Reduce 3 pecks to the decimal of a bushel. 
 
INTEREST. 93 
 
 17. Reduce .5 of a bushel to pecks. .75 of a bu 
 to pecks. 
 
 18. Reduce 15 minutes to the decimal of an hour. 
 
 19. Reduce 45 minutes to#the decimal of an hour. 
 
 20. Reduce to minutes .5 of an hour. .25 of an 
 hour. .75 of an hour. 
 
 21. Reduce 6 inches to the decimal of a foot. 
 9 inches to the decimal of a foot. 3 inches to the 
 decimal of a foot. 
 
 SECTION XIV. 
 
 INTEREST. 
 
 Interest is the sum paid by the borrower to the 
 lender for the use of money. The rate of interest 
 is established by law, and varies in different countries. 
 In England, it is 5 per cent., that is, 5 for the use of 
 100 for 1 year ; in the New England States, it is 6 
 per cent. ; in New York, it is 7 per cent. When no 
 particular rate is mentioned in this book, 6 per cent, 
 will be understood. 
 
 If I borrow 100 dollars for 1 year, at the end of the 
 year I owe the sum I borrowed, 100 dollars, and 6 
 dollars for the use of it, making 106 dollars. The 
 sum borrowed is the principal; the sum paid, for the 
 use of it is the interest; the principal and interest 
 added together make the amount. 
 
 1. What is the interest of 100 dols. for 2 years? 3 
 years? 4 years? 5 years? 6 years? 7 years? 
 
 2. What is the interest of 200 dols. for 2 years ? 3 
 years ? 4 years ? 5 years ? 6. years ? 
 
 3. What is the interest of 300 dols. for 2 years? 
 For 4 years ? Of 400 dols. for 3 years ? 
 
94 MENTAL ARITHMETIC. 
 
 4. What is the interest of 50 dols. for 1 year? For 
 
 3 years ? Of 25 dols. for 1 year ? 2 years ? 
 
 5. What is the interest of 100 dols. for 1 year? 
 What is the interest of^J.00 cents for 1 year? 
 What is the interest of 2 dols. for 1 year? Of 3 
 
 dols. ? Of 4 dols. ? 5 dols. ? 6 dols. ? 7 dols. ? 8 
 dols. ? 9 dols. ? 
 
 6. What is the interest of 36 dols. for 1 year? Of 
 47 dols. ? Of 57 dols. ? Of 34 dols. ? Of 62 dols. ? 
 Of 89 dols. ? Of 125 dols. ? Of 136 dols. ? Of 207 
 dols.? Of 561 dots.? 
 
 7. What is the interest of 50 cents for 1 year? Of 
 25 cents? Of 10 cents? Of 20 cents? Of 30 cents? 
 Of 40 cents? Of 50 cents? Of 70 cents? Of 80 
 cents? Of 90 cents? 
 
 8. What is the interest of 50 dols. 60 cents for 1 
 year? Of 84.30? Of 96.40? Of 112.25? Of 230.75? 
 
 9. What is the interest of 100 dols. for 6 months? 
 For 3 months? For 2 months? For 1 month? For 
 
 4 months? For 5 months? For 7 months? For 8 
 months? For 9 months? For 10 months? For 11 
 months ? 
 
 10. What is the interest of 10 dols. for 6 mo. ? 3 
 mo.? 2 mo.? 1 mo.? 4 mo.? 5 mo.? 7 mo.? 8 
 mo.? 9 mo.? 10 mo.? 11 mo.? 
 
 11. What is the interest of 1 dol. for 6 mo. ? 1 mo.? 
 The interest of 1 dollar for 1 month is half a cent, 
 
 and for any number of months, it is half as many 
 cents. 
 
 12. What is the interest of 1 dollar for 5 mo. ? 7 
 mo.? 8 mo.? 9 mo.? 11 mo.? 12 mo.? 15 mo.? 
 16 mo.? 17 mo.? 18 mo.? 
 
 The interest of any number of dollars for 1 month 
 is half as many cents. 
 
 13. What is the interest o( 12 dollars for 1 mo. ? 
 Of 15 dols.? 25 dols.? 37 dols.? 42 dols.? 67 dols.? 
 93 dols. ? 104 dols. ? 
 
 
INTEREST. 95 
 
 14. What is the interest of 12 dols. for 3 months? 
 What is the interest of 25 dols. for 6 months? 
 
 In computing interest, a month is reckoned 30 days. 
 As the interest on a dollar for 30 days is half a cent, 
 that is, 5 mills, the interest on a dollar for 1 fifth of 30 
 days will be 1 mill. One fifth of 30 is 6; the in- 
 terest, therefore, on 1 dollar for 6 days is 1 mill ; and 
 the interest on any number of dollars for 6 days will 
 be as many mills as there are dollars. 
 
 15. What is the interest of 15 dollars' for 6 days? 
 Of 25 dols. ? Of 40 dols. ? Of 65 dols. ? Of 75 dols. ? 
 Of 100 dols. ? Of 500 dols. ? Of 360 dols. ? Of 840 
 dols.? Of 1000 dols.? 
 
 As the interest of 1 dollar for 6 days is 1 mill, for 
 12 days it will be 2 mills, for 18 days 3 mills, &c. 
 
 16. What is the interest of 1 dol. for 24 days? Of 
 2 dols. for 6 days? Of 2 dols. for 12 days? Of 2 dols. 
 for 18 -days? Of 5 dols. for 6 days? For 12 days? 
 For 24 days? Of 36 dols. for 18 days? 
 
 17. What is the interest of 125 dols. for 1 year and 
 
 6 mo.? 
 
 18. What is the interest of 268 dols. for 1 year? For 
 2 years? For 3 years? 
 
 19. What is the interest of 45 dols. for 4 years 
 
 7 mo.? 
 
 20. What is the interest of 60 dols. for 1 year 3 
 mo. 18 days? 
 
 21. What is the interest of 100 dols. for 2 years 1 
 mo. 12 days? 
 
 22. What is the interest of 165 dols. for 3 years 2 
 mo. 6 days? 
 
 23. What is the interest of 50 dols. for one month 1 
 For 6 months ? For 1 vear and 7 months ? 
 
96 
 
 MENTAL ARITHMETIC. 
 
 24. What is the interest of 94 dols. for eight mo. 24 
 days? 
 
 25. What is the interest of 320 dols. for 8 mo. and 12 
 days ? 
 
 26. What is the interest of 84 dols. for 4 mo. and 15 
 days? 
 
 27. What is the interest of 196 dols. for 10 mo.? 
 
 28. What is the interest of 86 dols. for 9 days? 
 
 29. What is the interest of 340 dols. for 15 days? 
 
 30. What is the interest of 875 dols. for 22 days? 
 
 When interest is more or less than 6 per cent., first 
 find the interest at 6 per cent., and then make a pro- 
 portional addition or subtraction for the required per 
 cent. If it is 7 per cent., add one sixth; if 5 per 
 cent., subtract one sixth. 
 
 31. What is the interest of 140 dols. for 1 year, at 
 7 per cent. ? 
 
 32. What is the interest of 200 dols. for 1 year and 
 6 mo., at 5 per cent.? 
 
 33. What is the interest of 460 dols. for 1 year, at 
 A} per cent. ? 
 
 Remark. — 4£ is three fourths of 6. 
 
 34. What is the interest of 500 dols. for 1 mo., at 9 
 per cent.? 
 
 BANKING. 
 
 When money is obtained at a bank, the note which 
 is given for it promises to pay it at a certain time, as 
 60, 90, or 120 days. The interest on this note, in- 
 stead of being paid at the end of the time, when the 
 note is taken up, is paid beforehand; that is, it is 
 subtracted from the sum named in the note; so that, 
 
BANKING. 97 
 
 when you take up the note, you have only to pay the 
 face of it, as the interest has been paid already. 
 
 If you give a note to a bank for 100 dollars, to be 
 paid in 90 days, they subtract from the sum named in 
 the note the interest of the sum for 90 days, and three 
 days besides, called days of grace; the balance is the 
 sum you receive. The interest of 100 dollars for 90 
 days is $1.50; for 3 days, it is 5 cents. $1.55 sub- 
 tracted from $100.00 leaves a balance of $98.45, 
 which is the sum you will receive. 
 
 If the note is given for 60 days, the interest is cast 
 for 63 days, and subtracted from the sum named. 
 
 The interest, thus subtracted, is called the bank 
 discount; and the bank, when it lends money on such 
 a note, is said to discount the note. 
 
 35. What is the bank discount on a note of 100 
 dollars, payable in 30 days? And how much will be 
 received on such a note ? 
 
 The interest on 100 dollars for 30 days is 50 cents; 
 for 3 days, it is 5 cents; the discount. 55 cents, sub- 
 tracted from 100 dollars, leaves $99.45, the sum re- 
 ceived. 
 
 36. What is the bank discount on a note for 200 
 dollars for 60 days? And what is the cash value of 
 the note? 
 
 37. What is the bank discount, and what is the 
 cash value of a note for 150 dollars payable in 30 
 days? 
 
 38. What is the bank discount, and what is the 
 cash value of a note for 200 dollars payable in 90 
 days? 
 
 39. What is the bank discount, and what is the 
 cash value of a note for 300 dollars payable in 90 
 days? 
 
 9 G 
 
98 MENTAL ARITHMETIC. 
 
 DISCOUNT. 
 
 When money is paid by the debtor before it be- 
 comes due, an allowance is made, which is called 
 f discount. If I owe 100 dollars, to be paid in three 
 months from this time, and I pay it now, I ought not 
 to pay the full hundred dollars, for I am entitled to the 
 use of the money three months longer. The sum which 
 should be paid now, to cancel a debt due at some 
 future time, is called the present ivorth of the debt. 
 
 To find the present worth of a debt due at some 
 future time, first find the interest on the debt from the 
 time of payment to the time when the debt is due; 
 subtract this interest from the debt, and the remainder 
 will be the present worth. Thus, if I pay a debt of 
 100 dollars three months before it is due, I subtract 
 the interest of 100 dollars for three months (= $1.50) 
 from 100 dollars, leaving $98.50 for the sum which I 
 must pay. 
 
 This rule is not strictly equitable, because $98.50, 
 with three months' interest added, will not amount to 
 $ 100. The above method, therefore, gives the present 
 worth a little too small ; but it is the method uniformly 
 adopted in business, and the error is on the right side, 
 for it encourages the debtor to be prompt in his pay- 
 ments. 
 
 40. What is the present worth of 200 dollars paya- 
 ble in 1 year? 
 
 41. What is the present worth of 150 dollars paya- 
 ble in 2 years? 
 
 42. What is the present worth of 60 dollars paya- 
 ble in 6 months? 
 
 43. What is the present worth of 530 dollars paya- 
 ble in 1 year? 
 
 What is the present worth of 400 dollars payable in 
 1 year and 6 months? 
 
LOSS AND GAIN. 99 
 
 LOSS AND GAIN.— PER CENTAGE. 
 
 44. A boy bought a penknife for 25 cents, and sold 
 it for 28 cents. How many cents did he gain on a 
 quarter of a dollar? 
 
 45. Suppose he had bought 4 knives at the same 
 price each, and sold them at the same profit, he would 
 then have traded with a dollar. How much would he 
 have gained on a dollar? 
 
 This is called so much per ce?it., which only means 
 so much on a hundred. 
 
 46. A boy bought a bushel of apples for 50 cents, 
 and sold them for 59 cents. How much did he gain 
 per cent. ? 
 
 47. A bookseller bought a book for 75 cents, and 
 sold it for 84 cents. How much did he make per 
 cent. ? 
 
 As 75 is £ of 100, what he gained on the book will 
 be £ of what he would gain on a hundred, or what he 
 would gain per cent. 
 
 48. A boy bought some melons for 40 cents, and 
 sold them for 60 cents. What did he make per cent. ? 
 
 Ans. His gain was equal to half his outlay. 
 
 49. A grocer bought a lot of flour for 5 dollars a 
 barrel ; but, finding it damaged, he sold it for 4 dollar? 
 a barrel. What did he lose per cent. ? 
 
 50. A man bought a share in a bank for 80 dollars ; 
 and sold it for 82 dollars. What did he gain per cent. ? 
 
 51. A man bought a lot of apples for $1.50 a bar- 
 rel. What must he sell them for to gain 10 per cent. ? 
 
 52. A hatter bought some hats for $3.50 each. He 
 is willing to sell them at a profit of 4 per cent. At 
 what price will he sell them ? 
 
 53. A manufacturing company declare a dividend of 
 7i per cent. What ought a stockholder to receive 
 who owns 350 dollars in that factory? 
 
100 MENTAL ARITHMETIC. 
 
 54. A has a note against B for 140 dollars, which 
 he sells for cash at 4 per cent, discount. What does 
 he receive for the note? 
 
 55. A merchant buys 100 barrels of flour for 5 
 dollars a barrel, and sells it so as to lose 5 per cent. 
 What does he sell it for a barrel ? 
 
 He afterwards buys 250 casks of lime at 1 dollar a 
 cask. He wishes to sell it so as to make good his 
 loss on the flour. At what per cent, profit must he sell 
 it, and for how much a cask ? 
 
 You observe that the money invested in lime is 
 only one half as much as was invested in flour. 
 
 56. A lends B 10 dollars for 2 months, without in- 
 terest. Afterwards B lends A 5 dollars. How long 
 can A keep it to balance the favor he did to B ? 
 
 57. O lends D 100 dollars, without interest, for 4 
 months. Afterwards D lends C 25 dollars. How 
 long can C keep it to balance the favor? 
 
 In these cases, you will see that the money, mul- 
 tiplied by the time it was kept, must, in the two cases, 
 be equal. If 10 dollars is lent me by A, without in- 
 terest, for 6 months, I can balance the favor by lend- 
 ing A 5 dollars for 12 months, or 4 dollars for 15 
 months, or 15 dollars for 4 months, or 30 dollars for 2 
 months, or 2 dollars for 30 months, or 20 dollars for 3 
 months. 
 
 58. A lends B 60 dollars for three months, without 
 requiring interest. Afterwards B lends A 90 dollars. 
 How long may A keep the money to balance the 
 favor ? 
 
 59. A lends B 40 dollars for three months. After- 
 wards B lends to A, for two months, a certain sum, 
 the use of which should balance the favor. How 
 large must the sum be? 
 
 60. A lends B 150 dollars for 4 months. B after- 
 wards lends A 100 dollars. How long can A keep it. 
 to balance the favor ? 
 
SQUARE MEASURE. 
 
 ioi 
 
 SECTION XV. 
 
 SQUARE MEASURE. 
 
 Linear measure is measure in a straight line, having 
 length only. Square measure is the measure of sur- 
 face, having length and breadth. 
 
 Thus, a linear inch, ■ 
 
 A square inch. 
 
 *A line of 2 inches, when square, will therefore 
 make 4 square inches j thus, 
 
 
 1. A line of three inches, 
 how many square inches ? 
 
 when square, will make 
 
 Note 6. 
 
 9* 
 
iu% 
 
 MENTAL ARITHMETIC. 
 
 square 
 Of 7? 
 
 2.; The square, of .4 inches is how many- 
 inches ? The square of 5 inches ? Of 6 ? 
 Of 8? Of 9? Of 10? Of 11? Of 12? 
 
 3. How many square inches are there in a square 
 foot? 
 
 4. How many linear feet are there in a linear yard ? 
 How many square feet in a square yard ? 
 
 5. How many square inches are there in a piece of 
 board 12 inches long and 3 inches wide ? 
 
 6. How many square inches are there in a piece of 
 board 8 inches long and 6 inches wide ? In a board 5 
 inches long and 3 inches wide ? In a board 9 inches 
 iong and 5 inches wide ? 
 
 7. How many square feet are there in the floor of a 
 room 12 feet long and 10 feet wide ? 
 
 8. How many square feet are there in the floor of 
 an entry 15 feet long and 4 feet wide ? 
 
 9. How many square yards of carpeting will cover 
 a room 6 yards long and 5 yards wide ? 
 
 How many square rods are there in a piece of land 
 14 rods long and 8 rods wide ? 
 
 10. If a road 4 rods wide passes through my land 
 for the distance of 60 rods, how many square rods of 
 my land does it occupy ? 
 
 We will now return to the measure of an inch. If, 
 instead of squaring a linear inch, 
 we take only half an inch and 
 square it, we shall have but one 
 fourth of a square inch ; thus, 
 
 So if we square one third of an 
 inch, it will give us $ of a square 
 inch. If we square one fourth of 
 an inch, it will give us T V of a square inch. 
 
 11. What part of a square inch will £ of an inch, 
 when squared, be ? What part of a square inch will 
 £ of an inch be when squared ? } ? i ? £ ? T V ? tV ? 
 
 A* 
 
SQUARE MEASURE. 103 
 
 12. What part of a square inch is a piece of paper 
 1 inch long and half an inch wide ? One inch long 
 and three fourths of an inch wide ? 
 
 13. What part of a square foot is a board 1 foot long 
 and half a foot wide ? One foot long and 9 inches 
 wide ? 
 
 14. How many square inches will there be in the 
 square of a line 1£ inches long ? 
 
 [This and the following questions may be answered 
 by drawing the figure on a slate or on a board.] 
 
 How many square inches will there be in the square 
 of a line 2^- inches long ? 3£ ? 4£ ? H ? 6^ ? 7£ ? 8£? 
 9^? 1(H? 
 
 15. How many square feet in £ a yard squared ? 
 
 16. How many square inches are there in 1 square 
 foot? 
 
 17. How many square feet in 1 square yard ? 
 
 18. How many square yards in 1 square rod ? 
 
 19. How many square feet in 1 square rod ? 
 
 40 sq. rods make 1 rood ; 4 roods make 1 acre. 
 
 20. How many rods make 1 acre ? 
 
 21. If a piece of board is 6 inches wide, how long 
 must it be to contain a square foot ? 
 
 22. If a piece of board is 3 inches wide, how long 
 must it be to contain a square foot ? 
 
 23. How long must it be to contain a square foot, 
 if it is 2 inches wide ? If 1 inch wide ? If 4 inches 
 wide ? 
 
 24. If cloth is £ a yard wide, how much in length 
 will make a square yard ? 
 
 25. How much lining } of a yard wide will line 
 one yard of cloth one yard wide ? 
 
 26. If cloth is f of a yard wide, how much in 
 length will it take for a square yard ? 
 
104 MENTAL ARITHMETIC. 
 
 27. How much cloth £ a yard wide will it take to 
 line 7 yards of cloth £■ of a yard wide ? 
 
 How much -| wide will line 1} yards f wide ? 
 
 28. How long must a strip of land 1 rod wide be 
 to contain an acre ? How long, if 2 rods wide ? If 
 3 rods wide ? If 4 rods wide ? If 8 rods wide ? If 
 10 rods wide ? 
 
 29. How long must a piece of land be to contain £ 
 of an acre, if it is 4 rods wide ? 
 
 30. If a piece of land is 10 rods in length, how 
 wide must it be to contain £ an acre ? 
 
 31. A man has an acre of land 16 rods in length. 
 How wide is it ? 
 
 32. How many steps must the owner take to walk 
 round it, if he take 5 steps to a rod ? 
 
 33. A man has an acre of land 8 rods wide. How 
 long is it I How many rods of fence will it take to 
 fence it ? 
 
 34. If a road 4 rods wide is laid out through my 
 land, how much of the road will it take in length to 
 make an acre ? How many acres will there be in one 
 mile of the road ? 
 
 35. If it passes through my land for half a mile, 
 and I am paid at the rate of 30 dollars an acre for the 
 land occupied by the road, what will be the amount o, 
 damages due me ? 
 
 36. If land in the city is worth 45 cents a square 
 foot, what will be the cost of a building-lot 30 feet 
 front and 60 feet from front to rear ? 
 
 37. There are two pieces of land ; one of them 12 
 rods square, the other 13. Which is nearest an acre ? 
 
 38. There is a piece of land 12£ rods square. How 
 much does it fall short of an acre ? 
 
 39. A painter tells me it will cost 20 cents a square 
 yard to paint the floor of a room in my house. Sup- 
 posing the room is 5 yards wide and 6J- yards long, , 
 what will the painting of it come to? 
 
SQUARE MEASURE. 105 
 
 40. What will the painting of an entry cost, at the 
 same rate, that is l£ yards wide and 7 yards in length ? 
 
 41. A stonecutter agrees to lay a hammered stone 
 door-step for 50 cents for every square foot of ham- 
 mered surface. The stone is 5 feet long, 3| feet wide, 
 and 9 inches thick. What is the surface of the top, 
 the two ends, and the front edge, added together? 
 What will be the cost of the stone ? 
 
 42. How many men could stand on | of a mile 
 square, allowing each man 1 square yard to stand upon ? 
 
 There are various ways of finding the answer to 
 the above questions. To encourage the student's in- 
 vention, some of them will be here suggested. 
 
 First Method. — As there are 30 \ square yards in one 
 square rod, multiply 80 (the number of rods in one 
 fourth of a mile) by itself, and this product by 30£. 
 80X80 = 6400; 6400X30*= 180,000 + 12,000+ 1600 
 = 193,600, answer. 
 
 Second Method. — Multiply 80 by 5J, which will 
 give the number of men in one line one fourth of a 
 mile long. Multiply this product by itself. 80X5^ = 
 440 ; 440 2 = 160,000 + 32,000 + 1600 = 193,600, an- 
 swer. 
 
 There are still other ways of solving the question, 
 which the student may discover for himself. 
 
 Note. — For those students who have not time to go 
 through the book, the course of instruction in Mental 
 Arithmetic may properly close at this place. With a 
 similar view, the Second Part may be divided at the 
 close of Section XXXVI. The ground thus gone 
 over tvill be found to embrace all the principles and 
 practice needed in the transactions of ordinary busi- 
 ness. Those iv hose opportunities permit should have 
 fhe advantage of the higher discipline furnished in the 
 remainder of the book. 
 
106 MENTAL ARITHMETIC. 
 
 SECTION XVI. 
 
 CONSTRUCTION OF THE SQUARE. 
 
 If I place three dots in a row, and place three such 
 rows side by side, this will represent to the eye the 
 square of the number 3. 
 
 In the same way you may repre- 
 
 Thus, • • • sent the square of 4, 5, or any num- 
 
 © © © ber whatever. I will now ask your 
 
 © © © attention to the square of 4. We 
 
 may make it by making a row of 4 
 
 dots, and placing 4 such rows side by side. But there 
 
 is another way of coming at the square of 4. We 
 
 will take the square of 3, as shown above, and see 
 
 what additions we must make to it, in order to make 
 
 it the square of 4. You observe that it must be 
 
 wider by one row, and longer by one row, than it 
 
 is now. We will then add a row above the others, 
 
 and also a row on the right hand. 
 
 I have made the additions by 
 stars to distinguish them from 
 the dots. You now see there is 
 something wanting to complete 
 the square, — a single star in the 
 corner. 
 
 Thus, 
 
 * * * 
 
 • • • * 
 
 
 • • • * 
 
 
 © © © * 
 
 Thus, 
 
 * * * * 
 
 • • • * 
 
 
 • • • * 
 
 
 • • • * 
 
 You observe, therefore, that 
 you obtain the square of 4 by 
 adding to the square of 3 twice 
 3 plus 1. We will now take the 
 square of 4, and by additions to 
 it obtain the square of 5. Adding a row of 4 at the 
 top, and a row of 4 at the right hand, there will be 
 one wanting at the corner to complete the square.' 
 
CONSTRUCTION OF THE SOJJARE. 107 
 
 Adding this, which makes twice 4+1, we have the 
 square complete. If, therefore, we have the square of 
 any number, we can find the square of a number one 
 greater by adding twice the first number plus 1. 
 
 The square of 5 is 25. What must you add to this 
 square to make the square of 6? 
 
 What must you add to the square of 6 to make the 
 square of 7 ? What must you add to the square of 7 
 to make the square of 8 ? 
 
 What must you add to the square of 9 to make the 
 square of 10? 
 
 The square of 15 is 225. What is the square of 16, 
 by the above method ? 
 
 The square of 20 is 400. What is the square of 21 ? 
 
 The square of 30 is 900. What is the square of 31 ? 
 
 The square of 40 is 1600. What is the square of 41 ? 
 
 The square of 50 is 2500. What is the square of 51? 
 
 What is the square of 60 ? Of 61 ? Of 70 ? Of 71 ? 
 Of80? Of 81? Of90? Of91? 
 
 We will now return to the square of 3 ; and I ask 
 your close attention once more. Supposing we have 
 the square of 3 before us, and we wish to make such 
 additions to it as shall make the square of 5. As 
 5 is 2 greater than 3, we must add 2 rows instead 
 of 1. If we add 2 rows of 3 at the top, and 2 
 rows of 3 at the right hand, the figure will stand 
 thus, 
 
 * * * O O Here you see there are four 
 
 * * * O O stars wanting to complete the 
 
 square. I have marked their 
 
 * 9 © * * places by the circle, O. If you 
 
 * • • * * suppose these four to be added, 
 
 the square will be complete, 
 
 * ® * and will be the square of 5. 
 , The question is, now, What has been added to the 
 
 square of 3 in order to make the square of 5? You 
 
108 MENTAL ARITHMETIC. 
 
 observe there are added 6 stars, or two rows of 3 at 
 the top, 6 on the right hand, and 4 in the corner, to 
 make the square of 5. But we can express this in a 
 different way. We may consider 5 as consisting of 
 two parts, 3 and 2 added together. We will call 3 the 
 first part, and 2 the second part, of 5. Now, by the 
 figure, you perceive that the square of 5 is made up, 
 first, of the square of the first part, that is, the nine 
 dots ; then the stars at the top are the product of the 
 first part multiplied by the second; and adding to 
 these the stars on the right hand, we have twice the 
 product of the first part into the second; and, last, 
 we have, in the corner, the square of the second 
 part. 
 
 To state it briefly once more : Regarding 5 as 
 made up of the two parts, 3 and 2, the square of 5, 
 we find, is equal to the square of the first part + twice 
 the product of the two parts + the square of the second 
 part. 
 
 This is called expressing the amount of a square in 
 the terms of its parts. 
 
 Examine and answer the following questions: — 
 
 1. If we regard the number 6 as made up of two 
 parts, 4 and 2, how will you express the square of 6 
 in the terms of its parts? 
 
 2. Regard the number 7 as consisting of two parts, 
 5 and 2. What is the square of 7 in the terms of its 
 parts ? 
 
 You can draw the figure for yourself, and see the 
 application of the principle in the above cases. 
 
 It is of no consequence in what way the number is 
 divided. The operation will bring out the exact 
 square of the whole number in all cases. To show 
 this, we will take the number 10, the square of which 
 is 100. We will first divide 10 into the parts 7 and 
 3; then, by the formula given above, the sq. of 7 + 
 twice the product of 7 into 3 + sq. of 3, will be the* 
 
THE SQUARE OF FRACTIONAL NUMBERS. 109 
 
 sq. of the whole number. The sq. of 7 = 49; twice 
 7X3=42; the sq. of 3 = 9; 49 + 42 + 9=100. 
 
 We will now divide 10 into the parts 6 and 4, pro- 
 ceeding as above. We find 36 +48 + 16 = 100. 
 
 Again; we will divide 10 into the equal parts 5 
 and 5. 25 + 50 + 25=100. 
 
 Finally; divide 10 into the parts 8 and 2. 64 + 32 
 + 4=100. 
 
 We will now apply the above method to the pur- 
 pose of finding some squares of larger numbers. 
 
 3. What is the sq. of 25? Dividing into 20 + 5; 
 Ans. 400 + 200 + 25 = 625. 
 
 4. What is the sq. of 35, or 30 + 5? Ans. 900 + 
 300 + 25=1225. 
 
 5. What is the sq. of 46, or 40 + 6? Ans. 1600 + 
 480 + 36 = 2116. 
 
 6. What is the sq. of 55} Of 64? Of 75} Of 83? 
 Of 92? 
 
 7. What is the sq. of 125? Divide into 100 + 25. 
 100 sq. = 10,000; twice 100X25=5000; 25sq. = 625. 
 Ans. 15,625. 
 
 8. What is the square of 150? Of 230? Of 510? 
 
 The same formula will embrace the examples men- 
 tioned in the first part of this section, when the second 
 part of the number is 1. For example, 
 
 9. What is the sq. of 5, or 4 + 1 ? Here twice the 
 product of the two parts is merely twice the first part, 
 inasmuch as multiplying by 1 does not increase the 
 number; and the sq. of 1 is only 1. The answer, 
 therefore, by the formula, is 16 + 8 + 1 = 25. 
 
 10. This method will apply to the squaring a whole 
 number and a fraction, as follows : What is the sq. of 
 l+£? Ans. l + l+£ = 2£; for twice the product of 
 £ into 1 is 1, and the sq. of £ is £. 
 
 To test the correctness of this answer, we will per- 
 form the operation another way. 14- = £ ; g sq. = £ = 2£. 
 
 11. What is the sq. of 2} ? 3f?" 4£? 5£? 6£? 7 p. 
 
 10 
 
110 MENTAL ARITHMETIC. 
 
 The answer, in each of these cases, may be tested 
 by changing the mixed number to an improper frac- 
 tion ; as, 2f = £, &c. 
 
 We may in the same way square the sum ot two 
 fractions. 
 
 12. What is the sq. of £ + £? Ans. i + i + i=l. 
 Now, |-|-^z=l, the square of which is 1. 
 
 13. What is the sq. of £ + £ ? Ans. T %^ T \ + T V = 
 \% or 1. Now, f -f- j-= 1, the sq. of which is 1. 
 
 14. What is the sq. of f + £? The sum of these is 
 2, the sq. of which is 4 ; the answer, therefore, should 
 be 4. Applying the formula, the operation is as fol- 
 lows; £ + £ + £ = .^.= 4. 
 
 PRACTICAL QUESTIONS. 
 
 We will now apply the above principle to the solu- 
 tion of some questions which appear at first a little 
 difficult. 
 
 1. A boy had some apples. He placed part of them 
 in rows making a square, and found he had 6 apples 
 left. He placed another row on two sides, and found 
 he had enough to complete the square except one 
 apple at the corner. How many apples were there in 
 the first square ? How many apples had he ? 
 
 2. Three boys were playing at marbles. The first 
 says, " I have just marbles enough to make a square ; " 
 and he placed them in rows on the floor, forming a 
 square. The second boy says, " I have twelve mar- 
 bles, and I will put a row on two sides of yours, and 
 make your square larger;" but, on placing his marbles, 
 he found he wanted 3 more to complete the square. 
 Then the third boy says, " I have just three, and that 
 will make the square complete." 
 
 How many had the first boy ? How large was the 
 square which all the marbles made ? 
 
 3. The boys of a school thought, one day, at 
 
COMPLETING OP THE DEFECTIVE SQUARE. Ill 
 
 recess, they would form themselves into a square. A 
 part of them first formed a square, when it was found 
 that there were 15 boys left. These 15 then placed 
 themselves in a row on two sides of the square, when 
 it was found that it required 2 boys more to complete 
 the square. How many boys were there in the first 
 square ? How many in all ? 
 
 4. A general, drawing up his soldiers in a square 
 body, with the same number in rank and file, found 
 he had 55 men over and above. He placed these in a 
 row on two sides, and found that he now wanted 30 
 men to complete the square. How many men were 
 there on a side of the first square ? How many men 
 in the first square ? How many men had he in all ? 
 
 5. There is a certain square number expressed in 
 the terms of its parts ; that is, it is expressed in three 
 terms, the first of which is the square of the first part, 
 the second is twice the product of the two parts, and 
 the third is the square of the second part. Now, the 
 first two terms are 16+24. What is th*e third term? 
 What is the number ? 
 
 6. There is a square number expressed in the terms 
 of its parts. The first two terms are 9 + 24. What 
 is the third ? 
 
 7. The first and last terms of a square are 4 + 25. 
 What must be the middle term ? What is the number ? 
 
 8. The first and last terms of a square are 9 + 4. 
 What is the second term ? What is the square ? 
 
 9. Complete the square whose first two terms are 
 16 + 40 + D. 
 
 10. Complete the sq. 36 +24+ a. 
 
 11. Complete the sq. 4+4+ □. 
 
 12. Complete the sq. 9 + 6 + a. 
 
 13. Complete the sq. 16 + 8 + d 
 
 14. Complete the sq. 25 + 10 + Q 
 
 15. Complete the sq. 25 + 20 + a. 
 
 16. Complete the sq. 36 + 72 + a- 
 
1 12 MENTAL ARITHMETIC. 
 
 17. Complete the sq. 25 + 30 + L3 
 
 .18. Complete the sq. 25 + 40 + □• 
 
 The square root is the number which, multiplied 
 into itself, produces the square. Thus 3 is the sq. root 
 of 9, 2 is the sq. root of 4, 5 is the sq. root of 25. 
 
 The square root of a square of three terms, like 
 those given above, is the square root of the first term, 
 plus the square root of the third ; for these, multiplied 
 by themselves, will produce the square. Thus the 
 square root of the square 9+12 + 4, is 3 + 2, or 5. 
 3 is the sq. root of 9, and 2 the sq. root of 4. This 
 number, 3 + 2, multiplied by itself, will produce the 
 square of 9+12 + 4. 
 
 19. Complete the sq. 36 + 60 + d. What is its 
 root? 
 
 20. Complete the sq. 36+12 + CZI. What is its 
 root? 
 
 21. Complete the sq. 16 + 4 + a. What is its root? 
 
 22. Complete the sq. 9 + 12 + d What is its root? 
 
 23. What is the sq. root of 169? 
 
 Divide the number into three terms, 100 + 60 + 9. 
 We divide it so, because 100 is a square, and 60 is 
 twice the product of 10, the root of the 1st term, into 
 the root of what this way of dividing leaves us for the 
 3d term. That is, if we take 60 for the 2d term, we 
 leave 9 for the 3d term, and this is as it should be, for 
 60 is twice the product of 10 into 3. 
 
 The root is therefore 10 + 3, or 13. 
 
 24. What is the sq. root of 196 ? 
 
 We will take for the 1st term 100, whose root is 
 10. Now, as the 2d term is twice the product of the, 
 two terms of the root, if we divide half of it by the 
 1st, it will give the 2d term of the root ; or, what is the 
 same thing, if we divide the 2d term of the square by 
 twice the 1st term of the root, it will give the 2d of 
 the root. Now, 96 contains the 2d and 3d terms of 
 the square. We must separate it into two parts, such 
 
COMPLETING OF THE DEFECTIVE SQUARE. 113 
 
 that the first part, divided by twice 10, or 20, will 
 give for quotient the root of the second part. 
 
 Let us first try by dividing it into 60 and 36. Now, 
 60 divided by 20 gives 3, which is not the root of 36. 
 . Our division, therefore, was wrong. The 2d term was 
 too small, and the 3d too great. We will try again. 
 By dividing it into 80 and 16, we find that 80 divided 
 by 20 gives 4, which is the root of 16. The number 
 196, when arranged in the three terms of the square, 
 will be 100 + 80+16, and the root is 10 + 4, or 14. 
 
 25. What is the root of 225 ? Here we must not 
 take for our 1st term 200, for this is not a square. 
 We must take the largest square whose root is an even 
 10. This is 100. We have remaining 125. This 
 we must divide into two terms, such that the 1st di- 
 vided by twice 10 will give the root of the 2d term. 
 We will first divide into 80 and 45. 80 divided by 
 20 gives 4, which is not the square root of 45. We 
 will divide into 100 and 25. 100 divided by 20 
 gives 5, which is the exact root of 25. The number 
 225, therefore, when arranged in the three terms of a 
 square, stands 100+100 + 25, and its square root is 
 10+5, or 15. 
 
 26. What is the square root of 256 ? Taking for the 
 1st term 100, it remains to divide the remainder, 156, 
 according to the principle stated above. Now, 120 
 will contain 20, 6 times, which is the root of the re- 
 mainder, 36. The number stands, therefore, 100 + 
 120 + 36; square root, 10 + 6, or 16. 
 
 27. What is the square root of 484? Here we. 
 take for the 1st term 400, for that is the largest square 
 whose root is in even tens; its root is 20. The re- 
 mainder we may divide into 80 and 4. Dividing 80 
 by twice 20, or 40, we have for the quotient 2, which 
 is the root of the 3d term. The square, therefore, is 
 ,400 + 80 + 4; the root, 20 + 2, or 22. 
 
 10* H 
 
114 MENTAL ARITHMETIC. 
 
 28. What is the square root of 529 ? Of 576 ? Of 
 625? Of 676? 
 
 29. The following is a ready method of squaring a 
 mixed number whose fraction is £; as, 2£, 3£, 4£. 
 The fraction will be £. For the other number, mul- 
 tiply the whole number by a number greater than 
 itself by 1. Thus the square of 2£; the fraction is 
 i, the whole number 2X3, or 6; 6£. The square of 
 3^ ; 3X4, or 12, and £. The sq. of 4^ is 4X5, or 20, 
 and {. What is the sq. of 5±1 6£? 7£ ? 8£? 9£? 
 10i? 
 
 The same principle will apply to the square of whole 
 numbers whose last figure is 5 ; as, 25, 45, 55, &c. ; 
 for such a number consists of a certain number of 
 tens, and half of 10. As the right-hand figure is 5, 
 the two right-hand figures of the square must be 25. 
 Then multiply the number at the left of the 5 by itself 
 increased by 1 ; and this, read at the left hand of the 
 25, will be the square. Thus, for the square of 25, 
 the two right-hand figures will be 25 ; for the rest, 
 multiply 2 by 3, which is 6. Ans. 625. 
 
 30. What is the square of 35? 3X4 = 12. To this 
 annex 25. 1225, Ans. 
 
 What is the square of 45? Of 551 Of 65? Of 
 751 Of 85? Of 95? 
 
 SECTION XVII. 
 
 PRACTICAL QUESTIONS IN SQUARE MEASURE. 
 
 1. How many square rods are there in a square 
 mile ? 
 
 2. How many acres are there in a square mile? . 
 
MENSURATION. 115 
 
 3. Divide a square mile into 4 equal farms. How 
 many acres would there be in each? 
 
 4. How many acres are there in one fourth of a 
 mile square ? 
 
 5. How many acres are there in a town 6 miles long 
 and 5 miles broad ? 
 
 6. If half of the town is unfit for improvement, in 
 consequence of water and mountains, how many farms 
 of 100 acres might be made from the other half? 
 
 7. A man bought a rectangular piece of land con- 
 taining 40 acres. On going out with his son to meas- 
 ure round it, to ascertain how much fence it would 
 require to enclose it, they found the first side they 
 measured to be 160 rods. " We need not measure 
 any more," said the son, " for I can tell all the rest in 
 my head." How wide was the piece ? and how many 
 rods of fence would it take to go round it ? 
 
 8. A man bought 7 acres of land, in rectangular 
 form. The width of it was 28 rods. What was its 
 length ? 
 
 If a four-sided piece of land is rectangular, its con- 
 tents may be found by multiplying two adjacent sides, 
 or sides that meet and form a corner. 
 
 Thus, if a piece is 12 rods long and 4 rods wide, 
 12 c the two boundaries, 12 and 4, 
 
 which meet and form the right 
 4 angle at c, will, when multiplied 
 together, give the contents in 
 square rods ; 12X4 = 48. 
 
 If the opposite sides are parallel, but the angles are 
 not right angles, the distance between the two sides 
 must be measured by a perpendicular line, thus : 
 
 the length, 16 rods, multi- 
 
 plied by the width, 4 rods, 
 will give the contents. 
 
 If the piece is a triangle, there must be a perpen- 
 
116 MENTAL ARITHMETIC. 
 
 •dicular line drawn to the longest side from the an- 
 gle opposite to it. This perpendicular we may call 
 the height of the triangle, and the longest side, its 
 length ; and the height multiplied into the length, 
 will give double the area ; dividing this by 2, we get 
 the area. 
 
 To show the reason of this, take the following 
 
 iigure. By examining this, you will see that there are 
 
 16 in it two triangles just alike. 
 
 /p =: =^ZI~ -J The length of each is 16 
 
 / t -— ^^/ rods, and the height 4 rods. 
 
 Now, 16X4 will give, as in 
 the case above, the area of the whole figure, that is, 
 ■of both the triangles ; therefore it will give twice the 
 area of one Of them. 
 
 9. What is the area of a triangle whose longest side 
 is 16 rods, and the perpendicular, from the opposite 
 angle to this side, 12 rods ? 
 
 10. What is the area of a triangle whose longest 
 side is 24 rods, and its height 9 rods ? 
 
 11. What is the area of a triangle whose longest side 
 is 18 rods, and height 16 rods ? 
 
 12. A triangle contains just one acre. Its longest 
 side is 20 rods. How long must the perpendicular be 
 from the opposite angle to that side ? 
 
 13. A triangle contains 2 acres. Its longest side is 
 32 rods. How long is the perpendicular, from the op- 
 posite angle to this side ? 
 
 In a right-angled triangle, the longest side is called 
 the hypotenuse ; the sides containing the right angle 
 are called the legs, or one the base and the other the 
 perpendicular. In all right-angled triangles, the square 
 of the hypotenuse is just equal to the sum of the 
 squares of the two other sides. This important 
 principle is exhibited to the eye in the. following 
 figure : — 
 
MENSURATION. 
 
 117 
 
 The hypotenuse is 
 divided into 5 equal 
 parts, and its square 
 is therefore 25. The 
 base has 4 equal parts 
 of the same length, 
 making a square of 
 16. The perpendic- 
 ular is divided into 
 3 equal parts, of the 
 same length as the 
 others, which makes 
 a square of 9. The 
 square of the perpen- 
 dicular and of the 
 base, added together, 
 16 + 9 = 25, which is the square of the hypotenuse. 
 
 If we know the square of the hypotenuse, we know 
 the sum of the squares of the two legs. If we know 
 the sum of the squares of the two legs, we know the 
 square of the hypotenuse. If we know the square of 
 the hypotenuse and of one leg, we can find the square 
 of the other leg. And if we know the square of any 
 one of these sides, we can obtain the length of the 
 side by extracting the square root. 
 
 14. In a certain right-angled triangle, the square of 
 the hypotenuse is 100 feet. What is the length of the 
 hypotenuse? — In the same triangle, the square of the 
 base is 64 feet. What is the length of the base ? 
 
 In the same triangle, what must be the square of 
 the perpendicular ? What is the length of the perpen- 
 dicular? 
 
 15. A and B set out from the same place. A trav- 
 els east 6 miles. B travels north till he is 10 miles in a 
 straight line from A. How far north has B travelled ? 
 > 16. There is a triangle, the perpendicular of which 
 is 3 feet, the hypotenuse is 5 feet. How long is the 
 base? 
 
118 
 
 MENTAL ARITHMETIC. 
 
 17. A man had a piece of land, in the form of a 
 right-angled triangle, the two legs of which were equal 
 to each other, and the square of the hypotenuse was 
 128 rods. How many rods were there in the piece ? 
 
 The circumference of a 
 circle is 3 times and \ greater 
 than the diameter. If the 
 diameter is 1 foot, the circum- 
 ference will be 3f feet ; if the 
 diameter is 2 feet, the circum- 
 ference will be 6f feet. 
 
 18. If the diameter of a circle is 3 feet, what will 
 be the circumference ? If the diameter is 4 feet ? If 
 5 feet ? If 6 feet ? If 7 feet ? 
 
 If the diameter of a water-wheel is 16 feet, what is 
 the circumference ? 
 
 19. If the diameter of the earth is 8000 miles, what 
 is the circumference ? 
 
 To find the area of a sector of a circle, as a, e, b, 
 multiply the arc by half the radius. This figure may 
 be regarded as a triangle, the base of which is the arc, 
 and the radius the height ; and you have seen before, 
 that, in a triangle, the base multiplied by half the 
 height gives the area. From this we may obtain a 
 method of obtaining the area of the whole circle. 
 
 Multiply the circumference by half the radius. For 
 we may regard the circle as made up of a great num- 
 ber of small triangles, whose bases added together are 
 the circumference of the circle, and whose height is 
 equal to radius, being, in each case, the distance from 
 the circumference to the centre. 
 
 20. What is the circumference of a circle whose t 
 diameter is 14 feet ? 
 
ANALYSIS OF PROBLEMS. 119 
 
 What is its area ? 
 
 21. What is the circumference of a circle whose 
 diameter is 12 feet. 
 
 What is its area ? 
 
 22. What is the circumference of a circle whose 
 diameter is 20 feet ? 
 
 What is its area ? 
 
 23. What is the circumference of a circle whose 
 diameter is 28 feet ? 
 
 What is the area? 
 
 SECTION XVIII. 
 
 ANALYSIS OF PROBLEMS. 
 
 1. A boy spent one half the money he had, and had 
 1 dollar left. How much had he at first ? 
 
 2. A boy spent one third of the money he had, and 
 had 1 dollar left. How much had he at first ? 
 
 Ans. If he lost one third, he had two thirds left. If 
 1 dollar was two thirds, half a dollar must be one 
 third, and f of a dollar the whole. He had 1 dollar 
 and a half. 
 
 3. A boy spent J of his money, and had 1 dollar 
 left. How much had he at first ? 
 
 Let this and the following answers be given in form 
 of a fraction, like the preceding answer. 
 
 4. A boy spent | of his money, and had 1 dollar 
 left. How much had he at first ? 
 
 5. A boy spent £ of his money, and had 1 dollar 
 left. How much had he at first ? 
 
 6. A boy spent } of his money, and had 1 dollar 
 left. How much had he at first ? 
 
 7. A boy spent £ of his money, and had 1 dollar 
 left. How much had he at first ? 
 
120 MENTAL ARITHMETIC. 
 
 8. A boy spent £ of his money, and had 1 dollar 
 left ? How much had he at first ? 
 
 9. A boy spent T V of his money, and had 1 dollar 
 left. How much had he at first? 
 
 10. A man carried some corn to mill. The miller 
 took T V of it for toll, and then there was just a bushel. 
 How much did the man carry to mill ? 
 
 11. A man carried some cloth to be fulled. It 
 shrank two sevenths in its length, and was then just a 
 yard long. How long was it at first ? 
 
 12. A man drew a prize in a lottery. ? V °f the prize 
 was retained, and then the drawer received just 100 
 dollars. How much was the prize ? 
 
 13. If a stick of timber shrink T \ in weight in sea- 
 soning, and then weigh 100 pounds, how much did it 
 weigh at first ? 
 
 14. A teamster sold f- of a cord of wood, and then 
 had half a cord left. How much had he at first ? 
 
 15. A man had an estate left him by his father. He 
 lost | of it. He then received 1000 dollars, and then 
 he had 3500 dollars. How much had he at first ? 
 
 16. A merchant began trade with a sum of money, 
 and gained so as to increase his original stock by I of 
 itself. He then lost 500 dollars, and had 4500 dollars 
 left. How much did he begin with? 
 
 17. A man set out on a journey, and spent half the 
 money he had for a dinner. He then paid half of what 
 he had left for provender for his horse ; then, half of 
 what now remained for toll in crossing a bridge ; and 
 had 10 cents left. How much had he at first ? 
 
 18. A boy spent § of his money for a book, and f of 
 it for some paper, and had 8 cents left. How much 
 had he at first ? 
 
 19. A boy, playing at marbles, lost, in the first game, 
 I of what he had ; in the second game, i of what he 
 then had ; in the third, -\ of what he then had ; in the 
 fourth, 11; and then he had 16 marbles left. How 
 many had he at first? 
 
ANALYSIS OF PROBLEMS. 121 
 
 20. A boy, playing at marbles, wins, in the first 
 game, so as to double the number of marbles he had ; 
 in the second game, he loses £ of what he then had ; 
 in the third game, he loses 5, and then finds he has 
 just as many as at first. How many had he at first ? 
 
 21. A man had his sheep in three pens. In the 
 first, there were 10 sheep ; in the second, there were 
 as many as in the first, and half the number in the 
 third ; in the third, there were as many as in the first 
 and second. How many had he in all ? 
 
 22. In an orchard, i of the trees are plum-trees; 
 there are 20 cherry-trees ; and the apple-trees, which 
 constitute the remainder, are half as many as the plum 
 and cherry-trees added together. How many trees 
 are there in the orchard ? 
 
 23. John and William were talking of their ages. 
 John says, "lam twelve years old." William says, 
 " If half my age were multiplied by one fourth of 
 yours, and half your age plus one subtracted from the 
 product, that would give my age." How old was he ? 
 
 24. A man, talking of the age of his two children, 
 said the youngest was three years old ; the age of the 
 eldest was £ his own age ; if his own age was divided 
 by that of his youngest, and once and one third the 
 age of the youngest subtracted from the quotient, that 
 would give the age of the eldest. How old was the 
 eldest ? 
 
 25. The number of pupils in a school is such that, 
 if you take half of them, and increase that by 2 ; then 
 take one third of this last number, and increase it by 
 3; and from this number subtract 6; the remainder 
 will be 7. How many are there in the school ? 
 
 26. A boy plays three games at marbles. In the first, 
 he loses a certain number ; in the second, he gains 8 ; 
 in the third, he loses 4 ; and then he finds he has 2 
 
 j more than he began with. How many did he lose in 
 the first game ? 
 11 
 
122 MENTAL ARITHMETIC. 
 
 27. A boy, playing at marbles, first lost one third 
 of what he had ; he then doubled his number, when 
 he had 5 marbles more than he had at first. How 
 many had he at first ? 
 
 28. There is a certain number, one third of which 
 exceeds one fourth of it by 2. What is the number ? 
 
 29. There is a certain number, one fourth of which 
 exceeds one fifth of it by 1. What is the number ? 
 
 30. There is a certain number, one third of which 
 added to one fifth of it amounts to 16. What is the 
 number ? 
 
 31. There is a number, one third, one fourth, and 
 one fifth of which, added together, are 94. What is 
 the number ? 
 
 32. What is that number, a fifth of which exceeds 
 a sixth of it by 4 ? 
 
 33. What number is that, of which a fourth part 
 exceeds a seventh part by 9 ? 
 
 34. In a certain orchard there are apple, peach, and 
 pear-trees. The apple-trees are 2 more than half the 
 whole. The peach-trees are one third of the whole, 
 and are 14 less than the apple-trees. The rest are 
 pear-trees. How many are there of each kind ? and 
 how many in all ? 
 
 SECTION XIX. 
 
 SOLID MEASURE. 
 
 Whatever has length, and breadth, and thickness, is 
 a solid. A block of wood 1 inch long, 1 inch high, 
 and 1 inch wide, is a solid inch. A block 1 foot long, 
 1 foot wide, and 1 foot high, is a solid foot. A block. 
 1 yard long, 1 yard wide, 1 yard high, is a solid yard. 
 
SOLID MEASURE. 123 
 
 1. How many solid inches are there in a block 3 in. 
 long, 2 in. wide, and 1 in. high ? 
 
 2. How many in a block 4 in. long, 3 in. wide, and 
 2 m. high ? 
 
 3. How many solid feet are there in a block 5 feet 
 long, 3 feet wide, and 2 feet high ? 
 
 4. How many solid feet in a block 7 feet long, 2 feet 
 wide, and 2 feet high ? 
 
 When a solid has its length, height, and breadth, 
 equal to each other, it is called a cube ; and the linear 
 measure of its length, height, or breadth, is called the 
 root of the cube. We have seen what is a cubic inch, 
 a cubic foot, and a cubic yard. 
 
 Suppose, now, we have a pile of cubic inch blocks, 
 and we wish to construct from them a cube, each of 
 whose dimensions shall be 2 inches. We will first 
 take 2 blocks, and place them down side by side. 
 This will be as long as the required figure, but it will 
 not be wide enough nor high enough. To make it 
 wide enough, we will place 2 more blocks down by 
 the side of the former. The figure now contains 4 
 cubic inches, and is 2 inches long and 2 inches wide, 
 but it is only 1 inch high. To make it 2 inches high, 
 we must place upon this another layer of 4 blocks, 
 arranged just like the former. The figure will then 
 be 2 in. long, 2 in. wide, and 2 in. high : it contains 
 8 cubic inches, and is the cube of 2. 
 
 5. How many blocks will you require, and how will 
 you arrange them, to make the cube of 3 ? 
 
 6. How many blocks will you require, and how will 
 you arrange them, to make the cube of 4 ? 
 
 7. How many blocks will you require, and how will 
 you arrange them, to make the cube of 5? 
 
 The cube, when expressed in numbers, is the same 
 as the 3d power of the root. It is found by taking the 
 root 3 times as a factor. Thus, the 3d power of 2 is 
 
124 
 
 MENTAL ARITHMETIC. 
 
 2X2X2 = 8. The 3d power of 3 is 3X3X3 = 27; 
 of 4, is 4X4X4 = 64; of 5, is 5X5X5=125. 
 
 In this way we may find the 3d power of any 
 number. 
 
 8. How many blocks, of a cubic foot each, will it 
 take to form a cubic solid 6 feet on a side ? 
 
 9. How many blocks, of a cubic foot each, will it 
 take to form a cubic solid of 7 feet each way ? 
 
 10. How many cubic feet will it take to form a 
 cube of 8 feet ? 
 
 11. How many cubic feet will it take to form a 
 cube of 9 feet ? 
 
 12. How many cubic feet will it take to form a 
 cube of 10 feet? 
 
 13. How many cubic inches are there in a cubic 
 foot? 
 
 14. A pile of wood 8 feet long, 4 feet high, and 4 
 feet wide, makes a cord. How many cubic feet are 
 there in a cord ? 
 
 15. A pile of wood 4 feet long, 4 feet high, and 1 
 foot thick, makes what is called a cord foot. How 
 many cubic feet are there in a cord foot ? 
 
 16. How many cord feet are there in a cord ? 
 
 17. There is a pile of wood 40 feet long, 4 feet 
 wide, and 5 feet high. How many cords does it 
 contain? 
 
 18. There is a stick of hewn timber 25 feet long, 
 1 foot wide, and 1 foot thick. How many cubic feet 
 does it contain ? 
 
 19. There is a tree, from the but-end of which a 
 stick may be hewn 13 feet long, 2 feet wide, and 2 
 feet thick. How many cubic feet will it contain ? 
 
 20. It is estimated that 50 feet of hewn timber 
 weigh a ton. If 50 cubic feet weigh 20 cwt. net 
 weight, what will 1 foot weigh ? 
 
 21. If you divide a cubic inch into blocks meas- t 
 
CONSTRUCTION OF THE CUBE. 125 
 
 uring i an inch each way, how many such will there 
 be in a cubic inch ? 
 
 22. How many cubic half inches are in a cubic 
 inch ? 
 
 23. If you divide a cubic inch into cubes of i of 
 an inch each, how many such will there be ? 
 
 24. How many cubic quarter inches are there in a 
 cubic inch? 
 
 25. How many cubic inches are there in a cube of 
 one inch and a half ? 
 
 26. If a man digs a cellar at the rate of f of a dol- 
 lar for a cubic yard, what will the job come to, if the 
 cellar is 18 feet long, 12 feet wide, and 6 feet deep? 
 
 27. A stone-layer agreed to build a solid wall 30 
 feet long, 44 feet thick, and 6 feet high, for 2£ dollars 
 a cubic yard. What did the wall cost ? 
 
 CONSTRUCTION OF THE CUBE. 
 
 We have seen that the third power, or cube, of any 
 number, is obtained by taking the number three times 
 as a factor. The product is the cube, or third power. 
 
 In this way the cube of any number whatever may 
 be obtained. There is another way, however, of con- 
 structing the cube, the knowledge of which is very im- 
 portant in the operation of extracting the cube root. 
 
 Suppose we wish to find the cube of 5. Instead of 
 taking 5 three times as a factor, thus, 5X5X5 = 125, 
 we will regard the number 5 as consisting of two 
 parts, 3 and 2. We will call 3 the first part, and 2 
 the second part, of 5. 
 
 We will begin by making the cube of the first part, 
 3, thus, 3X3X3 = 27. 
 
 We will regard this as a cube of 3 inches, (that is, 
 »3 in. long, 3 in. wide, and 3 in. high,) and represent 
 it bv the following figure : — 
 11* 
 
126 
 
 MENTAL ARITHMETIC. 
 
 The question now is, How shall 
 we enlarge this cube of 3, so as 
 to make it the cube of 5 ? It is 
 evident, it must be 2 in. longer, 2 
 in. broader, and 2 in. higher, than 
 it now is. We will begin, then, 
 by putting 2 layers of inch blocks 
 on the front side, 2 layers on the 
 right side, and 2 layers on the top. The figure, thus 
 enlarged, is not a cube. There are several places not 
 filled up. It is nearer the cube of 5 than it was 
 before ; but something more must be added. Before 
 making that addition, however, let us see what we 
 have done. The figure is the cube of 3, which is the 
 first part of 5. To this there are 3 equal additions 
 made. Each of these additions is 3 in. square, and 
 2 in. thick. Now, 3 is the first part of 5. Each ad- 
 dition, therefore, contains the square of the first part, 
 3, multiplied by the second part, 2 or 3 2 X2. There- 
 fore the three additions will be 3 times the square of 
 the first part multiplied by the second. 
 
 The whole figure, therefore, after these three addi- 
 tions are made, contains 3 3 + 3 times 3 2 X2. 
 
 We will now see what additions must next be made 
 to the figure. 
 
 There are 3 places that need filling up, each 3 in. 
 long, 2 in. wide, and 2 in. high. Each of these new 
 additions is 2 in. square and 3 in. long. It consists, 
 therefore, of the first part of 5 multiplied by the square 
 of the second ; and the three together are 3 times the 
 first part multiplied by the square of the second. 
 There is one addition wanting to complete the cube ; 
 that is at the corner. It must be 2 in. long, 2 in. 
 wide, and 2 in. high; that is, the cube of 2; or, 
 in other words, the cube of the second part. 
 
 Remembering that the two parts of 5, as here di- 
 vided, are 3 and 2; the printed figure is the cube ot 
 
CONSTRUCTION UF THE CUBE. 127 
 
 the first part ; the first addition is 3 times the square 
 of the first part, multiplied by the second ; the second 
 addition is 3 times the first part, multiplied by the 
 square of. the second ; the third addition is the cube 
 of the second part. Let the letter a stand for the first 
 part, 3 ; and the letter b, for the second part, 2. The 
 printed figure will then be a 3 ; the first addition, 3a 2 b; 
 the second addition, 3a b 2 ; the third addition, b s . The 
 whole cube, therefore, will be a* + 3a 2 b-\-3ab 2 + b 3 . 
 Observe that the letters and numbers are to be mul- 
 tiplied together, though there is no sign of multiplica- 
 tion Between them; as, 3a 2 b is three times the square 
 of a multiplied by b. 
 
 These are called the four terms of the cube, when 
 the root is in two parts. 
 
 If we express the above in the numbers for the 
 cube of 5, it will stand thus; 
 
 1st. 2d. 3d. 4th. 
 
 ^+3X3 2 X2 + 3X3X2 2 +? 3 . 
 
 1. What number makes the first term of this cube ? 
 
 2. What number forms the second term? 
 
 3. What number forms the third term ? 
 
 4. What number forms the fourth term? 
 
 5. What do all the four terms amount to ? 
 
 6. Which of the four terms contains the third power 
 of the first part? Which contains the second power 
 of the first part? 
 
 7. Which contains the first power of the first part ? 
 Which term contains the first power of the second 
 part? Which the second power? Which the third? 
 
 8. If the fourth term of the above cube were not 
 given, how could you determine from the others 
 what it must be? fefejfe* 
 
 9. If the third term were gone, how could you restore 
 it? If the second was gone, how could you restore it ? 
 
 10. If you divide the number 5 into the two parts 
 
128 MENTAL ARITHMETIC. 
 
 4 and 1, and express the cube according to the above 
 rule, what will the first term be ? What will be the 
 second term? What will be the third term? What 
 the fourth? 
 
 Remember, here, that all powers of 1 are 1, — neither 
 more nor less. 
 
 Divide the number 6 into 4 + 2, and form the cube 
 according to the above rule. 
 
 11. What will the first term be? The second? 
 The third? The fourth? 
 
 12. What will they all amount to ? 
 
 Multiply 6 into itself 3 times, thus, 6X6X6, and 
 see if it amounts to the same. 
 
 13. Divide 6 into the parts 5 + 1, and form the 
 cube. What will be the first term? The second? 
 The third ? The fourth ? What do they all amount to ? 
 
 14. There is a cube in four terms, the first two 
 terms of which are 3 3 + 3X3 2 Xl. What must be 
 the third term? What the fourth term? What is the 
 number of the cube ? What is the root of the cube ? 
 This root is the cube root of the first term, added to 
 the cube root of the last. 
 
 15. There is a cube in four terms, the first two of 
 which are 2 3 + 3X2 2 X2. What is the third term? 
 What the fourth? What is the whole cube? What is 
 the first part of the root? What is the second part? 
 What is the whole root? 
 
 16. Complete the cube 4 3 +3x4 2 X2 + C3+2 3 . 
 What is the number of the cube? What the root? 
 
 17. Complete the cube 3 3 + 3X3 2 X3 + 3X3X 
 3 2 + t=3. 
 
 What is the number of the cube? 
 
 18. There is a cube in four terms, the first of which 
 is 1000. What is the first part of the root ? 
 
 19. The second term of the same cube is 300X6, 
 or 1800. What is the third term? 
 
 20. What is the fourth term of the same cube ? 
 
RATIO. PROPORTION. 129 
 
 21. What is the root of the above cube? 
 
 22. The first term of a cube is 1000 j the second is 
 300X8, or 2400. What is the third term? 
 
 23. What is the fourth term of the above cube? 
 
 24. The first term of a cube is 8000 ; the second is 
 1200X3. What is the third term? 
 
 Observe that 1200 is 3 times the square of the first 
 term; consequently, one third of it is the square of 
 the first term. 
 
 25. What is the fourth term in the above cube? 
 What is the root? 
 
 SECTION XX. 
 
 RATIO. — PROPORTION. 
 
 If we compare the two numbers 3 and 9 in order to 
 ascertain their relative magnitude, we may subtract 3 
 from 9. We find the difference to be 6. 
 
 There is another way of comparing the two num- 
 bers. We may see how many times 3 will go in 9. 
 We shall find the quotient to be 3. 
 
 The numbers we obtain in each of these compar- 
 isons is called the ratio of the two numbers ; but they 
 differ in kind. The former is called arithmetical ratio ; 
 the latter, geometrical ratio. 
 
 Arithmetical ratio, then, expresses the difference of 
 two numbers ; geometrical ratio expresses the quotient 
 of one of the numbers divided by the other. As we 
 shall speak only of geometrical ratio in what follows 
 here, the word ratio, whenever it is used, may be un- 
 derstood to mean geometrical ratio. The ratio of 4 
 to 2, written 4:2, is 2; for 2 will go in 4 twice. 
 >The ratio of 12 to 3, written 12 : 3, is 4; for 3 
 will go in 12, 4 times. 
 
 I 
 
130 
 
 MENTAL ARITHMETIC. 
 
 The two numbers compared are together called the 
 terms of the ratio, or simply the ratio. The first is 
 called the antecedent ; the second is called the conse- 
 quent. These two terms, you will perceive, corre- 
 spond exactly to the numerator and denominator of a 
 fraction ; for, in a fraction, the numerator is divided 
 by the denominator. A ratio is, then, another way of 
 expressing a fraction. The antecedent is the numer- 
 ator ; the consequent, the denominator. 4:2 is the 
 same as f ; 6:3 the same as f . 
 
 As a ratio is essentially the same as a fraction, 
 every thing is true of a ratio which is true of a 
 fraction. 
 
 1. What effect will it have on the value of the 
 ratio, if you increase the antecedent ? If you diminish 
 the antecedent ? If you double the antecedent ? If 
 you divide the antecedent by 2 ? 
 
 2. What effect will it have on the value of the 
 ratio, if you increase the consequent ? If you diminish 
 the consequent ? If you multiply the consequent ? 
 If you divide the consequent ? 
 
 3. Take the ratio 4 : 2. How can you multiply 
 it by 2 ? In what other way ? 
 
 4. How can you divide it by 2 ? In what other- 
 way ? 
 
 5. Take the ratio 6 : 3. How can you multiply it 
 by 2 ? Can you do it in more than one Avay ? If you 
 cannot, why ? 
 
 6. How can you divide it by 4? Can you do it in 
 more than one way ? If not, why ? 
 
 Take the ratio 4 : 2. Multiply both terms by the 
 same number, 3, for example. It will be 12 : 6. You 
 see the value is not altered. — Divide both terms 4 : 2 
 by 2. It will be 2:1. The value is not altered. It 
 is still 2. 
 
PROPORTION. 131 
 
 PROPORTION. 
 
 If there are four numbers, and the first has the same 
 ratio to the second that the third has to the fourth, the 
 four numbers are said to be in proportion. Thus the 
 numbers 2 : 1 : : 12 : 6 are in proportion. The first 
 has the same ratio to the second, that the third has to 
 the fourth. The ratio is 2. 
 
 A proportion, then, is the equality of two ratios. 
 
 The four dots : : between the two ratios, are the 
 same as the sign of equality, =. 
 
 In order to preserve the proportion, the two ratios 
 must always be equal to each other. You may make 
 any change you please in the terms, provided you do 
 not destroy this equality. 
 
 Let us take the proportion 4 : 2 : : 12 : 6. The 
 value of the two ratios is now equal. 
 
 1st. Multiply the antecedents by 2. 8 : 2 : : 24 : 6. 
 The numbers are still in proportion, for the value of 
 the two ratios is equal. 
 
 2d. Divide the antecedents by 2. 2 : 2 : : 6 : 6. 
 The value of the two ratios is equal. 
 
 3d. Multiply the consequents by 2. 4 : 4 : : 12 : 12. 
 The value of the two ratios is equal. 
 
 4th. Divide the consequents by 2. 4 ; 1 : : 12 : 3. 
 The value of the ratios is still equal. 
 
 5th. Multiply both terms of the first ratio by 2. 
 8 : 4 : : 12 : 6 ; or multiply the 2 terms of the second 
 ratio by 2 ; 4 : 2 : : 24 : 12 ; the ratios are still equal. 
 In the same way we might take any other number for 
 our operations instead of 2. The same operations 
 might be performed without destroying the proportion. 
 
 The two middle terms of a proportion are called the 
 means ; the first and last terms are called the extremes. 
 
 In a proportion, the product of the two means is 
 equal to the product of the two extremes. 
 
132 
 
 MENTAL ARITHMETIC. 
 
 Take the proportion 4 : 2 : : 6 : 3. The product of 
 the means, 2X6, is 12; and the product of the ex- 
 tremes, 4X3, is 12. 
 
 Take the proportion 6 : 2 : : 9 : 3. 2x9= 18, 
 3X6=18. 
 
 Take the proportion 10 : 2 : : 30 : 6. 2X30 = 6X10. 
 
 If we know, then, the product of the means, we 
 know the product of the extremes. 
 
 In a certain proportion, the product of the means is 
 30. What must be the product of the extremes ? 
 
 6. Further, if we know the product of the means, 
 and if we know one of the extremes, we can find the 
 other. If, as in the above case, the product of the 
 means is 30, and if one of the extremes is 3, what 
 must be the other ? 
 
 How do you find that number ? 
 
 7. If the product of the means is 30, and one of the 
 extremes is 10, what must the other be ? 
 
 How do you find the number ? 
 
 8. If the product of the means is 30, and one of the 
 extremes is 5, what is the other ? 
 
 If one of the extremes is 6, what is the other ? 
 
 If one of the extremes is 15, what is the other? 
 
 You see, therefore, that, if you multiply the means 
 together, and divide the product by one extreme, the 
 quotient will be the other extreme. 
 
 9. If the product of the means is 72, and one of the 
 extremes is 24, what will the other be ? 
 
 10. If the first three terms of a proportion are 
 9 : 6 : : 12, what must the fourth term be ? 
 
 11. What is the fourth term of the proportion 
 5:3:: 15? 
 
 12. Complete the proportion 8 : 6 : : 12. 
 
 13. Complete the proportion 14 : 8 : : 7. 
 
 14. Complete the proportion 10 : 4 : : 15. 
 
 By .ans of this rule, many interesting questions 
 may solved. 
 
PROPORTION. 133 
 
 15. If 8 yards of cloth cost 6 dollars, what will 20 
 yards of the same cloth cost ? 
 
 It is evident that the length of the shorter piece is 
 to the length of the longer, as the cost of the shorter 
 is to the cost of the longer. Now, we know all these 
 numbers except the last, and can express them in the 
 
 Yds. Yds. Dols. 
 
 form of a proportion, thus, 8 : 20 : : 6. 8 is the length 
 of the shorter piece ; 20, the length of the longer ; 6 
 is the cost of the shorter piece. The fourth term, that 
 is, the cost of the longer piece, we have not yet found. 
 You must discover that yourself. How can you 
 do it? 
 
 16. If 18 yards of cloth cost 15 dollars, what will 
 12 yards of the same cloth cost ? 
 
 What do you here seek, — the quantity of cloth, or 
 the price ? Is it the price of the longer, or of the 
 shorter piece ? 
 
 How can you make a proportion with the two quan- 
 tities of cloth, and the two sums they cost ? State 
 this proportion in general terms, putting the thing 
 sought as the fourth term. State the proportion in 
 figures, all except the fourth term. How will you 
 find the fourth term ? 
 
 17. If 5 yards of cloth cost 2 dollars, what will 7 
 * yards of the same cloth cost ? 
 
 State the proportion in general terms. 
 State the first three terms in figures, and find the 
 fourth. 
 
 18. If a horse travels 16 miles in 3 hours, how far 
 will he travel in 2 hours ? 
 
 As the longer time is to the shorter time, so is the 
 greater distance to the smaller distance. 
 
 Remember that things of the same kind should stand 
 t in the same ratio ; and that the quantity sought must 
 be the fourth term. Then inquire what the true pro- 
 12 
 
134 MENTAL ARITHMETIC. 
 
 portion must be, and state it in general terms, repeating 
 the trial, if necessary, till you perceive that you are 
 right. This is far better than any special rule, for it 
 leads you to reason on what you do. 
 
 19. If a certain number of cubic feet of timber 
 weighs a certain number of hundred weight, and if 
 we wish to know, without weighing, how many 
 hundred weight a certain smaller number of cubic 
 feet will weigh, what will be the proportion in general 
 terms ? 
 
 20. If 16 cubic feet of wood weigh 5 cwt., what 
 will 6 cubic feet weigh ? 
 
 21. If 3 barrels of flour last a family 7 months, how 
 many barrels will last them 12 months ? 
 
 22. If an iron rod, of equal size throughout, and of 
 a certain length, weighs a certain number of pounds, 
 and is broken into two parts, not in the middle, how 
 can you find the weight of one of the parts without 
 weighing it ? 
 
 23. If an iron roa* 7 feet long weighs 20 pounds, 
 what will 5 feet of it weigh ? 
 
 COMPARISON OF SIMILAR SURFACES. 
 
 As all the above questions may be answered by 
 analysis as well as by proportion, the rule of propor- 
 tion might be dispensed with for the purpose of solving 
 this kind of questions. It has, however, very impor- 
 tant and interesting applications in the measurement of 
 similar surfaces and solids. To prepare for this, you 
 must attend carefully to a few introductory statements. 
 
 Two surfaces are similar to each other when they 
 are shaped alike, though they may be unequal in size. 
 Thus a large circle is similar to a small circle, for they 
 are both shaped alike. 
 
 So one square is similar to another, though they 
 
COMPARISON OF SIMILAR SURFACES. 135 
 
 may be unequal in size. One equilateral triangle is 
 similar to another equilateral triangle. 
 
 If a rectangle is twice as long as it is wide, and a 
 larger or a smaller rectangle is twice as long as it is 
 wide, the two are similar. In the same way any two 
 surfaces, however irregular their shape, are similar, 
 provided they are shaped alike. 
 
 We will now come to a stricter definition of similar 
 surfaces. Similar surfaces are such as have their cor- 
 responding dimensions proportional. — Take the circle. 
 The dimensions of the circle are the diameter and the 
 circumference. The diameter of one circle is to its 
 circumference, as the diameter of a larger or a smaller 
 circle is to its circumference. For you have learned 
 before that the circumference of a circle is 3| times its 
 diameter. 
 
 Take the square. One side of a square is to another 
 side of it, as one side of a larger or smaller square is to 
 another side of it. 
 
 If a rectangle is twice as long as it is wide, another 
 rectangle, in order to be similar, whatever be its size, 
 must be twice as long as it is wide. 
 
 1. There are two similar rectangles. One is 8 feet 
 long and 6 feet wide. The other is 6 feet long. How 
 wide is it ? 
 
 2. There are two similar rectangles. One is 5 feet 
 long and 2 feet wide. The other is 11 feet long. How 
 wide is it ? 
 
 3. There are two similar right-angled triangles. 
 In the larger, the base is 9 ; the perpendicular, 4 ; — in 
 the smaller, the base is 8. How long is the perpen- 
 dicular ? 
 
 We will now come to the comparison of the areas 
 of similar surfaces. 
 
 4. There are two circles. One is 1 foot in diameter ; 
 the other, 2 feet. How much greater is the area of the 
 larger than the area of the smaller ? 
 
136 MENTAL ARITHMETIC. 
 
 It is clearly more than twice as large ; for you could 
 lay two of the smaller circles on the larger, and still 
 leave a considerable space uncovered. Before answer- 
 ing this question, we will take the simple case of two 
 squares, one of which measures 1 foot on a side, and 
 the other 2 feet. You perceive the larger one is 4 
 times as great as the smaller. 
 
 Let one square measure 2 feet on a side; the other, 
 4 feet. How much greater is the larger than the 
 smaller? The smaller contains 4 square feet ; the 
 larger, 16; it is therefore 4 times as large. 
 
 5. If one square measures 3 times as much on a 
 side as another, how much greater is its area than 
 that of the smaller? 
 
 Let one square measure 1 foot on a side ; the other, 
 3 feet. In what ratio are their areas? — Let one meas- 
 ure 2 feet on a side ; the other, 6. In what ratio are 
 their areas? 
 
 The area of the larger, you find, is 9 times as great 
 as that of the smaller. This may serve to suggest the 
 principle by which the areas of all similar surfaces 
 may be compared. 
 
 The areas of similar surfaces are to each other 
 as the squares of their corresponding dimensions. 
 
 Let one square measure 1 foot on a side; another, 
 
 2 feet. I 2 : 2 2 : : 1 : 4. 
 
 Let one square measure 2 feet, and another, 4 feet, 
 on a side. 2 2 : 4 2 : : 4 : 16. 
 
 Let one square measure 1 foot on a side j another, 
 
 3 feet. 1 2 :3 2 ::1 : 9. 
 
 Let one square measure 2 feet on a side ; another, 
 6 feet. 2 2 : 6 2 : : 4 : 36. 
 
 This principle applies to circles, triangles, and all 
 similar surfaces whatever. You can now recur to 
 question 4, and find the answer to it. 
 
COMPARISON OF SIMILAR SOLIDS, 137 
 
 6. There are two circles. The diameter of the 
 greater is 3 times that of the smaller. The area of 
 the smaller is 1 acre. What is the area of the 
 greater ? 
 
 7. There are two circles. The diameter of the 
 smaller is two thirds that of the greater, and the area 
 of the smaller is 4 acres. What is the area of the 
 greater ? 
 
 8. There are two similar triangles. The corre- 
 sponding dimensions are as 3 to 4, and the greater 
 contains 12 acres. What does the smaller contain? 
 
 9. A farmer fenced a triangular piece of ground for 
 a field ; but, finding it not large enough, he enlarged 
 it, making each side one third greater than before, and 
 it then contained 5 acres. How much did it contain 
 at first? 
 
 10. There is an irregular field containing 8 acres. 
 One of the sides measures 20 rods. If the field be 
 enlarged, retaining the same form, so that the above- 
 named side measures 25 rods, how much land will 
 it contain? 
 
 11. There are 2 circles. The smaller is 3 rods, the 
 larger 7 rods, in diametef. How much greater in pro- 
 portion is the area of the latter than that of the former ? 
 
 12. There are 2 circles, one with a diameter of 3 
 feet, the other of 8. How much greater is the area 
 of the larger than that of the smaller? 
 
 COMPARISON OF SIMILAR SOLIDS. 
 
 We now come to the comparison of similar solids. 
 
 1. Let there be 2 cubes, one of them measuring 1 
 inch on a side, the other 2 inches. How much greater 
 
 is one than the other ? 
 i 
 
 You will perceive, by thinking of the construction 
 12* 
 
138 MENTAL ARITHMETIC. 
 
 of the cube, that the cube measuring 2 inches has in 
 it 8 cubic inches, an<j is therefore 8 times as great as 
 the one measuring only 1 inch. 
 
 2. Take cubes measuring 1 inch and 3 inches. 
 How much greater is the latter than the former? 
 
 3. Let one measure 1 inch, tliQ other 4 inches. 
 How much greater will the larger cube be? 
 
 These examples may suggest the principle on which 
 all similar solids are compared. 
 
 Similar solids are to each other as the cubes of 
 their corresponding dimensions. 
 
 Take, now, the first of the above three examples. 
 The ratio of the corresponding dimensions is 2:1; 
 the cubes of these terms, or 2 3 : l 3 , are 8:1; and this 
 is the proportion of the one solid to the other. 
 
 In the second example, the ratio of the correspond- 
 ing dimensions is 3 : 1 ; the cubes of these, 3 3 : l 3 , are 
 27 : 1 ; and this is the ratio of the two solids. 
 
 In the third example, the ratio of the corresponding 
 dimensions is 4 : 1 ; the cubes of these terms, 4 3 : l 3 , 
 are 64 : 1, which is the ratio of the two solids to each 
 other. 
 
 4. There are 2 iron balls. The smaller is 1 inch, 
 the other 5 inches, in diameter. How much does the 
 larger weigh more than the smaller ? 
 
 5. There are 2 iron balls. Their diameters are 2 
 inches and 3 inches. What is the ratio of their 
 weight ? 
 
 6. If the diameter of 2 balls is respectively 3 inches 
 and 4 inches, what is the ratio of their weight ? 
 
 7. If a cubic inch of stone weigh 1 ounce, how 
 many ounces would a cubic stone, measuring 10 
 inches, weigh? f 
 
 8. How many ounces, if the cube measured 11 
 inches ? 
 
COMPARISON OF SIMILAR SOLIDS. 139 
 
 9. How many ounces, if the cube measured 12 
 inches ? 
 
 10. If there were a smaller pyramid, of the same 
 material and shape with the great pyramid of Egypt, 
 and of xV its height, how many such would it take to 
 equal in solid contents the great pyramid ? 
 
 11. A common brick weighs 4 pounds, and is 8 
 inches in length. How much will a similarly-shaped 
 brick weigh, that measures 16 inches in length? 
 
 12. If an axe 4 inches wide weighs 4| pounds, 
 what will be the weight of a similar axe 5 inches 
 wide ? 
 
 13. If a blacksmith's anvil 1 foot long weighs 200 
 pounds, how much will a similar anvil weigh that is 2 
 feet long ? 
 
 14. A farmer sells 2 stacks of hay of the same 
 shape and solidity. The smaller is 10 feet high, and 
 is found to weigh 3 tons. The larger is 15 feet high. 
 How can its weight be determined, without weighing 
 it ? and what will the weight be ? 
 
 15. There are 2 similar cisterns. The smaller is 6 
 feet deep, and holds 500 gallons. The larger is 8 feet 
 deep. How many gallons will it contain ? 
 
 These operations will be rendered more easy, if, in 
 every case, where the ratios may be reduced, you re- 
 duce them to their lowest terms. 
 
 16. If a coal-pit 8 feet high has required 10 cords 
 of wood, how much wood would be required for a 
 coal-pit of similar shape 10 feet high ? 
 
 17. If there are two trees shaped alike, the smaller 
 measuring 4 feet in circumference, the larger 5 feet, 
 how will the amount of wood in the one compare 
 with that in the other ? 
 
 The principle given above applies to all similar 
 solids, whether bounded by plain surfaces or by 
 curved surfaces. 
 
140 MENTAL ARITHMETIC. 
 
 18. If a dwarf measures 2 feet in height, and a 
 man of the same form and solidity, 6 feet high, weighs 
 180 pounds, how many such dwarfs would equal the 
 weight of the man? What would the dwarf weigh? 
 
 19. If a man 6 feet 2 inches in height weighs 200 
 pounds, what would be the weight of a giant of equal 
 solidity and similar form, 9 feet 3 inches in height ? 
 
 20. If an animal 4 feet high weighs 600 pounds, 
 what will an animal of the same form and equal 
 solidity weigh, whose height is 5 feet? 
 
 21. An artist in Europe has made a perfect model 
 of St. Peter's Church at Rome, representing every 
 part in exact proportion, on a scale of 1 foot to 100 
 feet. If the material of the model is of the same 
 solidity with that of the church, how many times 
 greater is the solid contents of the church than that 
 of the model? 
 
 22. If a granite obelisk were constructed in the 
 precise form of the Bunker Hill monument, of one 
 tenth its height, how many such obelisks would the 
 monument furnish material to construct? 
 
 The comparison of similar surfaces and solids by 
 proportion has various interesting applications in de- 
 termining the comparative strength of timbers and 
 materials used in building and in other arts. 
 
 Case First. — The strength of materials to resist a 
 strain lengthwise. 
 
 1. If an iron rod half an inch in diameter will hold 
 a certain weight suspended by it, how much greater 
 weight will a rod hold that is 1 inch in diameter? 
 
 Here the strength is in proportion to the size, with- 
 out regard to the length j that is, as the square of the 
 
 diameters. 
 
 « 
 
 2. If an iron rod half an inch in diameter will sus- 
 
STRENGTH OF MATERIALS. 141 
 
 pend 2 tons, what weight will a rod suspend that is 
 three fourths of an inch in diameter? 
 
 3. A builder finds that an iron rod 1 inch in diam- 
 eter will suspend a certain weight. He wishes, how- 
 
 ;ever, to add to the weight half as much more, and, 
 I in order to support it, substitutes for the inch rod 
 .another rod 1| inches in diameter. Will it sustain 
 I the required weight ? 
 
 4. There are two ropes of the same material ; one, 
 i 1£ inches in diameter j the other, 2 inches. What is 
 
 the ratio of their strength ? 
 
 Case Second. — The strength of beams to resist 
 fracture crosswise. In beams of the same material, 
 length, and width, but of different depth, the strength 
 varies as the square of the depth. 
 
 1. There are two beams of equal length ; but the 
 depth of one is 10 inches; of the other, 12 inches. 
 What is the ratio of their strength ? 
 
 2. There is a stick of timber 4 inches thick and 
 12 inches deep. If sawed into three 4-inch joists, 
 what part of the former strength of the whole stick, 
 when placed edgewise, will each part possess, allow- 
 ing nothing for waste in sawing? 
 
 3. There is a stick of timber 10 inches in depth. 
 If 4 inches of its depth be removed, what will be its 
 strength compared to what it was before ? 
 
 4. There are two sticks of timber equal in length 
 and width ; one, 7 inches deep ; the other, 5. What 
 is the ratio of their strength ? 
 
 5. If a stick of timber 6 inches deep have 2 inches 
 of the depth removed, will it be weakened more than 
 one half? 
 
 What is the exact ratio of its present, compared 
 w%ith its former strength ? 
 
 6. A builder went to a lumber-yard, wishing to 
 
142 MENTAL ARITHMETIC. 
 
 obtain an oak beam 5 inches wide and 10 inches 
 deep. The lumber-merchant said, " I have not such 
 a stick ; but I have two oak sticks of the right length 
 and width, and 7 inches deep. They will both, 
 placed side by side, be stronger than one beam 10 
 inches deep." " Not so strong," said the builder. 
 
 Which was right? And what is the ratio of 
 strength in the two cases? 
 
NOTES TO PART FIRST 
 
 Note 1. — Page 16. 
 
 This exercise should be often reviewed, till the pupils can 
 go through it with ease, and without mistake. No exercise 
 can be devised that will more rapidly increase the learner's 
 powers in addition. 
 
 Note 2. — Page 17. — To the Instructor. 
 
 The word complement means, something to fill up. In 
 arithmetic, the complement of a number, strictly speaking, 
 is that number which must be added to it, to make it up to 
 the next higher order. The complement of a number con- 
 sisting of units only, as 3, 7, 9, is the number that must be 
 added to make it up to 10, and consists of units only. If the 
 number consist of tens, as 20, 50, its complement is the 
 number that must be added to make a hundred, and consist 
 of tens. If the number is hundreds, its complement is so 
 many hundreds as will make up a thousand. 
 
 If the number consist of several orders, its full comple- 
 ment will consist of the same orders, of such an amount as to 
 raise the sum to the next order above the highest named in it. 
 The complement of 745 is 255, for 745 -\- 255 = 1000, which 
 is the order next above the highest named in the given sum. 
 
 The more restricted use of the word, as employed in the 
 text, is sufficient for the purposes here had in view. 
 
 A few suggestions will here be made in reference to the 
 best mode of conducting the accompanying recitation. The 
 object of the lesson is to cultivate the power of instantly 
 associating a number and its complement together. In con- 
 ducting the recitation, the answer to each question, as it is 
 given out, should be required simultaneously of the whole 
 
144 
 
 NOTES TO PART FIRST. 
 
 class. The teacher should stand before them, and require 
 that every eye be fixed on him. The questions should not 
 be hurried, but the class should be encouraged to answer 
 instantly on hearing the question. This will be easy in the 
 first class of numbers given, which are even tens. In regard 
 to the remaining numbers, however, which are not even tens, 
 something more will be necessary. Suppose the question is, 
 What is the complement of 37? It may be conducted as 
 follows : 
 
 Teacher. What is the complement of 30 ? 
 
 Class. 70. 
 
 Teacher. Now, listen to me without speaking. What is 
 
 the complement of 30 ? . You observe, I am going to 
 
 say something more. What will it be ? 
 
 Class. Something between 30 and 40. 
 
 Teacher. Well, then, whereabouts will the complement 
 be found ? 
 
 Class. Between 60 and 70. 
 
 Teacher. Very good. Now, when I say 30, and keep 
 my voice suspended, showing that that is not all, what 
 number can you think of, that you know will be a part 
 of the complement? 
 
 Class. 60. 
 
 Teacher. Very well. Now, listen. What is the comple- 
 ment of 30 ? What have you now in your mind? 
 
 Class. 60. 
 
 Teacher. Well. Now, once more listen, and all answer 
 as soon as you hear the question. What is the comple- 
 ment of 37 ? 
 
 Class. 63. 
 
 In the following questions, let the teacher always make a 
 short pause between pronouncing the tens and the units ; 
 and if the class hesitate or disagree in their answer, let the 
 question be resolved into its elements, and each one pre- 
 sented separately. Thus, if 64 is the number, and the class 
 have not answered promptly and alike, say thus; " What is 
 the complement of 60?" 
 
 Class. 40. 
 
 Teacher. What is the complement of 60 ? What 
 
 do you think of? t 
 
 Class. 30. 
 
NOTES TO PART FIRST. 145 
 
 Teacher. Now, answer all together. What is the com- 
 plement of 64? 
 
 Class. 36. 
 
 In the examples of addition that follow, the teacher should 
 make a pause between the two numbers, and see that every 
 member of the class is intent and eager to catch the second 
 number, and answer instantly. A few questions answered 
 by the whole class in this way, will benefit them more than 
 whole pages recited in an indolent and listless manner. 
 
 Note 3. — Page 17. 
 
 In these and all other examples, the large numbers should 
 be taken first. If the pupil begins with the units, as in 
 written arithmetic, he should be checked at once. Such a 
 method would only lead to a laborious imitation of the pro- 
 cess of written arithmetic, which is not the natural one, and 
 could give no new power to the pupil, nor awaken any new 
 interest in the study. Only a small portion of these ques- 
 tions should be recited at one lesson. 
 
 Note 4. — Page 25. 
 
 Care must be taken here, that the pupil does not imitate 
 the process of written arithmetic, but be required to re- 
 gard every number in its true value. Thus, in the ques- 
 tion, "What is one fifth of 250 1 " he must not say, "5 
 in 25 is contained 5 times ; and 5 in 0, no times ; " but, 
 "One fifth of 25 is 5; therefore, one fifth of 250 is 50." 
 
 Note 5. — Page 28. 
 
 In the higher as well as the lower numbers, let the pupil 
 grapple at once with the number as it stands. In this way 
 his interest will be very much increased. He will see, 
 throughout, the progress he is making. Whereas, in written 
 arithmetic, as usually studied, the pupil has no sooner begun 
 13 K 
 
146 NOTES TO PART FIKST. ' 
 
 an operation than he loses sight of the process, and goes on 
 in blind bondage to his rule, till he comes out at the end, 
 and then looks to the book, as to an oracle, for the answer. 
 
 Let the oldest class in arithmetic in a school be called up, 
 and one of them be required to perform on the board the 
 question, "What is one sixth of 43,248?" and when he 
 has obtained the first quotient figure, stop him, and ask him 
 what he has now done. He will most likely be unable to 
 tell. The answer he will give will probably be, that he has 
 divided 43 by 6; and no one of his class will probably have 
 a better answer to offer. If he says he has divided 43 
 thousand, he is still wrong; for he has divided only 42 
 thousand, leaving one thousand undivided. 
 
 In some of the examples given in this section, the large 
 numbers may be separated in different ways preparatory to 
 division. Thus, in the last example, 92,648 may be divided 
 80,000, 12,000, 600, 48; or 88,000, 4000, 640, 8; and in 
 still other ways. 
 
 Pupils should be encouraged to exhibit more methods than 
 one for obtaining the answer. If a scholar has two methods, 
 he should be allowed to give them both ; and if another has 
 a different one still, it should be brought forward ; and the 
 most lucid and easy one should receive the commendation of 
 the teacher. 
 
 Note 6. — Page 101. 
 
 Strictly speaking, there is no relation in quantity between 
 a line and a surface, but only between a line and the dimen- 
 sions of a surface. By the square of a line is meant a square 
 surface, each of whose sides has the same length as the 
 given line. 
 
PART SECOND; 
 
 CONTAINING 
 
 RULES AND EXAMPLES FOR PRACTICE 
 
 WRITTEN ARITHMETIC. 
 
 NUMERATION OF WHOLE NUMBERS. 
 
 In common arithmetic, there are 9 figures used for 
 the expression of numbers. 1, one ; 2, two ; 3, three ; 
 4, four ; 5, five ; 6, six ; 7, seven ; 8, eight ; 9, nine. 
 When one of these figures stands alone, it signifies so 
 many units, or ones ; when two figures stand side by- 
 side, the left-hand figure signifies so many tens ; when 
 three stand side by side, the left-hand figure signifies 
 so many hundreds ; and universally, as you advance 
 to the left, the figures increase in value tenfold at each 
 step, as will be seen in the table on the next n*«r- 
 
 The right-hand place is always tha' ' all * s \ w ^n 
 ., ° j 1 J . ,^i, 0, must stand u, 
 
 there are tens, and no units, a cip v ' ; , TT ^ tr . no 
 *i •*» i .i_ r»rv miiS merely serves to oc- 
 
 the unit's place, hus, 20 ^ * figure, 2, is 
 
 cupy the una's place, an£ hun | re(te and 
 
 m the place of tens " d T^ the 
 
 no tens nor units cWU ^ 1 P ucl ° "^ ' Qnn . 
 
 unit's place. -^ one in the place of tens; as, 200, 
 and so n*" all higher numbers. 
 
 To annex a cipher to a figure, therefore, is the same 
 as to multiply the number by ten ; for it removes the 
 figure from the unit's place to the place of tens. lo 
 aiinex two ciphers, is the same as to multiply the num- 
 ber by a hundred ; for it removes the figure from the 
 unit's place to that of hundreds. 
 
 j 
 
148 
 
 TABLE OF NUMERATION. 
 
 
 s'i 
 
 two. 
 
 two tens, that is, twenty. 
 
 two hundred. 
 
 enumerate. 
 
 enumerate. 
 
 i two tens of thousands, 
 \ that is, twenty thousand. 
 
 two hundred thousand. 
 
 enumerate. 
 
 twenty-two million. 
 
 enumerate. 
 
 enumerate. • 
 
 In writing numbers, every place not occupied by a figure* 
 must be occupied by a cipher. Otherwise the true value of 
 
NUMERATION. 
 
 149 
 
 the figures at the left hand of that place would not be pre- 
 served. Thus, if you wish to write in figures the number 
 three hundred and four, as there are no tens, a cipher must 
 stand in the place of tens, 304. Should you omit the cipher, 
 and write 14, the 3 would have slid into the tens' place, and 
 it would not express three hundred and four. 
 
 As, in advancing to the left, figures increase their value 
 tenfold at each step, so, if you begin at any place in a line of 
 figures, and move towards the right, the figures will diminish 
 in value tenfold at each step : that is, each figure will signi- 
 fy but a tenth part of what it would, if it stood in the next 
 left-hand place. This will prepare you to look at the 
 
 NUMERATION OF DECIMALS. 
 
 Whole Numbers. Decimals. 
 
 S = = 3- 
 
 7th place, millions. 
 
 9th place, tens of millions. 
 
 9ih plaoe, lunula, of mills. 
 
 r 5- S- 
 a" m 5" 
 
 i 3 s 
 [ft 
 
 S 
 s 
 
 tr 
 
 3 
 
 8 
 
 IS 
 
 J 
 
 2 
 
 1 
 
 2 
 
 4 
 
 2 
 
 f . 
 
 1 
 
 2 
 2" 
 
 2; 
 
 3;- 
 
 -i £ 
 
 i § 
 
 r b 
 
 ' 1 
 
 1 1 
 
 ir =~ 
 
 I 
 
 2 
 
 7th place, tens of mill'ths. 
 6th place, millionths. 
 5th place, hund. of thou'ths. 
 4th place, tens of thou'ths. 
 
 
 
 1 
 
 3 
 
 6 
 
 7 ] 
 
 2; 
 2; 
 2^ 
 
 L4 
 
 16 
 
 4 
 
 >2 
 >2 
 
 [7 
 J 9 
 
 1 
 
 17 
 
 2 
 2 
 22 
 
 7 
 6742 
 
 two, and two tenths. 
 
 5 four, and two tenths, and five 
 
 ( hundredths, or 25 hundredths. 
 
 { twenty-two, and twenty- 
 ) two hundredths. 
 
 two thousandths, 
 enumerate. 
 
 four hundred and seventeen 
 hundred-thousandths. 
 
 enumerate. 
 
 13* 
 
150 ADDITION. 
 
 SECTION I. 
 
 ADDITION. 
 
 Addition is the uniting of several sums into one, to 
 show their amount. 
 
 Rule. — Set down the numbers, units under units, 
 tens under tens, and so on. Add the column of units; 
 set down the units of the amount, and carry the tens, if 
 there are any, to the column of tens. Add the column 
 of tens, and set down the unit figure of the amount, car- 
 rying the figure of tens to the next column ; and so on. 
 In adding the last column, set down the whole amount. 
 
 To prove the work, repeat the operation, beginning 
 at the top, and adding downwards. 
 
 Examples. 
 
 8. 871 + 934 + 340. 
 
 9. 516 + 617 + 713. 
 
 10. 685 + 937 + 742. 
 
 11. 840 + 931 + 672. 
 
 12. 963 + 847 + 784. 
 
 13. 421 + 317 + 844. 
 
 1. 472 + 842. 
 
 2. 376 + 421 + 645. 
 
 3. 431 + 843 + 794. 
 
 4. 821 + 954 + 359. 
 
 5. 267 + 549+121. 
 
 6. 834 + 682 + 762. 
 
 7. 468 + 912 + 683. 
 
 14. 6342 + 1896 + 4741 + 8962 
 
 15. 3249 + 856 + 8007 + 4990. 
 
 16. 3819 + 42 + 906 + 1728. 
 
 17. 1645 + 2718 + 92 + 1807. 
 
 18. 1543 + 1899 + 3054 + 26. 
 
 19. 1854+1962 + 2168 + 666. 
 
 20. 1062 + 6300 + 9071 + 7001, 
 
 21. 2593 + 1801 + 9201 + 2113. 
 
 22. 9064 + 2118 + 1802 + 3076. 
 
 23. 1001 + 9016 + 7990 + 26. 
 
 24. 106 + 2307 + 9436+108. 
 
 25. 1214 + 6403 + 7113 + 4009, 
 
ADDITION. 151 
 
 26. In 1840, the population of the Ne\^ England 
 States was as follows: Maine, 501,793; New Hamp- 
 shire, 284,574 ; Vermont, 291,948 ; Massachusetts, 
 737,699; Connecticut, 309.978; Rhode Island, 108,850. 
 What was the population of all the New England 
 States ? 
 
 27. The population of the Middle States, in 1840, 
 was as follows : New York, 2,428,921 ; New Jersey, 
 373,306; Pennsylvania, 1,724,033; Delaware, 78,085; 
 Maryland, 469,232; Virginia, 1,239,797. What was 
 the total population of the Middle States ? 
 
 28. The population of the Southern States, in 1840, 
 was — North Carolina, 753,419 ; South Carolina, 
 594,398; Georgia, 691,392 ; Alabama, 590,756 ; Ten- 
 nessee, 829,210 ; Mississippi, 375,651 ; Arkansas, 
 97,574; Louisiana, 352,411. What was the total pop- 
 ulation of these states ? 
 
 29. In 1840, the population of the Western States 
 was as follows: Ohio, 1,519,467; Indiana, 685,866 ; 
 Illinois, 476,183; Michigan, 212,267; Kentucky, 
 777,828; Missouri, 383,702. What was the total 
 population of these states ? 
 
 30. What is the total population of all the United 
 States, as set down in the four preceding examples ? 
 
 Whenever, in adding a column, two figures occur 
 together, which amount to 10, as 8 and 2, 7 and 3, 
 take them both together, and call them 10. This will 
 make the addition more rapid and easy. 
 
 When you have become familiar with the operations 
 in addition, you may occasionally vary your method, 
 by taking two columns of figures at a time. If you 
 have been thorough in the mental part of this work, 
 you will be able to do this. It will furnish an agree- 
 able variation in your method of work, and greatly 
 increase your power of rapid calculation. 
 
 31. This method is seen in the following example: 
 
152 SUBTRACTION. 
 
 3124 ^ 42 an d 81 are 123, and 24 are 147. Set 
 7681 I down the 47, and carry the 1 hundred to the 
 4942 . column of hundreds. 50 and 76 are 126, 
 15747 J an( i 31 are 157. 
 
 It will be well often to adopt this as your method 
 of proof. After performing the w^>rk by taking one 
 column at a time, prove it by taking two columns ; or 
 perform it first in the latter way, and prove it in the 
 other. 
 
 32. 1467 + 894 + 1721 + 8396. 
 
 33. 9461 + 8134 + 2016 + 4317. 
 
 34. 84161 + 9632 + 78167+43180. 
 
 35. 109761+20671 + 437674 + 963. 
 
 36. 26431 + 184097 + 467124 + 84321. 
 
 37. 43126 + 91434 + 237210 + 127. 
 
 38. 1235467+1096 + 34271 + 4081. 
 
 39. 10467 + 31762 + 10921 + 9634. 
 
 40. 37193+10634 + 206721 + 104367. 
 
 SECTION II. 
 
 SUBTRACTION. 
 
 Subtraction is the taking of a smaller number from 
 * a larger, to show the difference. The larger number 
 is called the minuend; the smaller, the subtrahend; 
 the difference is called the remainder. 
 
 Rule. — Set down the numbers, the larger number 
 uppermost, units under units, tens under tens. Sub- 
 tract the units of the lower number from the unit 
 figure above, and set down the difference. Proceed in 
 the same way, with the tens and higher orders, to the' 
 
SUBTRACTION. 
 
 153 
 
 close. If, in any case, the figure of the minuend is 
 less than the figure below it, increase it by 10, by bor- 
 rowing 1 from the next higher figure of the minuend ; 
 remembering, at the next step, that the figure in the 
 minuend has already been diminished by 1. 
 
 To prove the work, add the remainder and the sub- 
 trahend together, and, if the work is correct, the sum 
 will agree with the minuend. 
 
 Examples. 
 
 1. 
 
 748 — 365. 
 
 14. 
 
 8990—7096. 
 
 2. 
 
 674—582. 
 
 15. 
 
 8243 — 6492. 
 
 3. 
 
 849—634. 
 
 16. 
 
 784—96. 
 
 4. 
 
 347—267. 
 
 17. 
 
 210 — 100. 
 
 5. 
 
 431 — 249. 
 
 18. 
 
 681 — 504. 
 
 6. 
 
 867—312. 
 
 19. 
 
 901 — 75. 
 
 7. 
 
 419—224. 
 
 20. 
 
 16432—14968. 
 
 8. 
 
 519 — 499. 
 
 21. 
 
 195864—137461. 
 
 9. 
 
 318—201. 
 
 22. 
 
 228476 — 13962. 
 
 10. 
 
 856 — 106. 
 
 23. 
 
 740016 — 116799 
 
 11. 
 
 3416—2999. 
 
 24. 
 
 86400 — 199. 
 
 12. 
 
 4162—4091. 
 
 25. 
 
 10006 — 4364. 
 
 13. 
 
 7089—3007. 
 
 
 
 26. America was discovered in 1492. Plymouth 
 was settled in 1620. How long was that after the 
 discovery of America? 
 
 27. The Independence of the United States was 
 declared in 1776. How long was that after the settle- 
 ment of Plymouth ? 
 
 f 28. George Washington was born in 1732. He 
 took command of the American armies in 1776. How 
 old was he then ? 
 
 29. General Washington became President of the 
 United States in 1789. How old was he then ? 
 
 30. In 1820, the population of Maine was 298,335. 
 ^In 1830, it was 399,955. What was the increase in 
 
 10 years ? 
 
L 
 
 154 SUBTRACTION. 
 
 31. The population of Maine, in 1840, was 501,973. 
 What was the increase from 1830 to 1840 ? 
 
 32. The population of Massachusetts, in 1810, was 
 472,040 ; in 1820, 523,487. How much had it in- 
 creased from 1810 to 1820? 
 
 33. The population of Massachusetts, in 1830, was. 
 610,408. How much had it increased from 1820 to 
 1830? 
 
 34. In 1840, the population of Massachusetts was 
 737,699. How much had it increased from 1830 to 
 1840? 
 
 35. The population of the state of New York, in 
 1810, was 959,949. In 1820, it was 1,372,812. What 
 was the gain ? 
 
 36. The population of New York, in 1830, was 
 1,918,608. What was the gain from 1820 to 1830 ? 
 
 37. In -1840, it was 2,428,921. What was the gain 
 from 1830 to 1840 ? 
 
 38. The population of Ohio, in 1810, was 230,760. 
 In 1820, it was 581,434. What was the gain from 
 1810 to 1820 ? 
 
 39. In 1830, the population of Ohio was 937,903. 
 What was the increase from 1820 to 1830? 
 
 40. In 1840, the population of Ohio was 1,519,467. 
 What was the increase from 1830 to 1840 ? 
 
 Another method of performing subtraction, often 
 more convenient than the former, is the follow- 
 ing:— 
 
 Regard the subtrahend as a round number, one 
 greater than the figure of its highest order ; that is, if 
 the subtrahend is 43, call it 50; if 251, call it 300. 
 Subtract this round number from the minuend, and 
 then to the remainder add the complement required 
 to make up the subtrahend to the round number ; as 
 follows : — 
 
MULTIPLICATION. 155 
 
 41. 674—381. 400 from 674 leaves 274. Add 19, 
 the complement of 381. 274+19 = 293, Ans. 
 
 Apply this method to example 11, above. 
 
 42. 3416 — 2999. The first remainder, you see, is 
 416, and the complement is 1. Ans. 417. 
 
 This example shows how much shorter the work 
 often becomes by adopting this method. 
 
 One of the above methods may be used as a proof of 
 the other. 
 
 43. 384—219. 
 
 44. 1260—984. 
 
 45. 1679—291. 
 
 46. 2496 — 954. 
 
 SECTION III. 
 
 MULTIPLICATION. 
 
 In multiplication, a number is repeated a certain 
 number of times, and the result thus obtained is called 
 the product. 
 
 Rule. — Set down the smaller factor under the 
 larger, units under units, tens under tens. Begin with 
 the unit figure of the multiplier. Multiply by it, first, 
 the units of the multiplicand, setting down the units 
 of the product, and reserving the tens to be added to 
 the next product. Proceed thus through all the figures 
 of the multiplicand. If there are more figures than 
 one in the multiplier, take, next, the tens, and multiply 
 the figures of the multiplicand as before, setting the 
 figures of the product one degree farther to the left 
 than before. 
 
 Add the several partial products, and the amount 
 will be the whole product. 
 
156 
 
 MULTIPLICATION. 
 
 1. 342X64. 
 
 2. 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 14. 
 15. 
 16. 
 17. 
 18. 
 19. 
 
 Examples. 
 
 Thus, 342 i 
 64 
 
 1368 
 2052 
 
 346 X 34. 
 579 X 82. 
 976 X 38. 
 826 X 91. 
 376 X 121. 
 345 X 243. 
 798X114. 
 6181X35. 
 6821 X 82. 
 7413 X 96. 
 7921x22. 
 8964 x 85. 
 9056 X 43. 
 8007X41. 
 4559 X 741. 
 9642 X 864. 
 8721x317. 
 1841 X 134. 
 
 Ans. 21888 - 
 
 20. 
 21. 
 22. 
 23. 
 24. 
 25. 
 26. 
 27. 
 28. 
 29. 
 30. 
 31. 
 32. 
 33. 
 34. 
 35. 
 36. 
 
 Proof. 
 
 The best practical 
 " method of proof is 
 carefully to repeat 
 the operation. 
 
 13763 X 26. 
 97623X318. 
 1172671X216. 
 1874215X341. 
 742634X912. 
 189423 X 62. 
 14376281 X 194. 
 17284265X36. 
 671234X427. 
 1895453 X 28. . 
 3469528 X 672. 
 906421384 X 923. 
 713489605 X 84. 
 843469537 X 906. 
 236749024X516. 
 443754262X916. 
 1123496113X413. 
 
 37. The average length of the state of Massachu- 
 setts is 150 miles ; its breadth, 50 miles. How many- 
 square miles does it contain ? 
 
 38. The average length of Pennsylvania is 275 
 miles ; its breadth, 165 miles. How many square 
 miles does it contain? 
 
 39. The state of Ohio averages 223 miles in length, 
 180 in breadth. How many square miles does it con- 
 tain? 
 
MULTI PLICATION. 157 
 
 40. The state of Illinois averages 245 miles in 
 length, 147 in breadth. How many square miles 
 does it contain? 
 
 41. If there are 365 days in one year, how many 
 days are there in 25 years? 
 
 42. If the wages of a soldier is 8 dollars a month, 
 what will be the wages of 7867 soldiers for 12 months ? $ 
 
 43. There are 320 rods in 1 mile. How many rods 
 are there in 278 miles ? 
 
 44. 741X84. 
 
 45. 19643 XS92. 
 
 46. 246731X9210. 
 
 47. 946734X496. 
 
 48. 1623X198. 
 
 49. 9336X1998. 
 
 When the multiplier is a composite number, you 
 may multiply first by one of its factors, and the 
 product thus obtained by the other factor, or by the 
 others in succession, if there are more than two. 
 
 Apply this method to the following examples : — 
 
 50. 8476X45. 51. 1371X125. 52. 7465X108. 
 
 If a figure in the multiplier is a factor of the figure 
 in the next higher place, you may shorten the opera- 
 tion by multiplying the partial product of the lower 
 figure by the other factor of the higher. ' Thus, in 
 example 44, above, having found 4 times 741, you 
 know that 8 times the same is twice as many, and 80 
 times is 20 times as many. You need, therefore, only 
 double the line of the first partial product, setting it 
 one degree farther to the left, to express the tenfold 
 higher value. The same may be done if the right- 
 hand figure is a factor of the number expressed by 
 the next two higher figures. 
 
 Apply the process to the following examples: — 
 
 53. 
 
 947X639. 
 
 56. 
 
 27934X369. 
 
 54. 
 
 13674X4812. 
 
 57. 
 
 67514X64164. 
 
 55. 
 
 19742X568. 
 
 58. 
 
 259385X13212 
 
 If the multiplier is 10, or any power of 10, annex 
 14 
 
158 DIVISION. 
 
 to the multiplicand, for the answer, as many ciphers 
 as there are in the multiplier. 
 
 If the multiplier consists of 9's, add as many ciphers 
 to the multiplicand as there are 9's in the multiplier, 
 and from the product subtract the multiplicand. The 
 remainder will be the product sought ; for, by adding 
 the ciphers, you multiply by a number greater by one 
 than the multiplier. The multiplicand, therefore, will 
 be found in the product once too many times. So, if 
 the multiplier is 2 or 3 less than some power of 10, 
 you may do the same, remembering to take the multi- 
 plicand out as many times as the multiplier is units 
 less than a power of 10. 
 
 In this way perform the following examples: — 
 
 59. 3847X99. 
 
 60. 4572X999. 
 
 61. 54327X98. 
 
 62. 45314X997. 
 
 SECTION IV. 
 
 DIVISION. 
 
 In division, two numbers are given, in order to find 
 how many times one contains the other, or in order 
 to separate one number into as many equal parts as 
 there are units in the other. 
 
 The number to be divided is the dividend; the 
 number it is divided by is the divisor ; the answer 
 is the quotient. 
 
 To perform the operation, set down the divisor at 
 the left of the dividend. Take as many figures on 
 the left of the dividend as will contain the divisor 
 one or more times. See how many times the divisor 
 is contained in these figures, and set down the num- 
 ber as the first figure of the quotient. Multiply the' 
 
DIVISION. 
 
 159 
 
 divisor by the quotient figure, and subtract the product 
 from the number taken. To the remainder bring 
 down another figure of the dividend ; and proceed as 
 before. 
 
 1. 13276-^122; thus, 
 
 ' 122 ) 13276 ( 108, quotient. 
 122 
 
 1076 
 976 
 
 100, remainder. 
 
 To prove the work, multiply the divisor and the 
 quotient together, and add the remainder, if there be 
 any ; and the amount, if the work be right, will be 
 equal to the dividend. 
 
 Thus, in the above example, 122 
 
 108 
 
 976 
 122 
 
 100 
 
 13276 
 
 11. 3059-^214. 
 
 12. 700601 -f- 34. 
 
 13. 643817-^150. 
 
 14. 300796-^145. 
 
 15. 3264291-^27. 
 
 16. 18947633-^-181. 
 
 17. 384628910 -r 26. 
 
 18. 137900-^62. 
 
 19. 3946908 -M 72. 
 
 20. A man divided 35,785 dollars equally among 
 five children. How much did each receive? 
 
 21. In one barrel of flour there are 196 lbs. How 
 many barrels of flour are there in 13,916 lbs. ? 
 
 * 22. In 1840, the population of Maine was 501,793. 
 
 2. 
 
 11764-^34. 
 
 3. 
 
 47478-1-82. 
 
 4. 
 
 37088-^38. 
 
 5. 
 
 75116-^91. 
 
 6. 
 
 45496^-121. 
 
 7. 
 
 83835-^243. 
 
 8. 
 
 90972 — 114. 
 
 9. 
 
 89743 — 17. 
 
 10. 
 
 7426831 -t-141. 
 
160 DIVISION. 
 
 The state contained then 30,000 square miles. How 
 many inhabitants were there, on an average, to a 
 square mile? 
 
 23. The state of Massachusetts contained, in 1840, 
 737,699 inhabitants. Its territory is 7500 square miles. 
 How many inhabitants are there to a square mile ? 
 
 24. The population of Ohio, in 1840, was 1,519,467. 
 Its territory is 40,000 square miles. How many in- 
 habitants to a square mile? 
 
 If the divisor is less than 12, the multiplication and 
 subtraction may be carried on in the mind, and only 
 the quotient set down. This may most conveniently 
 be written directly under the dividend. 
 
 25. 7846 -=- 3. Operation, 3 ) 7846 
 
 2615+1; remainder. 
 26. 964385 -=-5. 
 
 27. 346218 -=-7. 
 
 28. 214681 -=-9. 
 
 29. 684219 -=-8. 
 
 30. 9640279 -=-4. 
 
 31. 146710063 -=-6. 
 
 32. 1143762 -Ml. 
 
 33. 1964217 -=-12. 
 
 34. 4691382 -=-4. 
 
 It is well to adopt the method of short division 
 sometimes when the divisor is larger than 12. 
 
 35. 33467-=- 15. 
 
 36. 46943 -M5. 
 
 37. 81743 4-16. 
 
 38. 91674 -=-21. 
 
 39. 673845 -=-22. 
 
 Miscellaneous Examples on the foregoing Rules. 
 
 1. A merchant began to trade with 4325 dollars. 
 He gained in one year 784 dollars. What was he 
 then worth? 
 
 2. A man's income is 948 dollars a year. His ex- 
 penses are 762 dollars. How much does he save of 
 his income in one year? 
 
 3. How much will he save in 9 years ? 
 
MISCELLANEOUS EXAMPLES. 161 
 
 4. A man bequeathed his property, 3882 dollars, 
 one third to his wife, and the remainder, in equal 
 shares, to his four children. What was each child's 
 share ? 
 
 5. A merchant buys 643 barrels of flour at 5 dol- 
 lars a barrel. He pays in addition, for freight, 65 dol- 
 lars ; for insurance, 17 dollars. What does the whole 
 cost him then ? What does each barrel cost him ? 
 
 6. A drover bought 7 oxen for 46 dollars a head, 
 12 cows for 32 dollars a head, 96 sheep for 3 dollars a 
 head. How much did they all come to ? 
 
 7. A drover buys 48 head of cattle at 32 dollars a 
 head. The whole expense of driving them to market 
 and selling them is 72 dollars. He sells them for 38 
 dollars a head. What does he gain ? 
 
 8. A laborer receives 16 dollars for every four 
 weeks' labor. He works 48 weeks. What do his 
 earnings amount to? 
 
 9. A man buys 7 tons of hay in the field for 13 
 dollars a ton. The cost of carrying it all to market is 
 48 dollars. He sells it for 15 dollars a ton. Does he 
 gain, or lose? and how much? 
 
 10. A man receives a salary of 950 dollars. He 
 spends for groceries 154 dollars; for milk, 21 dollars; 
 for meat, 75 dollars ; for wood, 67 dollars ; for cloth- 
 ing, 184 dollars ; for horse-hire, 38 dollars ; for jour- 
 neying, 93 dollars ; for repairs, 19 dollars ; for hired 
 help, 132 dollars ; for attendance of the physician, 26 
 dollars; for furniture, 51 dollars; for house rent, 184 
 dollars : and 86 dollars in charity and other incidental 
 expenses. Has he spent more than his salary, or less ? 
 and how much ? 
 
 11. A man works five months for 23 dollars a 
 month, and boards himself. He pays for 24 weeks' 
 board at 2 dollars a week. He expends, besides, for 
 
 ,a hat, 3 dollars ; for boots, 4 dollars ; and for other 
 articles of clothing, 6 dollars. How much does he 
 14* L 
 
162 MISCELLANEOUS EXAMPLES. 
 
 lay up, deducting the above expenses from the amount 
 of his earnings ? 
 
 12. A ship sails from Boston with a crew of 19 men. 
 At New Orleans, 3 of the crew are discharged, and 7 
 new hands taken on board. The ship then sails to 
 Liverpool, where 5 of the crew desert, 2 are left on ac- 
 count of sickness, and 3 new hands taken. At Havre, 
 the ship's next port, 4 men desert, and 3 new hands 
 are taken, when the ship sails for Boston. What is 
 the number of her crew on the return ? and what do 
 the whole wages of the men amount to, reckoning the 
 time from Boston to New Orleans one month ; from 
 New Orleans to Liverpool, one month and a half; from 
 Liverpool to Havre, half a month ; from Havre home, 
 one month and a quarter ; allowing 9 dollars a month 
 to each man as far as Liverpool, and 8 dollars a month 
 for the remainder of the voyage ? 
 
 13. What is the sum of 3 times 694 divided by 2 ; 
 9 times 1836 divided by 27 ; and 14 times 923 ? 
 
 14. What is the amount of the following bill of 
 provisions, namely, 18 barrels of beef, at 6 dollars a 
 barrel ; 19 hundred weight of ham, at 8 dollars a hun- 
 dred weight ; 173 barrels of flour, at 5 dollars a barrel ; 
 and 73 bushels of rye, at 92 cents a bushel ? 
 
 15. The area of Pennsylvania is 46,000 square miles. 
 If the population is 1,740,000, how many does that 
 give, on an average, to each square mile ? How many 
 to a township containing 30 square miles ? 
 
 16. New York has the same area as Pennsylvania 
 If the population is 2,440,000, how many is that to a 
 square mile ? How many to a township containing 30 
 square miles ? 
 
 17. Virginia has an area of 64,000 square miles. 
 If the population is 1,240,000, how many are there to 
 a square mile ? How many, on an average, to a county 
 containing 420 square miles ? 
 
 18. How much greater was the increase of popula- 
 tion in Massachusetts, (see Section II.) from 1820 to 
 
REDUCTION. 163 
 
 1830, than from 1810 to 1820 ? How much greater 
 from 1830 to 1840, than from 1820 to 1830 ? 
 
 19. The area of Massachusetts is 7500 square miles. 
 What was the average population on a square mile in 
 1810? In 1820? In 1830? 
 
 20. What was the average population on a square 
 mile, in the state of New York, in 1810? In 1820? 
 In 1830 ? 
 
 2J.! Greenland has an area of 840,000 square miles, 
 and 20,000 inhabitants. How many square miles does 
 it take to support one person ? How many would live 
 on a township of 30 square miles ? 
 
 22. Siberia contains 5,317,000 square miles. How 
 many states, of the size of Massachusetts, is its area 
 equal to? It contains 2,650,000 inhabitants. How 
 many live, on an average, on every 30 square miles ? 
 
 SECTION V. 
 
 REDUCTION. 
 
 The object in reduction is to change a quantity, in 
 one denomination, to another, which shall have the 
 same value. (See Sec. VI., Part I.) Higher denomi- 
 nations are reduced to lower by multiplication. 
 
 Examples. 
 
 1. Reduce 3 yards to feet. 
 
 2. Reduce 42 yards to feet. 
 
 3. Reduce 4 feet to inches. 
 
 4. Reduce 17 feet to inches. 
 
 5. Reduce 132 feet to inches. 
 % 6. Reduce 16 yards to inches. 
 
 7. Reduce 21 yards to inches. 
 
1G4 
 
 REDUCTION. 
 
 8. In 112 feet 7 inches how many inches ? 
 
 9. In 165 feet 4 inches how many inches ? 
 
 10. In 5 yards, 2 feet, 9 inches, how many inches? 
 
 11. In 24 rods how many feet ? 
 
 12. In 87 rods how many feet ? 
 
 13. In 567 rods how many inches ? 
 
 14. In 7 rods 4 feet how many feet ? 
 
 15. In 31 rods, 2 feet, 6 inches, how many inches? 
 
 16. In 131 £ how many shillings? 
 
 17. Reduce 781 £ to shillings. 
 
 18. Reduce 758 £ to shillings. 
 
 19. Reduce 19 shillings to pence. 
 
 20. Reduce 7£ 11 shillings to pence. 
 
 21. Reduce 141 £ 16 shillings, 4 pence, to pence. 
 
 22. Reduce 4£ 7 shillings, 3 pence, to farthings. 
 
 23. Reduce 14 lbs. 8 oz. avoirdupois, to oz. 
 
 24. Reduce 3 qrs. 9 lbs. 13 oz. to oz. 
 
 25. Reduce 44 cwt. 3 qrs. 19 lbs. to lbs. 
 
 26. Reduce 13 T. 12 cwt. 2 qrs. to lbs. 
 
 27. Reduce 3 lbs. 6 oz. 17 dwt. Troy, to dwt. 
 
 28. Reduce 13 lbs. 2 oz. 14 dwt. to dwt. 
 
 29. Reduce 4 oz. 16 dwt. to grs. 
 
 30. Reduce 6 lb. 7 oz. 9 dwt. 4 grs. to grs. 
 
 31. Reduce 27 gallons wine measure to pints. 
 
 32. Reduce 7 hhds. 13 gals. 2 qts. to qts. 
 
 33. Reduce 1 hhd. to gills. 
 
 34. Reduce 174 bushels to quarts. 
 
 35. Reduce 73 bushels to pints. 
 
 36. Reduce 231 bushels to quarts. 
 
 37. In 13 square feet how many square inches ? 
 
 38. In 84 square rods how many square feet ? 
 
 39. Reduce 13 square rods to inches. 
 
 40. Reduce 3 R. 17 rods to feet. 
 
 41. Reduce 5 A. 2 R. 14 rods to feet. 
 
 42. Reduce 17 solid feet to inches. 
 
 43. Reduce 19 s. yards 14 feet to inches. 
 
 44. Reduce 24 s. yards, 8 feet, 504 inches, to inches. 
 
REDUCTION. 165 
 
 45. Reduce 6 cords 13 s. feet to feet. 
 
 46. Reduce 27 cords 28 s. feet to feet. 
 
 47. Reduce 45 cords 13 s. feet to feet. 
 
 48. In 75 E. e. of cloth, how many quarters ? 
 
 49. Reduce 78 yards 3 qrs. to quarters. 
 
 50. Reduce 194 yds. 1 qr. to nails. 
 
 51. Reduce 11 yds., 3 qrs., 2 nails, to inches. 
 
 52. Reduce 174 Fr. e. to nails. 
 
 53. Reduce 4 m., 5 fur., 13 rods, to feet. 
 
 54. Reduce 17 m., 6 fur., 20 rods, 8 feet, to inches. 
 
 55. Reduce 21 £ 17s. 3d. to pence. 
 
 56. Reduce 24 £ to sixpences. 
 
 57. Reduce 95 £ 3 s. to sixpences. 
 
 58. Reduce 45 £ 5 s. to threepences. 
 
 59. Reduce 84 £ to fourpences. 
 
 60. In 1 cwt. 3 qrs. how many times 7 lbs. ? 
 
 61. In -8 bis. of flour, at 7 qrs. each, how many par- 
 cels of 14 lb. each ? 
 
 62. In 13 bis. of cider, at 31 J gals, each, how many 
 times 8 gallons ? 
 
 63. In a town 5 miles wide and 6 long how many 
 acres ? 
 
 64. How many acres in 7500 square miles ? 
 
 SECTION VI. 
 
 REDUCTION. 
 
 Lower denominations are reduced to higher by 
 division. 
 
 Examples. 
 
 1. In 348 shillings how many £ ? 
 
 2. Reduce 5000 shillings to £. 
 
 3. Reduce 13680 shillings to £. 
 
 4. Reduce 11040 pence to £. 
 
166 REDUCTION. 
 
 5. Reduce 11292 pence to £. 
 
 6. Reduce 20220 pence to £. 
 
 7. Reduce 1405 pence to £. 
 
 8. In 678 sixpences how many £ ? 
 
 9. Reduce 549 threepences to £. 
 
 10. Reduce 974 threepences to shillings. 
 
 11. Reduce 1776 hours to days. 
 
 12. Reduce 13841 hours to days. 
 
 13. Reduce 1964210 minutes to days. 
 
 14. Reduce 3742196 seconds to hours. 
 
 15. Reduce 964 gills to gallons. 
 
 16. Reduce 84672 gills to gallons. 
 
 17. Reduce 6794 gallons of wine to barrels. 
 
 18. Reduce 3469 quarts to pecks. 
 
 19. Reduce 96431 pints to bushels. 
 
 20. Reduce 3846 qts. to bushels. 
 
 21. Reduce 5674 rods to furlongs. 
 
 22. Reduce 38961 rods to miles. 
 
 23. Reduce 76381 feet to rods, 
 
 24. Reduce 7960 inches to rods. 
 
 25. Reduce 7126734 inches to miles. 
 
 26. How many steps of 2J- feet each are there in 1 
 mile ? 
 
 27. A man walks 30 miles. How many steps does 
 he take, 2f feet each ? 
 
 28. Reduce 179 lbs. avoirdupois to cwt. 
 
 29. Reduce 413 lbs. to cwt. 
 
 30. Reduce 1048 oz. to quarters. 
 
 31. Reduce 4352 drams to lbs. 
 
 32. Reduce 6130 oz. to cwt. 
 
 33. Reduce 1280 dwt. Troy to lbs. 
 
 34. Reduce 1511 dwt. to lbs. 
 
 35. Reduce 17812 grs. to lbs. 
 
 36. Reduce 720 square inches to square feet. 
 
 37. Reduce 1029 square feet to yards. 
 
 38. Reduce 2203 square inches to yards. 
 
 39. Reduce 3267 square feet to rods. 
 
COMPOUND ADDITION. 167 
 
 40. Reduce 5631 square feet to rods. 
 
 41. Reduce 86 solid feet to yards. 
 
 42. Reduce 191934 solid inches to feet. 
 
 43. Reduce 2333 solid inches to feet. 
 
 44. Reduce 876 solid feet to yards. 
 
 45. Reduce 2293 solid inches to yards. 
 
 46. In 92 qrs. cloth, how many E. e. ? 
 
 47. Reduce 361 nails to quarters. 
 
 48. Reduce 467 nails to yards. 
 
 49. Reduce 3741 inches to E. e. 
 
 50. Reduce 467 yards to E. e. 
 
 51. In 27 acres, 2 roods, 17 rods, how many lots 
 of 32 rods each ? 
 
 52. How many times does a carriage wheel 11£ feet 
 in circumference go round in one mile ? 
 
 53. In 35 tons weight how many wagon loads of 
 22 cwt. each ? 
 
 54. In 186 £ 12s. how many guineas of 21 shillings 
 each ? 
 
 55. In 75 yards how many E. e. ? 
 
 56. How many cannon balls, at 24 lbs. each, will it 
 take to weigh 1 ton gross weight ? 
 
 57. How many times must you apply a pole 12 feet 
 long to the ground, to measure 1 mile ? 
 
 58. How many 10 gallon kegs may be filled from 
 17 hhds. wine measure ? 
 
 SECTION VII. 
 
 COMPOUND ADDITION. 
 
 When numbers are used without being applied to 
 any particular kind of quantity, as 67, 84, they are 
 called abstract numbers; when they are applied to 
 
168 
 
 COMPOUND ADDITION. 
 
 some particular quantity, as 67 yards, they are called 
 denominate numbers. 
 
 When several numbers of different denominations 
 are to be added, as 3 £ 7s. +7 £ 4s., it is called Com- 
 pound Addition. 
 
 Examples. 
 
 Operation. 
 
 £ s. d. 
 
 3 14 9 
 
 14 11 6 
 
 18 6~ ¥ Ans. 
 
 1. 3£14s. 9d.+14£lls. 6d. 
 
 Set down numbers of the same denomination under 
 each other. Add first the numbers of the lowest de- 
 nomination. If the sum amounts to more than one 
 of the next higher, set down what is over, and carry 
 the number of the higher to the next column. So 
 proceed through the whole. In adding the last col- 
 umn, set down the whole amount. 
 
 2. 5£ 15s. 4d.+14£ 17s*. lld.+2£ 6s. 5d. 
 
 3. 13 £ 14s. 6d. 2qr. + 65£ 17s. lOd. lqr. 
 
 4. 48 £ 16s. + 73£ 10s. + 91£ 15s.+16£ 17s. 
 
 5. 17s. 3d.+15s. 7d. 2qr.+14s. Od. 2qrs. 
 
 6. Troy Weight. 12 lbs. 1 oz. 16 dwt. 14 grs. -f 
 3 dwt. 17 grs. 
 
 7. 17 lbs. 2 oz. 16 dwt. 5 gr. + 6 lbs. 14 dwt. 17 grs. 
 
 8. 21 lbs. oz. dwt. 3 grs.+19 dwt. 19 grs. + 
 13 dwt. 
 
 9. 21 lbs. 3 oz. 16 dwt. 15 grs. + 4 lbs. 4 oz. 17 dwt. 
 
 13 grs. 
 
 10. Avoirdupois Weight, gross. 3 cwt. qr. 17 lbs. 
 
 14 oz. +12 cwt. 3 qrs. 13 lbs. 12 oz. 
 
 11. IT. 14 cwt. 3 qrs. 17 lbs. + 3 T. 17 cwt. 1 qr. 
 21 lbs. 
 
 12. 3 T. 16 cwt. 1 qr. 20 lbs. 6 oz. + 5 cwt. 3 qrs. 
 19 lbs. 13 oz. 
 
COMPOUND ADDITION. 169 
 
 13. 14 cwt. 1 qr. 20 lbs. +18 cwt. 1 qr. 16 lbs. + 
 17 cwt. 1 qr. 11 lbs. 
 
 14. 1 m. 3 fur. 17 r. 6 ft. + 3 m. 5 far. 36 r. 12 ft. 
 
 15. 65 m. 7 fur. 31 r. +18 m. 19 fur. 23 r. +19 m. 
 
 4 fur. 17 r. 
 
 16. 5 r. 15 ft. + 27 r. 14 ft. +16 r. 11 ft. + 21 r. 12 ft. 
 
 17. 15 r. 9 ft. 6 in. +17 r. 3 ft. 4 in. +25 r. 15 ft. 
 11 in. 
 
 *18. 3 fur. 17 r. 4 ft. 5 in. +5 fur. 16 r. 14 ft. 9 in. 
 
 19. 7 fur. 16 r. 3 ft. 2 in. + 6 fur. 34 r. 12 ft. 10 in. 
 
 20. 13 m. 7 fur. 31 r. + 6 m. 3 fur. 22 r.+ll m. 
 
 5 fur. 8 r. 
 
 21. Square Measure. 2A.3R.6 p. +15 A. 1 R. 
 17 p. 
 
 22. 16 A. 2 R. 21 p.+8 A. 3 R. 33 p. + 9 A. 2 R. 
 9 P . 
 
 23. 2 R. 15 p. 63 ft. 29 in.+l R. 17 p. 31. in. + 
 37 p. 18 inches. 
 
 24. 13 p. 45 ft. 18 in. +19 p. 3 ft. 23 in. +17 p. 64 
 ft. 71 inches. 
 
 25. Solid Measure. 3 yds. 17 ft. 126 in. + 4 yds. 
 23 ft. 64 in. 
 
 26. 19 yards, 3 feet, 61 inches + 2 yards, 26 feet, 
 1650 inches + 4 yards, 18 feet, 91 inches. 
 
 27. 16 gals. 3 qts. 1 pt. + 84 gals. 2 qts. 1 pt. 
 
 28. 13 bushels, 3 pks. 4 qts. + 76 bushels, 3 pks. 5 
 qts. 
 
 29. 19 bu. 1 pk. + 76 bu. 3 pks. + 18 bu. 2 pks. 
 
 30. 14 yds. 3 qrs. 1 n. +21 yds. 2 qrs. 3 n. 
 
 31. 3 days, 16 hours, 23 minutes, +17 days, 13 h. 
 51m. 
 
 32. 1 year, 11 weeks, 4 days, + 3 y. 14 w. 2 d. 
 
 33. 7 deg. 14 min. 34 sec. + 19 deg. 20' 30". 
 
 34. 21 deg. 7' 1 1"+14 deg. 18' 19". 
 
 35. 61 deg. 26' 14" + 34 deg. 1' 8". 
 
 15 
 
170 COMPOUND SUBTRACTION. 
 
 SECTION VIII. 
 
 COMPOUND SUBTRACTION. 
 
 Compound subtraction is the subtraction of num- 
 bers of different denominations. 
 
 Rule. — Set numbers of the same denomination 
 under each other. Begin at the right hand, setting 
 down the remainder found by subtraction, under its 
 own denomination. If, in any case, the minuend is 
 less than the subtrahend, borrow one from the next 
 higher denomination of the minuend. 
 
 Examples. 
 
 1. 15£ 8s. 9d. — 11£ lis. 4d. 
 
 2. 22£ 19s. 8d.— 18£ 15s. 9d. 
 
 3. 13£4s. 6d. — 9£ 15s. lOd. 
 
 i 
 
 Operation. 
 
 £ s. d. 
 15 8 9 
 11 11 4 
 
 3 17 5 Ans. 
 
 4. 18 bushels, 3 pecks, 4 quarts, — 16 bushels, 2 
 pks. 5 qts. 
 
 5. 44 bushels, 1 peck, 3 quarts, — 20 bushels, 2 
 pks. 6 qts. 
 
 6. 4 years, 3 months,* 14 days, — 2 years, 4 months, 
 18 d. 
 
 7. 28 years, 8 months, 5 days, — 19 years, 11 months, 
 2 days. 
 
 8. 18 gallons, 3 quarts, 1 pint, — 10 gallons, 1 quart, 
 1 pint. 
 
 9. 14 gallons 1 quart, — 2 quarts 1 pint. 
 
 10. 4 miles, 3 furlongs, 17 rods, — 3 miles, 4 fur- 
 longs, 21 rods. 
 
 11. 19 miles, 7 furlongs, 11 rods, — 9 miles, 6 fur- 
 longs, 13 rods. 
 
 * Allow 30 days to a month. 
 
COMPOUND SUBTRACTION. 171 
 
 12. 5 cwt. 3 quarters, 14 pounds, — 4 cwt. 1 quarter, 
 20 lbs. 
 
 13. 12 cwt. 2 qrs. 21 lbs., — 9 cwt. 3 qrs. 23 lbs. 
 
 14. The battle of Bunker Hill was on June 17, 
 1775; the battle of Long Hand, August 27, 1776. 
 What was the length of time between them ? 
 
 • - 15. The battle of the Brandywine was September 
 11, 1777. How long was that after the battle of Long 
 Island? 
 
 16. The battle of Monmouth was June 28, 1778. 
 How long was that after the battle of the Brandy- 
 wine ? 
 
 17. The army of Burgoyne was captured October 
 17, 1777; that of Cornwallis, October 9, 1781. How 
 long between these events ? 
 
 18. If I give a note on interest, June 5, 1839, and 
 pay it March 10, 1841, for how long a time must the 
 interest be cast ? 
 
 19. If I give a note on interest, August 17, 1841, 
 and pay it June 9, 1843, for what time must the inter- 
 est be cast ? 
 
 20. How long is it from December 17, 1843, to 
 June 6, 1844? 
 
 21. How long from September 9, 1842, to August 
 3, 1844 ? 
 
 22. How long from January 16, 1840, to July 17, 
 1843? 
 
 23. How long from November 14, 1841, to August 
 21, 1844? 
 
 24. Boston is in longitude 71° M W. ; New York, 
 74° 1'. What is the difference of longitude ? 
 
 25. Cincinnati is in longitude 84° 27 7 . How many 
 degrees W. from Boston ? 
 
 26. How many degrees of longitude is Cincinnati 
 west from New York ? 
 
 • 27.- How many degrees of longitude is Cincinnati 
 west from Philadelphia, whose longitude is 75° ll 7 ? 
 
172 COMPOUND MULTIPLICATION. 
 
 SECTION IX. 
 
 COMPOUND^IVIULTIPLICATION. 
 
 Multiply, first, the lowest denomination. If the 
 product amounts to more than one of the next higher, 
 set down what is over, and carry the number of the 
 next higher to the next product. Multiply the next 
 denomination in the same way, and so on. 
 
 Examples. 
 
 Operation, 
 
 1. 3T. 7cwt. 3qr. multiplied by 7. 
 
 T. cwt. qr. 
 3 7 3 
 
 7 
 
 2. Multiply 4 £ 5 s. 6 d. by 2. 
 
 3. 6£ 4s. 3d. 1 qr.X3. 
 
 4. 6 hours, 43 min. 15 sec.X4. [23 14 1 Ans. 
 
 5. 9 h. 11 m. 41 sec.X6. 
 
 6. 14 days, 17 hours, 15 minutes, 3 seconds, X 8. 
 
 7. 8 bushels, 3 pecks, 1 quart, X 16. 
 
 8. 15 bushels, 2 pecks, 3 quarts, 1 pint, X 12. 
 
 9. 19 bushels, 1 peck, 2 quarts, X 5. 
 
 10. 1 mile, 5 furlongs, 13 rods, 12 feet,X2. 
 
 11. 13 miles, 2 furlongs, 4 rods, 6 feet, 3 inches, X 3. 
 
 12. 2 cwt. 3 qrs. 16 lbs.X7. 
 
 13. 14 cwt. 3 qrs. 14 lbs. X 4. 
 
 14. What is the weight of 12 casks of lime, each 
 weighing 3 cwt. 1 qr. 17 lbs. ? 
 
 15. How many yards in 9 pieces of calico, each 
 measuring 23 yards 3 qrs. ? 
 
 16. Troy Weight. 6 lbs. 11 oz. 5 dwt.X7. 
 
 17. 9 lbs. 8 oz. 16 dwt. 4 grs.X9. 
 
 18. Square Measure. 7 acres, 2 roods, 17 rods, X 9. 
 
 19. 15 acres, 1 rood, 34 rods,Xl4. 
 
COMPOUND DIVISION. 173 
 
 SECTION X. 
 
 COMPOUND DIVISION. 
 
 Divide the highest denomination first, and set the 
 quotient under it. Reduce the remainder, if any, to 
 the next lower denomination ; add it to those of the 
 same in the dividend, and divide again. And so on 
 to the end. 
 
 Examples. 
 
 Operation. 
 
 bu. pk. qt. 
 2)7 3 5 
 
 3 3 6| Ans. 
 
 1. Divide 7 bu. 3 pks. 5qts.by2. 
 
 2. 3£ lis. 6d.-^2. 
 
 3. 18£ 12s. 9d.^3. 
 
 4. 26 hours 53 minutes -f- 4. 
 
 5. Bo hours, 10 minutes, 6 seconds, -f- 6. 
 
 6. 114 days, 18 hours, minutes, 24 seconds, -r 8. 
 
 7. 140 bushels 2 pecks -r 16. 
 
 8. 187 bushels, 1 peck, 2 quarts, -f- 12. 
 
 9. Seven men are entitled to equal shares of 67 £ 
 
 13 s. 4d. What is each man's share ? 
 
 10. Three men are to receive equal shares of 114£ 
 19s. 9d. What is each man's share? 
 
 11. What is one fourth of 13 lbs. 6 oz. 17 dvvt., 
 Troy? 
 
 12. A teamster has 7 T. 11 cwt. 3 qrs. of mer- 
 chandise, which he loads on three wagons, giving 
 an equal load to each. How much is each load? 
 
 13. If you divide 7 bushels and 3 pecks of oats 
 equally among 5 horses, how much will each receive? 
 
 14. If a piece of land, containing 35 acres, 3 roods, 
 
 14 rods, be divided into 4 equal parts, how much will 
 .each part be ? 
 
 15* 
 
174 MISCELLANEOUS EXAMPLES. 
 
 SECTION XI 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. A teamster loads a quantity of merchandise 
 equally on three wagons, putting on each 1 T. 11 
 cwt. 2 qrs. Finding these loads too heavy, he 
 takes a fourth wagon. How much must he load 
 on each, to divide the whole equally among the 
 four ? 
 
 2. A man's estate amounts to 784 £ 10 s. His 
 wife is to receive 214 £ 15s., and the remainder is 
 to be divided equally among 4 children. What will 
 be each child's share? 
 
 3. Three men have equal shares in a scaffold of 
 hay, the whole of which weighs 5 T. 11 cwt. What 
 is each man's share ? 
 
 For the following examples see pp. 47, 48, Part I. 
 
 4. Rome is in longitude 12° 28' E. from London. 
 What time is it at Rome when it is noon in London ? 
 
 5. Petersburg is in longitude 29° 48' E. What 
 time is it at Petersburg when it is noon in London ? 
 
 6. Paris is in longitude 2° 20' E. What time is it 
 at Paris when it is noon in London ? 
 
 7. Boston is in longitude 71° 4' W. What time is 
 it in Boston when it is noon at London ? 
 
 8. New York is in longitude 74° 1' W. What 
 time is it in New York when it is noon at London ? 
 
 9. What time is it in Cincinnati, 84° 27' W. } when 
 it is noon in Boston, which is 71° 4' W. ? 
 
DIVISIBILITY OF NUMBERS. 175 
 
 SECTION XII. 
 
 DIVISIBILITY OF NUMBERS. 
 
 In order to ascertain if a number is divisible by 
 either of the following numbers, 2, 3, 4, 5, 6, 8, 9, 10, 
 or any combination of these, see Sec. VIII., Part I. 
 
 To ascertain if a number is divisible by any other 
 number than the above, make trial of other prime 
 divisors, as 7, 11, 13, 17, &c, beginning with the 
 smallest, till you find one that will divide the given 
 number, or find that it is indivisible. 
 
 Remember, that, in making trial by these numbers, 
 you need not go higher than the square root of the 
 given number ; for, if a number is divisible, one of 
 the factors will certainly be as small as the square 
 root. Let us take the number 1079. What are its 
 prime factors ? By inspection you may see it is not 
 divisible by 2, 3, 5, or 11, consequently not by 4, 6, 8, 9, 
 10, or 12. On trying it by 7, it is found not divisible 
 by 7. The next number is 13. This divides it, giv- 
 ing a quotient, 83, which is prime. Its only factors, 
 therefore, are 13 and 83. 
 
 Examples. 
 
 1. What are the prime factors of 667? 
 
 2. What are the prime factors of 406 ? 
 
 # 3. What are the prime factors of 419? Of 361? Of 
 742? Of 281? Of 316? 
 
 4. Prime factors of 941 ? 812? 749? 1116? 246? 
 8104? 
 
 5. Prime factors of 266 ? 884 ? 1917 ? 376 ? 
 
176 REDUCTION OF FRACTIONS. 
 
 SECTION XIII. 
 
 REDUCTION OF FRACTIONS. 
 
 [See Section VIII., Part I.] 
 
 1. Reduce f-f to its lowest terms. Ans. £. 
 
 2. Reduce f-f to its lowest terms. 
 
 3. Reduce -$& to its lowest terms. 
 4 Reduce f § to its lowest terms. 
 
 5. Reduce T Vs to its lowest terms. 
 
 6. Reduce £fi to its lowest terms. In this exam- 
 ple, it is not evident on inspection whether the two 
 terms of the fraction have any common divisor. In 
 such cases, you may adopt the following 
 
 Mule to find the Greatest Common Divisor. 
 
 Divide the greater number by the less ; and then 
 take the divisor for a new dividend, and divide it by 
 the remainder ; and so on, till there is no remainder. 
 The last divisor will be the greatest common divisor. 
 
 Apply the above rule to the sixth example. 
 
 187)221(1 
 187 
 
 ~34)187 
 170 
 
 17)34(2 
 34 
 
 00 
 
 The greatest common di- 
 visor is, therefore, 17 ; and, 
 dividing the terms of the 
 fraction by this, we have 
 for the lowest terms, i£. 
 
 Demonstration of the Rule. 
 
 If the larger number is a multiple of the smaller, it 
 is evident that the smaller is a common divisor of the^ 
 two numbers. It is also the greatest common divisor ; 
 
THE GREATEST COMMON DIVISOR. 177 
 
 for a number cannot be divided by any number greater 
 than itself. The answer, therefore, is found by the 
 first division. But if there is a remainder, next find 
 whether the remainder will exactly divide the divisor. 
 , If it will, it will divide both the original numbers ; 
 for, if it will divide the divisor, it will divide any 
 multiple of the divisor ; and, as it will of course 
 divide itself, it will divide any multiple of the divisor 
 plus itself. Now, the larger of the original numbers 
 is a certain multiple of the smaller plus the remainder. 
 If, therefore, after the first division, the remainder will 
 divide the divisor, it is a common divisor, or measure, 
 of the two numbers. 
 
 It is also the greatest common divisor; for, as it 
 will exactly measure the smaller of the two numbers, 
 it will exactly measure any multiple of the smaller. 
 Now, the greater number is a certain multiple of the 
 smaller plus the remainder. The remainder, therefore, 
 in measuring the larger number, is obliged to measure 
 itself. No number greater than itself can do this. 
 Therefore the remainder is the greatest common 
 divisor. If the work has to be carried on farther 
 than the second division, the same reasoning in the 
 demonstration will apply. 
 
 Examples. 
 
 7. What is the greatest common divisor of 874 
 and 437? 
 
 8. What is the greatest common divisor of 497 
 and 451? 
 
 \ . 9. What is the greatest common divisor of 817 
 and 913? 
 
 10. What is the greatest common divisor of 1007 
 and 1219? 
 
 11. What is the greatest common divisor of 608 
 ,and 192? 
 
 12. What is the greatest common divisor of 869 
 
 and 1343 ? 
 
 M 
 
178 REDUCTION OF FRACTIONS. 
 
 When there are more than two numbers, first find 
 the greatest common divisor of two of them, and then 
 of that divisor and the third number. 
 
 13. What is the greatest common divisor of 608, 
 941, and 451 ? 
 
 Whenever it is possible, by inspection, to separate 
 the numbers into their prime factors, this method 
 should be adopted. 
 
 14. What is the greatest common divisor of 94, 
 804, and 126 ? 
 
 15. What is the greatest common divisor of 1274, 
 896, and 580 ? 
 
 Apply the above rules to the reduction of the fol- 
 lowing fractions : — 
 
 16. Reduce f£| to its lowest terms. 
 
 17. Reduce to their lowest terms, f If- ; iff ; Mil- 
 
 18. Reduce to their lowest terms, f f g ; £ff ; f f £. 
 
 19. Reduce to their lowest terms, T Wg-?' sWuJ Iff- 
 
 To reduce an Improper Fraction to a Whole or Mixed 
 Number. 
 
 Perform the division indicated by the fraction as far 
 as possible. If there is a remainder, express that part 
 of the division by placing the denominator under the 
 remainder. 
 
 20. Reduce T £ to a whole or mixed number. 
 Ans. 1 T V 
 
 21. Reduce S J- to a whole or mixed number. 
 Ans. 3f. 
 
 22. Reduce to a whole or mixed number, - 3 ff 9 - ; 
 
 if; ff 
 
 23. Reduce to a whole or mixed number, V>' 
 
 if ; W- 
 
 24. Reduce the improper fractions, -W 9 - ; x f a ; W I 
 if a 
 
CHANGE OF NUMBERS TO HIGHER TERMS. 179 
 
 SECTION XIV. 
 
 CHANGE OF NUMBERS AND FRACTIONS TO HIGHER 
 TERMS. 
 
 It is sometimes convenient to express whole numbers 
 in the form of fractions, and to express fractions in 
 higher terms without altering the value. Thus 3 = f, 
 or- 2 ^; 10 = ^°-, or-% -. 
 
 Examples. 
 
 1. In 4 how many fifths ? Ans. 20. 
 
 2. Express the value of 4 in fifths. Ans. 2 f- 
 
 3. Express 7 in thirds. 
 
 4. Express 19 in the form of sevenths. 
 
 5. In 13 how many eighths ? 
 
 6. Express 21 in thirds. 
 
 7. Express 7 in eighteenths. 
 
 8. Express 41 in fourths. 
 
 9. In 3£ how many halves ? 
 
 10. Change 4i to an improper fraction. 
 
 11. Change 17£ to an improper fraction. 
 
 12. Change 24£ to an improper fraction. 
 
 13. Change to an improper fraction, 18£ ; 112^- ; 
 318^. 
 
 14. Change f- to eighths, without altering its value. 
 
 15. Change f to fifteenths. 
 
 16. Change £ to 24ths ; T 3 <y to 80ths ; f to 99ths. 
 
 17. Change T \ to 26ths ; T 9 ¥ to 56ths ; if to 60ths. 
 
 18. Change f to 7ths. 
 
 This example presents a difficulty, because the re- 
 quired denominator, 7, is not, as in the preceding exam- 
 ples, a multiple of the given denominator, 4. We have 
 seen, however, that, if we multiply or divide both terms 
 of a fraction by the same number, the value will not 
 be altered. We must, then, multiply and divide both 
 
180 MULTIPLICATION AND DIVISION OF FRACTIONS. 
 
 terms by such numbers as will give us, in the end, 7 
 for the denominator. The question, then, is, How 
 can we, by multiplication and division, change 4 into 
 7? We can multiply it by 7, which will give 28, and 
 then divide by 4, giving 7 for the quotient. Thus the 
 denominator has been changed, by multiplication and 
 division, from 4 to 7. Now, whatever has been done 
 to the denominator must be done to the numerator, to 
 preserve the value of the fraction. Multiplying 3 by 
 7, we have 21 ; dividing this by 4, we have 5£ for the 
 
 51 
 required numerator. The answer, therefore, is — . 
 
 This fraction, as one of its terms contains a fraction in 
 itself, is called a Cornplex Fraction. 
 
 19. Change \ to 8ths ; £• to 9ths ; T 3 T to 7ths. 
 
 20. In f how many 4ths ? How many 5ths ? 
 6ths? 
 
 21. Change 4| to 5ths ; 84 to llths ; 7£ to 4ths. 
 
 22. Change 22f to 4ths ; I8i to 7ths ; 31^ to 5ths. 
 
 23. In 8£ how many 3ds? 4ths? 5ths? 9ths? 
 
 24. In 19£ how many 5ths ? 4ths ? 7ths ? 
 
 25. In 9± how many 3ds? 5ths? 8ths? 
 
 26. In 13£ how many 14ths ? 15ths ? 
 
 27. In 8£ how many 17ths? 13ths? 
 
 28. In 20| how many 7ths ? 8ths ? 
 
 29. In 16i how many 4ths ? 5ths ? 
 
 ' 30. In ll£ how many 37ths? 19ths? 
 
 SECTION XV. 
 
 MULTIPLICATION AND DIVISION OF FRACTIONS. 
 
 [See Section VUL, Part I.] 
 
 1. A man worked 72 days for | of a dollar a day. 
 What did his wases amount to? 
 
MULTIPLICATION AND DIVISION OF FRACTIONS. 181 
 
 2. Multiply £ by 46. 
 
 3. A man bought 139 bushels of apples for f of a 
 dollar a bushel. What did they come to ? 
 
 4. Multiply £ by 341 ; fXl27. 
 
 5. A garrison of 700 soldiers are allowed f of a 
 pound of flour a day for each man. How much 
 would they consume in 1 day ? How much in 7 days ? 
 
 6. If a horse eats T \ of a bushel of oats in a day, 
 how many bushels will he eat in 365 days ? 
 
 7. If a horse eats %£ cwt. of hay in a week, what 
 part of a cwt. will he eat in one day ? How many 
 cwt. will he eat in a year? 
 
 8. Three men gain, by an adventure, 56£ dollars, 
 which they are to share equally. .What is each man's 
 share ? 
 
 9. What is | of 187± ? What is * of 91§ ? 
 
 10. T %X141? T \X97? |X140? 
 
 11. If 3£ cwt. of flour be divided into 5 equal 
 parts, what part of a cwt. will each share be ? 
 
 12. Divide 17J by 8; 37£ by 14; 18+ by 9. 
 
 13. Divide 74±-=-5; 81l£-^-7; 38H-^-9. 
 
 14. A man and his son agree to work on the follow- 
 ing terms : The father is to receive | of a dollar a day, 
 and the son f as much as the father. The father 
 works 18 days, the son works 2£ times as long as the 
 father. What do the wages of both amount to ? 
 
 15. A carpenter agrees to work on a house, charg- 
 ing If dollar a day for himself, and f as much for his 
 hired man. He works himself 38 days, and his hired 
 man long enough to have his wages amount to 7£ dol- 
 lars more than his own. How many days does the 
 hired man work? 
 
 16. A, B, and O, work on the following terms : A 
 is to receive l£ dollar a day ; B, f as much as A ; and 
 C, T 7 u as much as B. A works 13 days ; B, twice as 
 many days as A ; and C, once and a half as many as 
 B. What do the wages of the three amount to ? 
 
 16 
 
JjK& MULTIPLICATION AND DIVISION OF FRACTIONS. 
 
 17. Multiply half of 1672 by J, and then find f of 
 the product. 
 
 18. If you multiply 9f by 643, and subtract 1572 
 from the product, what will T V of the remainder be ? 
 
 19. 18 times 72£ are how many halves of 16 times 
 84i? 
 
 20. A bought one third of a quarter ticket in a lot- 
 tery. The price of a whole ticket was 12 dollars. 
 The whole ticket drew 1000 dollars. Of the prize 
 money, 5 3 <y is deducted before it is paid over. How 
 much did A receive for his share of the prize ? How 
 much did he gain, after deducting the cost of his 
 ticket ? 
 
 21. Encouraged by his good luck, A then buys 18 
 whole tickets in the same lottery, and draws in all 25 
 dollars, subject to the same deduction as before. 
 Does he gain or lose by his whole operation in the 
 lottery, and how much ? 
 
 SECTION XVI. 
 
 MULTIPLICATION AND DIVISION OF FRACTIONS. 
 
 [See Section IX., Part I.] 
 
 1. In 63 gallons, how many bottles of T 2 ^ of a gal- 
 lon each ? 
 
 2. From 7 lbs.^f flour, how many loaves of bread 
 may be made, each containing § of a pound of flour ? 
 
 3. Divide U + i; JH-»; H4-J* 
 
 4. A man left 5846 dollars ; § of the whole to go to 
 his wife, and the remainder to be equally divided 
 among four children. What was each child's share ? 
 
 5. From a piece of cloth 32 yards long, how many 
 coats can be made, each requiring 2f- yards ? 
 
MULTIPLICATION AND DIVISION OF FRACTIONS. 183 
 
 6. From a stick of timber 26£ feet long, how many 
 blocks can be cut, each T 9 ^ of a foot long ? 
 
 7. If a family consume 22£ pounds of flour in a 
 week, how much is that a day ? How much will they 
 use in a year ? 
 
 8. Divide 134-^ ; 18} -Ml*; 86*-=-9£. 
 
 9. Multiply 251X16* ; 23*X2f; 31§X19£. i 
 
 A fraction of a fraction, as £ of f , is called a Com- 
 pound Fraction. This is reduced to a simple fraction 
 by multiplying the numerators together for a new 
 numerator, and the denominators for a new denomi- 
 nator. It is, in fact, the same as the multiplication of 
 two fractions together. 
 
 10. What is i of § of 76 ? 
 
 11. What is £ of f of 12? What is f of f of 
 18}? 
 
 12. What is A of f of 8* ? What is f of § 
 of 34? 
 
 13. Divide | of f by f Divide f of # by 8£. 
 
 14. Multiply i of 37 by 19*. Multiply | of 18 
 by 19*. 
 
 15. What is the value of 32* yards of cloth, at 4£ 
 dollars a yard ? 
 
 16. What do 17£ tons of hay come, to, at 11* dol- 
 lars a ton ? 
 
 17. What is the value of 2l£ cords of wood, at 4£ 
 dollars a cord ? 
 
 18. What is the value of 24£ barrels of apples, at 
 If dollar a barrel ? 
 
 19. What is the amount of 12£ shares bank stock, 
 at 64 1£ dollars a share? 
 
 20. A man paid away £ of his money for a horse, 
 and * as much for a saddle, 3 dollars for a bridle, and 
 had 1 dollar left. How many dollars had he at 
 first? 
 
 21. The width of a certain room is f- of its length. 
 Its height is } as great as its width. Its door is 6 feet 
 
184 ADDITION AND SUBTRACTION OF FRACTIONS. 
 
 high, which is £ of the height of the room. What is 
 its length ? 
 
 22. Of a certain room, the width is f of the length ; 
 the height is £ of the width, and also twice and one 
 seventh as great as the height of the door, which is 7 
 feet. How long is the room ? 
 
 23. Multiply 137| by 31 T 4 T . Multiply 45f by 
 183f. 
 
 24. What is the quotient of 163f divided by 29 T 4 F ? 
 
 25. How many times is £f of 57 contained in f of 
 1147*? 
 
 26. One half of f of f of 511 is how many fifths 
 of 19*? 
 
 27. Wliat is the cost of 31* tons of hay at 9f dol- 
 lars a ton ? 
 
 28. What is the cost of 109* gallons of molasses at 
 29* cts. a gallon ? 
 
 29. What is the cost of 56* cwt. of beef at 4$ dol- 
 lars per cwt. ? 
 
 30. *f of 73 are how many fifteenths of 61* ? 
 
 31. How many times is § of 181* contained in 
 1156* ? 
 
 SECTION XVII. 
 
 ADDITION AND SUBTRACTION OF FRACTIONS. 
 
 If the fractions have a common denominator, per- 
 form the required operation on the numerators, and 
 place the result over the common denominator. 
 
 1. Add A + A + f; ^t + tV + t*; tV + tV + tV 
 
 2. Subtract f-f; A~&J «— Uj if— «• 
 
 If the fractions have not a common denominator, 
 reduce them to a common denominator, (Sec. IX., 
 
ADDITION AND SUBTRACTION OF FRACTIONS. 185 
 
 Pt. I.) and then add or subtract, as the question re- 
 quires. 
 
 3. Addi + i + f; f + x 4 T + if; i* 6 +f + i 
 
 4. Add # + £ + *£; ! + fr + i§; £ + t 3 t+iV 
 
 5. Subtract 5^ — f; 8 T \ — $; f— A« 
 
 ; 6. Add3H + 57i + 18i; 191 + 21* + 3^. 
 
 In cases like the above, it is easiest to add the 
 whole numbers first. 
 
 7. Add 67 T V + i of 7i + i of 22; 4 T 3 T + * of 
 18f + | of 13. 
 
 8. A man spent for various articles i of a dollar, 
 | of a dollar, ^ of a dollar, T 5 ^ of a dollar. What 
 part of a dollar did he spend in all? 
 
 9. From 56£ bushels llf bushels were taken. 
 How much remained? 
 
 10. From a firkin of butter containing 42£ lbs. 
 18-J-i lbs. were taken. How much remained? 
 
 11. Add 1 Ai + 4 f ; Ui+gj 15* + 7* + |t 
 
 12. Add &tf+2*j 9i +?I; 13J + S+11*. 
 
 5 11 7 
 
 13. Add I9f+21«j 8f+g; 26i + «+g. 
 
 14. Add 4f + ^; 8i+^; 12i+3*+18A. 
 
 15. Subtract 19* — |g: 22£ — g; 18ft— 13A* 
 
 lo 4o 
 
 16. Add 3i+8* + iy+A ; 41£ + ?* + — . 
 
 40 Ui' ¥ ^ 7 28£ 
 16* 
 
186 REDUCTION OF DENOMINATE FRACTIONS. 
 
 SECTION XVIII. 
 
 REDUCTION OF DENOMINATE FRACTIONS. 
 
 Denominate fractions are fractions of numbers when 
 applied to a particular denomination. See Sec. XL, 
 Part I. 
 
 Examples. 
 
 1. What part of a bushel is £ of a quart ? f of a 
 quart ? 
 
 Consider, first, what part of a bushel one whole 
 quart is. 
 
 2. What part of 2 bushels is i of a quart ? f of a 
 peck ? 
 
 3. What part of 7 bushels is f 'of a peck ? £ of a 
 peck? 
 
 4. What part of 3 £ is f of a shilling ? T 8 T of a 
 shilling ? 
 
 5. What part of a shilling is i of a penny ? -ft of 
 a penny ? 
 
 6. What part of a mile is £ of a rod ? f of a rod ? 
 
 7. What part of 3 furlongs is f of a rod ? £ of a 
 rod? 
 
 8. What part of a mile is 3 furlongs 19 rods ? 
 
 9. What part of a mile is 1 foot ? 2 feet ? 
 
 10. What part of a ton is 6 cwt. 3 qrs. 7 lbs. ? 
 
 11. What part of a ton is 18 cwt. 1 qr. 6 lbs. ? 
 
 12. What part of a square rod is 17 feet? 3l£ 
 feet? 
 
 13. What part of a cord is 12£ cubic feet ? 25£ 
 cubic feet ? 
 
 14. What part of a cord is 18J cubic feet ? 34£ 
 cubic feet ? 
 
REDUCTION OF DENOMINATE FRACTIONS. 187 
 
 15. What part of a bushel is | of a quart ? § of a 
 quart ? 
 
 16. What part of a bushel is 7£ quarts ? 1 1| quarts ? 
 
 17. What part of a week is 4£ hours ? 5 T 2 T hours ? 
 
 18. What part of a week is 7± hours ? 8£ hours ? 
 
 19. What part of 3 hours is 12 minutes ? 15£ 
 minutes? 
 
 20. What is the value, in shillings and pence, of f 
 of a£? 
 
 If it were 3 £, there would be 60 shillings ; but it is 
 not 3£, but one seventh of that ; therefore, it is } of 
 60 shillings = 8f shillings. To find the value, in 
 pence, of f of a shilling, pursue the same reasoning. 
 If it was 4 shillings, it would be 48 pence ; but it is 
 not 4 shillings, but j- of that £= 6f pence. The answer, 
 then, is 8 s. 6f d. 
 
 21. What is the value, in shillings and pence, of f- 
 of a£? 
 
 22. Value, in hours, minutes, and seconds, of £ of 
 a week ? 
 
 23. How many minutes in T 5 S of a week ? 
 
 24. How many minutes and seconds in A of an 
 hour ? 
 
 25. Value of f of a gallon ? Value of ■£> of a bar- 
 rel of wine? 
 
 26. How many oz., dwt., and grs., in t* of a pound 
 Troy? 
 
 27. What is the value, in oz., dwt., and grs., of fV 
 of a pound Troy ? 
 
 28. How many square rods in f of an acre ? 
 
 29. How many square feet and inches in T \ of a 
 square yard? 
 
188 CHANGE OF DENOMINATE INTEGERS. 
 
 SECTION XIX. 
 
 CHANGE OF DENOMINATE INTEGERS TO FRACTIONS. 
 
 [See Section XL, Part L] 
 
 Examples. 
 
 1. What part of a furlong is 5 feet ? 7i feet ? 
 Consider, first, what part of a furlong 1 foot is. 
 
 2. What part of a mile is 17 feet ?' 28^ feet ? 
 
 3. What part of a mile is 3l£ feet ? 47f feet ? 
 
 4. What part of a ton is 25J- lbs. ? 82£ lbs. ? 
 
 5. What part of a ton is 107i lbs. ? 130£ lbs. ? 
 
 6. What part of a ton is 3 qrs. 9 lbs. 8 oz. ? 
 17* lbs.? 
 
 7. What part of a ton is 4 lbs. 13 oz. ? 19 T % lbs. ? 
 
 8. What part of an acre is 4 rods 17 feet ? 139 T V 
 feet? 
 
 9. What part of an acre is 113 rods 5l£ feet? 
 
 10. What part of a hogshead of wine is 17 gals. 3 
 qts. 1 pt. ? 
 
 11. What part of 10 gallons is 3£ pints ? 8{- pints? 
 
 12. What part of a guinea is 3 s. 7d. ? 14 s. 3£d. ? 
 
 13. What part of a £ is 8s. 5£d. ? 10s. 9f-d. ? 
 
 14. What part of a week is 3^ hours ? 7 T V hours ? 
 
 15. What part of 5 days is 1 hour, 41 m. 15 sec. ? 
 
 16. What part of a month of 31 days is 17 h. 
 18J m. ? 
 
 17. What part of an ell English is 2 n. 1 in. ? 
 
 18. What part of 21 yards is 3£ qrs.? 9 T 3 T yards? 
 
 19. What part of 37£ yards is l£ ell English ? 
 
 20. What part of 7 cords of wood is 2l£ cubic feet ? 
 
 21. What part of 31 £ is 5s. 3£d. ? 14s. 2£d. ? 
 
 
PRACTICAL EXAMPLES. 189 
 
 SECTION XX. 
 
 PRACTICAL EXAMPLES. 
 
 1. What is the cost of If yds. broadcloth at 4 J dols. 
 a yard ? $ 
 
 2. What is the cost of 2£ yds. cloth at £ of a dollar 
 a yard ? 
 
 3. What is the value of 12 bbls. of flour at 4f dols. 
 a barrel ? 
 
 4. What is the value of 31 casks of lime at £ of a 
 dollar a cask ? 
 
 5. What is 5£ cwt. of beef worth at 4f dols. per 
 hundred weight? 
 
 6. What is the cost of 43£ bushels corn at f- of a 
 dollar a bushel ? 
 
 7. If a horse eat l£ bushel of oats in a week, how 
 much will he eat in 52 weeks ? 
 
 8. What will be the cost of the oats for 52 weeks 
 at | of a dollar a bushel ? 
 
 9. What part of a cwt. is f of a barrel of flour con- 
 taining 196 lbs. ? 
 
 10. What is the weight of f of a lot of hay weigh- 
 ing 4 T 3 <y tons ? 
 
 11. How many cubic feet are there in f of £ of a 
 cord of wood ? 
 
 12. A man sold f of a lot of wood, the whole of 
 which was 17f cords. How much did he sell ? 
 
 13. What part of 9 rods in length is 10 feet ? 
 
 14. What part of a square rod is 3 square yards ? 
 
 15. What part of a square rod is 5 square feet ? 
 
 16. What is 36 square feet of land worth at 9£ 
 dollars a* square rod? 
 
 17. How many gallons are there in i£ of a barrel 
 of wine ? 
 
 18. Two men bought a lot of hay for ll£ dollars. 
 
190 DECIMAL FRACTIONS. 
 
 One took 13 cwt. ; the other, the remainder, which 
 was 8£ cwt. What ought each to pay ? 
 
 19. Two men divided a lot of wood, which they 
 purchased together for 27£ dollars. One took 5£ 
 cords ; the other, 8 cords. What ought each to pay ? 
 
 20. The main-spring of a watch weighs about 1 
 dwt. 12 grs. Troy weight. Estimating its worth at £ 
 of a dollar, what would a pound Troy of steel be 
 worth, after it was manufactured into watch main- 
 springs, allowing nothing for waste in manufacturing? 
 
 21. A hair-spring of a watch weighs j- of a grain 
 Troy. Estimating its value at 3 cents, what would 
 be the value of 1 lb. Troy of steel, made into hair- 
 springs, allowing nothing for waste ? 
 
 22. Two men hired a horse one week for 6£ dol- 
 lars. One rode him 70 miles ; the other, 84. How 
 much ought each to pay ? 
 
 23. A stack of hay is bought by two men for 76£ 
 dollars, to be paid for in proportion to the amount of 
 hay each one takes. One takes 3£ tons ; the other, 
 the remainder, which was 2| tons. How much ought 
 each to pay ? 
 
 SECTION XXI. 
 
 DECIMAL FRACTIONS. 
 
 Addition and Subtraction. Sec. XII., Pt. I. 
 
 Rule. — Set down figures of the same order undei 
 each other, units under units, tenths under tenths, &c* 
 The figures in the answer will be, each, of the same 
 order with the perpendicular line of figures over it. 
 With this in mind, insert the point in its proper place. 
 
DECIMAL FRACTIONS. 191 
 
 Examples. 
 
 1. 24.5 + 68.3+17.14 + 87.96 + 3.125. 
 
 2. 165.3 + 96.45 + 8.43 1+ .641+ 9412.5. 
 
 3. 450.61+27.134 + 89.4216 + . 984. 
 
 4. 64.25 + 3.125 + 87.25+181.7. 
 
 5. 125. 17+ 34.27+. 125 + 3761.5. 
 
 6. 186.4 — 27.31; 800.4 — 21.67. 
 
 7. 34.21—18.525; 94.31—81.167. 
 
 8. 167.51—35.125; 204.5 — 31.09. 
 
 9. 20.41—3.817; 601.4—517.24. 
 
 10. 648.62 — .541; 346.4 — 91.324. 
 
 11. 5.1—1.324; .5 — .0067. 
 
 12. .81— .126; .94 — .3816. 
 
 Multiplication and Division. 
 
 Rule. — Perform the operation as in whole num- 
 bers ; and, in inserting the point, remember that every 
 product or dividend has as many decimal places as 
 both the factors which produce it. 
 
 13. 124.3X87; 321.67X24.3. 
 
 14. 97.125X6; 31.4X.125. 
 
 15. 37.5X.94; 18.4X64. 
 
 16. 21X.106; 312X.05. 
 
 17. 31. IX. 004; 18.61x.03. 
 
 18. 641X.41; 843.5X.95. 
 
 19. 184.2X.121; 35.6X.025. 
 
 20. .625X71; .875X31.5. 
 
 21. 84-4-. 012; 965-K 15. 
 
 22. 1.65 -M5; 846 -h 3.4. 
 
 23. 1640 -K 96; 425-^.055. 
 
 24. 1-K001; 2 -4-. 0002. 
 
 25. .001H-2; 384-K0012. 
 
 26. 96-K024; 64-^.016. 
 
 27. 1827^.9; 34-f-.17. 
 
192 REDUCTION OF FRACTIONS TO DECIMALS. 
 
 28. .634-8; .154-14. 
 
 29. .484-. 9; .334-16. 
 
 30. 1814-.41; 414-.6. 
 
 31. 354- .36; 484-.47. 
 
 32. .174-31; .264-.013. 
 
 33. 43 4-. 06; 45 4-. 003. 
 
 34. 75 4-. 125; .954-. 04. 
 
 35. .18 4-. 0045; 114- .34. 
 
 36. 9 4-. 0225; .74-. 035. 
 
 37. 804-.18; 514- .031. 
 
 38. .55X.031; 71.4X.13. 
 
 39. 8.44-.021; .65-T-.8. 
 
 40. 1.21X.09; .14X.03. 
 
 41. .64X.31; .08X.009. 
 
 42. 364-. 13; 284-11.4. 
 
 43. 40.1^-8; 644-.9. 
 
 44. 81.4 4- .03; 74-. 4. 
 
 45. 94- .5; 154-.7. 
 
 46. 80.2X.03; 164-.9. 
 
 47. 105.44-37.15. 
 
 48. 118.75 4-. 0044. 
 
 SECTION XXII. 
 
 REDUCTION OF VULGAR FRACTIONS TO DECIMALS. 
 
 [See Section XIII., Part I.] 
 
 Examples. 
 
 1. Reduce f to a decimal. 
 
 Rule. — Reduce the numerator to tenths, and divide 
 the result by the denominator, pointing off as required 
 
REDUCTION OF FRACTIONS TO DECIMALS. 193 
 
 in division of decimals. If there is a remainder, reduce 
 it to the next lower order, and divide again. 
 
 2. .Reduce f to a decimal. 
 
 3. Reduce to decimals, £ j £. 
 
 4. Reduce to decimals, T V > A J tV 
 
 5. Reduce to decimals, T \ ; ft ; if. 
 
 If the fractions are reducible to decimals without a 
 remainder, obtain the answer exactly. If they are 
 irreducible, obtain the proximate answer to four places, 
 and annex the fractional remainder. In order to know 
 if a fraction is exactly expressible in decimals, see Sec- 
 tion XIII. , Part I., as directed above. 
 
 6. Reduce to decimals, £f ; |J ; § J. 
 
 7. Reduce to decimals, ^ T ; u 9 T ; rfc. 
 
 8. Reduce to decimals, T £ ; y^V; ttt- 
 
 9. Reduce to decimals, ¥ Vt J Ht J M- 
 
 10. Reduce to decimals, T ^ ; T f F ; ^f T . 
 
 In ordinary transactions, it is usual to carry the deci- 
 mal answer to three or four places. The remainder is 
 then so small in value, that it may be dropped as of no 
 importance. At whatever place you stop, however, 
 the decimal obtained, and the fractional remainder, 
 when added together, will exactly equal the original 
 fraction. 
 
 11. In order to show this, we will take }. Redu- 
 
 7)10 ■ ;. 
 
 eing it, — we obtain, at the first step, 1 tenth 
 
 I~TT> 
 
 -H of 1 tenth. Adding these, ^ + T \ — fg- == f, which 
 is the original fraction. 
 
 We now carry the reduction one step farther: 
 7)100 
 
 — , 2 We obtain 14 hundredths + ? of a hun- 
 dredth. Adding these, tWf+t&tf = +&{* = *, the original 
 fraction. 
 
 17 „ 
 
194 REDUCTION OF DENOMINATE INTEGERS. 
 
 We will carry the redaction one step farther ; 
 7)1000 
 
 — — We obtain 142 thousandths + 1 of a 
 
 142 + f. ^ T . 
 
 thousandth. Adding these, by using the common 
 denominator 7000, AVs + *A* = $ U% = h the original 
 fraction. 
 
 12. Reduce T 2 T to a decimal of one figure, with the 
 remainder; carried to 2 places, with the remainder; 
 carried to 3 places, with the remainder. 
 
 13. Reduce T 4 F to a decimal of 7 places. 
 
 14. Reduce T 7 3 to a decimal of 9 places. 
 
 15. Reduce F 8 T to a decimal of 10 places. 
 
 Repeating and Circulating Decimals. 
 
 When a fraction is irreducible, the decimal figure 
 will either repeat, as, £ = .333 + ; or the decimal fig- 
 ures obtained by the partial reduction will, after a time, 
 recur again, in the same order as at first. Thus, T \ 
 gives .090909 +, and so on, without end. When the 
 same figure is repeated continually, it is called a repeat- 
 ing decimal ; when the same series of different figures 
 recurs, it is called a circulating decimal. 
 
 SECTION XXIII. 
 
 REDUCTION OF DENOMINATE INTEGERS TO 
 DECIMALS. 
 
 1. Reduce 5s. lid. to the decimal of a £. 
 
 First, reduce the quantity to the vulgar fraction of 
 a £. Then reduce that vulgar fraction to a decimal. 
 
 2. Reduce 3s. 2£d. to the decimal of a £. 
 
INTEGRAL VALUE OF DENOMINATE DECIMALS. 195 
 
 3. Reduce 5d. to the decimal of a guinea. 
 
 4. Reduce 3 qts. to the decimal of a bushel. 
 
 5. Reduce 2£ pints to the decimal of a gallon. 
 
 6. Reduce 3 feet 5 inches to the decimal of a rod. 
 
 7. Reduce 7 feet 8 inches to the decimal of a rod. 
 
 8. Reduce 15 rods 9£ feet to the decimal of a fur- 
 long. 
 
 9. Reduce 23 rods 13 feet to the decimal of a mile. 
 
 10. Reduce 5 hours 18 minutes to the decimal of 
 a day. 
 
 11. Reduce 21 hours 6 minutes to the decimal of a 
 week. 
 
 12. Reduce 12£ square rods to the decimal of an 
 acre. 
 
 SECTION XXIV. 
 
 TO FIND THE INTEGRAL VALUE OF DENOMINATE 
 DECIMALS. 
 
 1. What is the value of .7 of a rod? 
 
 Supposing the quantity was 7 rods, its value in feet 
 would be found by multiplying it by 16£; 16£X7 = 
 115£, or 115.5. But it was ftot 7 rods, but 7 tenths of 
 a rod, whose value we wish to find. The answer ob- 
 tained, therefore, is 10 times too large. Dividing by 
 10, it is 11.55=11 feet and 55 hundredths. In order 
 to find the value in inches of 55 hundredths of a foot, 
 we will call it 55 feet; the answer is, 55X12 = 660 = 
 660 feet. But, as we regarded the 55 as 100 times 
 greater in value than it is, the answer is 100 times too 
 large. Dividing it by 100, the answer is 6.60 inches, 
 = 6 inches and 60 hundredths or 6 tenths. 
 
 The above analysis shows the nature of the opera- 
 tion in all cases. 
 
196 PRACTICAL EXAMPLES. 
 
 2. What is the value, in feet and inches, of .3 of a 
 rod ? 
 
 3. What is the value of .94 of a rod? 
 
 4. What is the value of .26 of a rod? 
 
 5. How many shillings and pence are there in .65 
 of a£? 
 
 6. How many shillings and pence are there in .8 
 of a£? 
 
 7. How many pence are there in .7 of a shilling ? 
 
 8. How many pence are there in .16 of a shilling? 
 
 9. What is the value of .19 of a £ ? 
 
 10. What is the value of .74 of a bushel ? 
 
 11. What is the value of .9 of a bushel ? 
 
 12. What is the value, in rods and feet, of .7 of an 
 acre ? 
 
 13. What is the value of .9 of an acre ? . 
 
 14. What is the value of .12 of an hour ? 
 
 15. How many minutes and seconds in .15 of an 
 hour ? 
 
 16. Find the value of .34 of a week. 
 
 17. Find the value of .162 of a week. 
 
 18. Find the value of .84 of a minute. 
 
 19. How many feet in .761 of a cord? 
 
 20. How many feet and inches in .2 of a cord ? 
 
 21. How many feet in .74 of a cord ? 
 
 22. How many feet in .13 of a cord? 
 
 SECTION XXV. 
 
 PRACTICAL EXAMPLES. 
 
 1. Add $1.50 + $.375 + $.0625 + $.1875 + $5.00. 
 
 2. Add $34.75 + $6.00 + $.375 + $.08. 
 
 3. A man had $50, and spent $.375 of it. How 
 much had he left ? 
 
DECIMAL FRACTIONS. 197 
 
 4. A man had $10.00, and spent $.875 of it. How 
 much had he left ? 
 
 5. A watch cost $45,675; the chain and key, 
 $4,845. What did the whole cost ? 
 
 6. The owner then sold the watch, chain, and key, 
 for $48,375. How much did he lose? 
 
 7. A man set out on a journey with $10.00. The 
 first day, he spent $1,125. How much had he left? 
 
 8. The second day, he spent $1,425. How much 
 had he left? 
 
 9. The third day, he spent $1.67. How much 
 had he left ? 
 
 10. The fourth day, he spent $.875. How much 
 had he left ? 
 
 11. What is the cost of 21 11*5. of flour at $.05 
 per pound? 
 
 Why do you point off two decimals in the answer ? 
 
 12. What is the cost of 35 lbs. of flour at $.045 
 per pound ? 
 
 Why do you point off three decimals? 
 
 13. What is the cost of 12.5 lbs. of flour at $.05 
 a pound ? 
 
 14. What is the cost of 15.5 lbs. of flour at $.045 
 a pound ? 
 
 15. What is the cost of 26.25 lbs. of flour at $.0375 
 a pound? 
 
 16. What is the cost of 13.75 lbs. of flour at $.0425 
 a pound ? 
 
 17. What is the cost of 15 barrels of flour at $4.75 
 a barrel ? 
 
 18. What is the cost of 17.5 barrels of flour at 
 $5.25 a barrel? 
 
 19. What is the cost of 3 tons of hay at $7.56 a ton ? 
 
 20. What is the cost of 13.5 tons of hay at $9.00 
 a ton? 
 
 21. What are 17 barrels of cider worth at $1.75 
 a barrel ? 
 
 17* 
 
198 PRACTICAL QUESTIONS. 
 
 22. What cost 16 gallons of molasses at $.345 a 
 
 gallon ? 
 
 23. Divide $1.05 into 21 equal parts. What will 
 each part be ? 
 
 24. How many pounds of flour will $15.75 buy, at 
 $.045 a pound? 
 
 25. How many times is $.05 contained in $.625? 
 
 26. How many pounds of flour can be bought for 
 $.6975 at $.045 per pound? 
 
 27. How many times is $.0375 contained in 
 $984,375? 
 
 28. How many times is $.0425 contained in 
 $584,375? 
 
 29. How many barrels of flour will $71.25 buy, 
 at $4.75 per barrel? 
 
 30. How many barrels of flour will $91,875 buy, 
 at $5.25 per barrel? 
 
 31. How many tons of hay can be bought for 
 $22.68, at $7.56 per ton? 
 
 32. How many times is $9.00 contained in 
 $121.50? 
 
 33. A shipmaster paid $29.75 for ballast, giving 
 $1.75 a ton. How many tons did he buy? 
 
 34. How many times is $.345 contained in $5.52? 
 
 35. What cost 14 lbs. of flour at $.045 a pound, 
 and 28 lbs. of sugar at $.095 a pound? 
 
 SECTION XXVI. 
 
 PRACTICAL QUESTIONS IN VULGAR AND DECIMAL 
 FRACTIONS. 
 
 1. Bought 7 cwt. 15 lbs. sugar at $6.62£ per cwt., 
 and sold it at 7 cents per pound. What was the gain ? 
 
VULGAR AND DECIMAL FRACTIONS. 199 
 
 2. Bought 156 gallons of wine at 93 cents per gal- 
 lon, and sold it at 34 cents per quart. What was the 
 gain ? 
 
 3. Bought 7 cwt. 1 qr. 11 lbs. coffee at $12.50 per 
 cwt., and sold it at 14 cents per pound. What gain? 
 
 4. Bought 37 yards broadcloth at $5.25 per yard. 
 Sold 20 yards of it at $7.00 per yard, and the remain- 
 der at $6.31 per yard. What was the gaiu ? 
 
 5. Bought 24 yards broadcloth at $6.40 per yard. 
 Sold 22£ yards at $7.25 per yard, and the remnant for 
 5 dollars. What was the gain? 
 
 6. Bought 87 E. e. calico at 17 cents per E. e., and 
 sold it at 21 cents per yard. What gain ? 
 
 7. Bought 4 dozen books at $1.50 per dozen, and 
 sold them at 16 cents each. What gain ? 
 
 8. Bought 13 dozen brooms at $1.04 per dozen, 
 and sold them at 15 cents each. What gain? 
 
 9. Bought 5£ dozen mats at $3.40 per dozen, and 
 sold them at 36 cents each. What gain ? 
 
 10. Bought 17 bushels of salt at 65 cents per bushel, 
 and sold it at 21 cents per peck. What gain ? 
 
 11. Bought one barrel of wine at 78 cents per gal- 
 lon, and sold it at 16 cents per pint. What gain? 
 
 12. Bought 3 dozen baskets at $2.05 per dozen, 
 and sold 1 dozen at 31 cents, 1 dozen at 37 cents, and 
 1 dozen at 42 cents each. What gain? 
 
 13. Bought 48 yards broadcloth at $5.62 per yard. 
 Lost 17 yards by fire, and sold the remainder at $6.25 
 per yard. How much gain or loss ? 
 
 14. Bought a hogshead molasses, containing 131 
 gallons, at 34 cents per gallon. 16 gallons leaked out. 
 Sold the remainder at 37 cents per gallon. What gain 
 or loss ? 
 
 15. Bought 3£ dozen axes at $6.80 per dozen, and 
 sold them at 92 cents each. What gain ? 
 
 16. Bought 7 dozen pails at $1.42 per dozen, and 
 sold them at 21 cents each. What gain ? 
 
200 REDUCTION OF CURRENCIES. 
 
 17. Bought 8£ dozen shovels at $9.25 per dozen, 
 and sold them at $1.00 each. What gain? 
 
 18. Bought 74 yards carpeting at 73 cents per yard, 
 and sold it at 87£ cents per yard. What gain ? 
 
 19. Bought 164 bushels corn at 54 cents per bushel. 
 Sold 93 bushels at 67 cents, and the remainder at 50 
 cents, per bushel. How much loss or gain? 
 
 20. Bought 75 barrels apples at $1.37 per barrel. 
 Lost 15 barrels by decay, and sold what remained at 
 $2.12 per barrel. What loss or gain? 
 
 21. Bought 13 dozen oranges at 7 cents per dozen. 
 Lost by decay 2£ dozen, and sold the remainder at 2£ 
 cents each. What gain ? 
 
 22. Bought 15 dozen pairs of shoes at $4.87 per 
 dozen, and sold them at 63 cts. per pair. What gain ? 
 
 23. Bought 18£ thousand of boards at $9.50 per 
 thousand. Sold 6 thousand at $12.25 per thousand, 
 and the remainder at $8.42 per thousand. What gain? 
 
 24. Bought 21£ cords wood at $4.75 per cord. 
 Sold 8 cords at $5.50 per cord, and the remainder at 
 $4.25 per cord. What gain or loss? 
 
 25. Bought 209 bushels apples at 27 cents per 
 bushel. Sold 46 bushels at 49 cents per bushel, and 
 the remainder at 25 cents per bushel. What gain ? 
 
 SECTION XXYII. 
 
 REDUCTION OF CURRENCIES. 
 English Currency. 
 
 1. Reduce 67 £ to dollars and cents. 
 As 4s. 6d. or 54 d. = $1.00, (see Table, p. 40,) and 
 20 s. or 240 d, = 1 £, 1 dollar is ^fc of a £. Reducing 
 
REDUCTION OF CURRENCIES. 201 
 
 this fraction to its lowest terms, it is T 9 D . The ques- 
 tion, therefore, is this: In 67 £ how many -^ of a £? 
 Dividing 67 by the fraction, we have 297£ dollars 
 for the answer. The fraction £ gives 77 cents and 
 7 mills. 
 
 2. Reduce 87 £ to dollars and cents. 
 
 3. Reduce 104 £ to dollars and cents. 
 
 4. Reduce 64 £ to dollars and cents. 
 
 5. Reduce 167 £ to dollars and cents. 
 
 6. Reduce 520 £ to dollars and cents. 
 
 7. Reduce 84 £ 6 s. to dollars and cents. 
 
 First reduce the 6 s. to the decimal of a £ j ^ = .3. 
 The sum then is, 84.3 £. Reduce it in the same way 
 as the cases above. 
 
 8. Reduce 124 £ 13 s. to Federal money. 
 
 9. Reduce 36 £ 9 s. 6 d. to Federal money. 
 
 10. Reduce 71 £ 18s. 4d. to Federal money. 
 
 To Reduce Federal Money to Sterling. 
 
 11. In 684 dollars how many pounds, shillings, and 
 pence ? 
 
 As * 1.00 = A of a £, 1£=- 4 / of $1.00. The 
 question therefore is, In 684 dollars how many 4 ^°- of 
 $1.00? Dividing by the fraction, we have for the 
 answer, £153.9, or 153£ 18s. 
 
 12. In $74.25 how many pounds, shillings, and 
 pence ? 
 
 13. Reduce $186.40 to Sterling money. 
 
 14. Reduce $564.35 to Sterling money. 
 
 15. Reduce $640.15 to Sterling money. 
 
 The comparative value of the dollar and the pound 
 sterling, as given above, is called the nominal par 
 value. The actual value of the pound is higher than 
 is here given. This difference is usually estimated in 
 
202 REDUCTION OF CURRENCIES. 
 
 trade by adopting the nominal par value, given above, 
 as the basis of the calculation, and then adding or 
 subtracting a certain per cent., as 8 or 10 per cent., to 
 compensate for the inequality of value. 
 
 Canada Currency. 
 5s. = 60d.= *1.00. 
 
 16. In 74 £ 15 s. how many dollars and 'cents? 
 
 As * 1.00 = 60 d., and l£=240d. $1.00 is Jft, or 
 I of a pound ; multiplying by 4, the answer is 
 $299.00. 
 
 17. In £ 126 12 s. Canada currency how many dol- 
 lars and cents ? 
 
 18. Reduce $841.50 to Canada currency. 
 
 / New England Currency. 
 
 6s.=72d. = $1.00. 
 
 19. In 64 £ 8s. how many dollars and cents? 
 $1.00 = ^= T \ of a £. Reduce the 8s. to a 
 
 decimal of a pound, and divide by the fraction ; we 
 have $214.66f. 
 
 20. Reduce 120 £ 12 s. 6d. to Federal money, 
 
 New York Currency. 
 8s. = 96d.= $1.00. 
 
 21. Reduce 146 £ 6 s. 4d. to Federal money. 
 
 As $1.00 = 2Vff = fV of a pound, reducing the shil- 
 lings and pence to the decimal of a pound, and divid- 
 ing by the fraction, we have $365.75. 
 
 22. Reduce 54 £ 10 s. 6d. to Federal money. 
 
 Pennsylvania Currency. 
 7s. 6d. = 90d.= *1.00. 
 
 23. Reduce 16 £ 5 s. 6d. to Federal money. 
 $1.00 = 2 9 ^=f of a £. 
 
 24. Reduce 7£ 8 s. 9d. to Federal money. 
 
INTEREST. 203 
 
 Miscellaneous Examples. 
 
 25. Reduce 187 £ 8 s. sterling to Federal money. 
 
 26. Reduce 964 £ 16 s. sterling to Federal money. 
 
 27. Reduce 1643 dollars to Sterling money. 
 
 28. Reduce 1600 dollars to Sterling money. 
 
 29. Reduce 167 £ 14s. Canada, to Federal money. 
 
 30. Reduce $196.50 to Canada currency. 
 
 31. Reduce $1674.40 to New England currency. 
 
 32. Reduce $744.15 to New York currency. 
 
 33. Reduce 142 £ 14s. 6d. New York, to New 
 England currency. 
 
 34. Reduce 643 £ 15 s. 9d. Pennsylvania currency 
 to Federal money. 
 
 35. Reduce $172.31 to Pennsylvania currency. 
 
 SECTION XXVIII. 
 
 INTEREST. 
 [See Section XIV., Part I.] 
 
 Rule. — Find the interest of 1 dollar for the given 
 time ; multiply the principal by it, and point off as in 
 the multiplication of decimals. 
 
 1. What is the interest of $156.34 for 11 months 
 
 and 20 days ? 
 
 As the interest of 1 dollar for 2 
 months is 1 cent, for 10 months it 
 will be 5 cents, .05. As the inter- 
 est of 1 dollar for 6 days is 1 mill, 
 for 30 days it will be 5 mills, and 
 for 20 days 3 mills and £, making 
 8 mills and £ Set down the 8 at I $9.11983, Ans. 
 the right hand of the .05, and for the £ divide by 3. 
 
 3)156.34 
 
 .058 
 
 125072 
 
 78170 
 5211 
 
204 INTEREST. 
 
 Observe, the before the 5 must be retained ; other- 
 wise it would be 5 tenths of a dollar, or 50 cents, and 
 the answer would be 10 times too great. If there are 
 no cents, there must be two ciphers at the left hand of 
 the mills. The number of cents for the multiplier is 
 always equal to half the greatest even number of 
 months ; the number of mills is one sixth of all the 
 days over and above the greatest even number of 
 months. 
 
 2. Interest of $384.18 for 7 months and 10 days? 
 
 3. Interest of $147.19 for 5 months 15 days? 
 
 4. Interest of $568.25 for 9 months 13 days? 
 
 5. Interest of $81.40 for 10 months 14 days? 
 
 6. Interest of $56.32 for 12 months 24 days? 
 
 7. Interest of $75.30 for 14 months 18 days? 
 
 8. Interest of $644.46 for 15 months 24 days? 
 
 9. Interest of $831.00 for 1 year, 4 months, 12 
 days? 
 
 10. Interest of $380.00 for 1 year 7 months? 
 
 11. Interest of $500.00 for 1 year, 5 months, 6 
 days? 
 
 12. Interest of $27.42 for 4 months 17 days? 
 
 13. Interest of $13.18 for 6 months 23 days? 
 
 14. Interest of $1000.00 for 5 months 4 days? 
 
 15. Interest of $65.48 for 30 days, or 1 month? 
 
 16. Interest of $94.00 for 30 days? 
 
 17. Interest of $840.60 for 18 days? 
 
 18. Interest of $632.00 for 18 days? 
 
 19. Interest of $349.40 for 12 days? 
 
 20. Interest of $267.62 for 12 days? 
 
 21. Interest of $384.92 for 15 days? 
 
 22. Interest of $811.19 for 20 days ? 
 
 23. Interest of $673.94 for 5 months 11 days? 
 
 24. Interest of $460.00 for 8 months 18 days? 
 
 25. Interest of $460.00 for 8 months 18 days, at 12 
 per cent. ? 
 
 26. Interest of $460.00 for 8 months 18 days, at 8 
 per cent. ? 
 
INTEREST. 205 
 
 27. Interest of $460.00 for 8 months 18 days, at 7 
 per cent. ? 
 
 28. Interest of $460.00 for 8 months 18 days, at 5 
 per cent. ? 
 
 29. Interest of $1500.00 for 15 months, at 4 per 
 cent. ? 
 
 30. Interest of $145.80 for 7 months 11 days, at 8 
 per cent. ? 
 
 31. Interest of $341.18 for 2 years, 9 months, 18 
 days? 
 
 As the interest of a dollar for 30 days is 5 mills, for 
 | of 30 days, or 6 days, it is 1 mill. As 1 mill is T sW) 
 one thousandth of a dollar, it follows that the interest 
 of 1 dollar for 1 day is one sixth of a thousandth, or 
 stjW °f a dollar. For two days, therefore, it will 
 be sttW, for 1 5 days s^vi of a dollar. 
 
 A convenient rule, therefore, when the time is 
 short, is the following : — 
 
 Multiply the sum by the number of days, and divide 
 the product by 6000. 
 
 This is often the shortest method. You divide by 
 1000, by removing the decimal point three places to 
 the left. It only remains, then, after doing this, to 
 multiply by the number of days, and divide by 6. 
 
 32. What is the interest of $348.25 for 18 days? 
 Dividing by 1000, you have $0.348£, thirty-four 
 
 cents eight mills and a quarter. Instead, now, of mul- 
 tiplying by 18 and dividing by 6, you may multiply 
 by 3, for 18 is 3 times 6. 
 
 3 times 0.348] is $1.044f, Arts. 
 
 33. Interest of $725.80 for 24 days? 
 
 34. Interest of $341.18 for 36 days? 
 
 35. Interest of $67.45 for 54 days? 
 
 36. Interest of $641.18 for 42 days? 
 
 37. Interest of $84.16 for 15 days? 
 
 18 
 
206 PARTIAL PAYMENTS. 
 
 » 
 
 To find the amount, add the interest to the principal j 
 or, find the amount of $ 1.00 for the given time, and 
 multiply the principal by it. 
 
 38. What is the amount of $560.50 for 8 months 
 12 days ? 
 
 39. Amount of $964.25 for 15 months 18 days? 
 
 40. Amount of $460.00 for 1 year 6 months? 
 
 41. Amount of $120.50 for 2 years 4 months? 
 
 42. Amount of $68.40 for 1 year, 6 months, 24 
 days? 
 
 43. Amount of $500.00 for 2 years 3 months? 
 
 44. Amount of $730.50 for 6 months 12 days? 
 
 45. Amount of $840.25 for 4 months 18 days? 
 
 46. Amount of $40.50 for 8 months 12 days? 
 
 SECTION XXIX. 
 
 PARTIAL PAYMENTS. 
 
 When partial payments are made on a note, the 
 amount due on the final payment of the note may be 
 found by the following rule : — 
 
 Find the interest on the note up to the time of the 
 first payment. If the payment exceeds the interest, 
 deduct it from the amount, regarding the remainder as 
 a new principal. On this, calculate the interest to the 
 time of the next payment ; and so on. If any payment 
 is less than the interest then due, reserve it, and com- 
 pute the interest on to the time when the payments, 
 added together, shall exceed the interest due. Then 
 subtract the. sum of the payments from the amount 
 then due, and proceed as before. 
 
PARTIAL PAYMENTS. 207 
 
 1. A note of 200 dollars is given July 1, 1834, on 
 which are the following partial payments : — 
 
 Dec. 15, 1834, . . $25.00. 
 
 March 1, 1835, 2.50. 
 
 Aug. 10, 1835, . . . 45.00. 
 What was due Dec. 31, 1835 ? 
 
 2. A note of $340.25 is given Aug. 1, 1840. 
 Endorsements, — Jan. 10, 1841, . .$28.40. 
 
 July 1, 1841, 9.00. 
 
 March 14, 1842, . . 74.00. 
 What was due Jan. 1, 1843? 
 
 3. A note of $480.00 is given June 9, 1841. 
 Endorsements, — Sept. 11, 1842, . $60.00. 
 
 Jan. 3, 1843, 95.00. 
 
 March 12, 1844, . 100.00. 
 What was due Dec. 1, 1844 ? 
 
 4. A note of $675.40 is given July 3, 1843. 
 Endorsements, — Jan. 4, 1844,. . . $65.00. 
 
 April 17, 1844, . . 29.50. 
 Nov. 18, 1844,.. . 74.00. 
 What is due Jan. 1, 1845 ? 
 
 5. A note of $345.40 is given April 1, 1843. 
 Endorsements, — Dec. 1, 1843,. . .$40.00. 
 
 - June 10, 1844, . . . 90.00. 
 
 Oct. 4, 1844, 31.50. 
 
 Feb. 6, 1845, 17.00. 
 
 What is due Jan. 1, 1846 ? 
 
 The rule given above is the legal rule. When, 
 however, the note is paid within a year from the time 
 when' it was given, the following rule is usually em- 
 ployed : — 
 
 Find the amount of principal and interest of the 
 whole note, from the time it was siven till the final 
 
208 
 
 ANNUAL INTEREST. 
 
 payment ; find the amount of each payment, from the 
 time it was paid till the final payment ; and the sum 
 of these amounts subtract from the amount of the 
 whole note. The remainder will be the balance due. 
 
 6. A note of $525.00 is given Sept. 1, 1844. 
 Endorsements, — Dec. 30, 1844, . . $58.75. 
 
 March 4, 1845, . . 104.20. 
 
 June 8, 1845, .... 63.40. 
 What is due Aug. 21, 1845? 
 
 ' 7. A note of $784.50, given July 7, 1844, has the 
 following endorsements : — 
 
 Sept. 5, 1844,. . .$54.00. 
 
 Nov. 10, 1844, . . . 60.00. 
 
 Jan. 12, 1845, 75.00. 
 
 March 17, 1845, . 100.00. 
 What is due May 1, 1845 ? 
 
 ANNUAL INTEREST. 
 
 When a note is given payable at a longer period 
 than a year from the date, it is usual to express in the 
 note that the interest shall be paid annually. At the 
 end of a year, the holder cf the note may compel the 
 payment of the interest. In such cases, the debtor, 
 instead of paying the interest that is due, sometimes 
 renews the note, adding the interest to the principal. 
 Thus, at the end of each year, the interest due is added 
 in, and goes to make a new principal for the following 
 year. This is called Compound Interest; but the 
 computation of it is the same as in simple interest ; 
 for, if the interest is not computed every year, and 
 either paid or put into the note by renewal, that in- 
 terest cannot draw interest.* The law regards it as 
 
 * In some of the states, the interest, after it falls due, draws simple 
 interest till it is paid. 
 
ANNUAL INTEREST. 209 
 
 the duty of the creditor to remind the debtor of his 
 debt, by exacting the payment of the interest every 
 year. If he does not do this, he can derive no advan- 
 tage from the promise in the note to pay the interest 
 . annually. 
 
 ILLUSTRATION. 
 
 
 $100.00. Boston, March 1, 1845. 
 
 For value received, I promise to pay to John Jones, 
 or order, one hundred dollars, in five years, with inter- 
 est annually. a 
 
 J Samuel Barton. 
 
 If John Jones does not exact the interest till the 
 end of the five years, and if he obtains no renewal of 
 it, the amount of the note will be only $ 130.00 ; for 
 the interest of 100 dollars, for five years, is 30 dollars. 
 
 If, however, he obtains a renewal of the note at the 
 end of each year, the principal of the note, for the 
 second year, will be $106.00. 
 
 8. What will the principal of the note for the third 
 year be ? 
 
 9. What will the principal of the note for the fourth 
 year be ? 
 
 10. What will the principal of the note for the fifth 
 year be ? 
 
 11. What will be due, principal and interest, at the 
 end of the fifth year? 
 
 12. How much would the holder of the above note 
 lose by omitting to obtain any renewal of it, or any 
 payment of annual interest ? 
 
 $250.00. New York, July 1, 1845. 
 
 For value received, I promise to pay John Foss, or 
 
 order, two hundred and fifty dollars, in four years, 
 
 with interest annually. , ~ 
 
 18 * Amos Carr. 
 
210 DISCOUNT. 
 
 13. If no interest is paid on this note till the prin- 
 cipal is due, and if no renewal of the note is made, 
 what will be the amount of the note at the time of 
 payment ? 
 
 14. If the note is renewed each year, what will be 
 the principal of the note for the second year ? 
 
 15. What will be the principal of the note for the 
 third year ? 
 
 16. What will be the principal of the note for the 
 fourth year ? 
 
 17. What will be the amount of the last note at 
 the time of payment ? 
 
 18. How much would the holder, John Foss, lose, 
 by neglecting to obtain any annual payment of inter- 
 est, or any renewal of the above note ? 
 
 SECTION XXX. 
 
 DISCOUNT. 
 
 [See Section XIV., Part I.] 
 
 Examples. 
 
 1. What is the present worth of $475.50, payable 
 in 3 months ? 
 
 Rule. — Deduct from the given sum its interest for 
 the specified time. The remainder will be the present 
 worth. 
 
 2. What is the present worth of $341.00, payable 
 in 65 days ? 
 
 3. Present worth of $940.25, payable in 4 months? 
 
 4. Present worth of $156.30, payable in 96 days? 
 
 5. Present worth of $312.60, payable in 35 days? 
 
BANKING. 
 
 211 
 
 6. Present worth of $500.00, payable in 41 days? 
 
 7. Present worth of $814.67, payable in 65 days? 
 
 8. Present worth of $46.30, payable in 20 days? 
 
 9. Present worth of $124.45, payable in 5 months? 
 10. Present worth of $360.20, payable in 4^ months ? 
 
 SECTION XXXI. 
 
 BANKING. 
 
 [See Section XIV., Part I.] 
 
 To find the present worth of a note given to a 
 bank, payable at some future time, find the present 
 worth of 1 dollar for the given time, and multiply 
 the sunt named in the note by it. 
 
 1. What is the present worth of a note for 100 dol- 
 lars, discounted at a bank, for 60 days ? 
 
 Interest of 1 dollar for 63 days is .0105. This 
 subtracted from 1 dollar, leaves for the present worth 
 .9895. 
 
 2. What is the present worth of a note for $450.00, 
 discounted at a bank, for 90 days ? 
 
 3. I give my note to a bank for $250.00, for 60 
 days. What do I receive ? 
 
 4. I give my note to a bank for $520.00, for 120 
 days. What do I receive ? 
 
 5. Present worth of a bank note for $600.00, dis- 
 counted for 60 days ? 
 
 6. Present worth of a bank note for $150.00, dis- 
 counted for 120 days ? 
 
 7. Present worth of a bank note for $75.00, dis- 
 counted for 30 days? 
 
212 LOSS AND GAIN. PER CENTAGE. 
 
 8. Present worth of a bank note for -$1000.00, 
 discounted for 60 days? 
 
 9. Present worth of a bank note for $560.00, 
 discounted for 120 days? 
 
 10. Present worth of a bank note for $150.00, 
 discounted for 30 days? 
 
 To find what must be the face of a note given to 
 a bank, in order to obtain a certain sum, — Find the 
 present ivorth of 1 dollar for the given time, and di- 
 vide the sum you wish to obtain by it. The quotient 
 will express the sum that must be named in' the note. 
 This, you observe, is just the reverse of the preceding 
 case. 
 
 11. For what sum must I give my note to a bank, 
 payable in 60 days, in order to receive $98.95? 
 
 12. For what sum must I give my note to a bank, 
 payable in 120 days, in order to receive $509.34? 
 
 13. For what sum must I give my note to a bank, 
 payable in 60 days, in order to receive $593.70? 
 
 14. For what sum must I give my note to a bank, 
 payable in 30 days, in order to receive $10000.00? 
 
 SECTION XXXII. 
 
 LOSS AND GAIN. — PER CENTAGE. 
 
 [See Section XIV., Part I.] 
 
 1. A man bought a horse for 75 dollars, and sold 
 him for $82.50. What did he gain per cent.? 
 
 2. A man bought a chaise for $178.00, and sold it 
 for $154.50. What did he lose per cent. ? 
 
 3. A merchant bought a lot of flour at $4.62 a 
 
PER CENTAGE. 213 
 
 barrel, and sold it at $5.15 a barrel. What was 
 his gain per cent. ? 
 
 4. A merchant bought a piece of broadcloth for 
 $4.30 per yard. What must he sell it for to gain 
 12 per cent. ? 
 
 5. A man has $1200.00 invested in a manufactory. 
 He receives, for his half-yearly dividend, 30 dollars. 
 What per cent, is that on his stock ? 
 
 6. A merchant fails, owing $8540.00, and can pay 
 but $2700.00. How much will that be on a dollar? 
 
 7. A man, failing in business, agrees to pay his 
 creditors 87 cents on a dollar. What must a cred- 
 itor receive whose claim is $740.30? 
 
 The pupil should be encouraged habitually to reason 
 upon the operations he performs ; so that his method 
 of procedure may be suggested by the relations of the 
 numbers, and not dictated by a special rule. To aid 
 in this important habit, a few remarks will be made 
 on some of the foregoing examples. These may 
 serve as specimens of analysis, and suggest to the 
 student a similar course of reasoning in other cases. 
 
 Example 1. The whole gain is $7.50. If this 
 gain were made on an outlay of 1 dollar, the gain 
 would be seven hundred and fifty per cent. But the 
 gain is made on an outlay of 75 dollars. The gain per 
 cent., therefore, is one seventy-fifth of the whole gain. 
 
 Example 4. If the cost was 1 dollar a yard, he 
 must add 12 cents ; if 2 dollars, he must add 24 
 cents; &c. 
 
 Example 5. If 30 dollars had been the gain upon 
 1 dollar, it would have been 30 hundred per cent. 
 But the gain was upon 1200 dollars. The per cent., 
 therefore, must be one twelve-hundredth of 30 dollars. 
 
 8. I invest in a factory 1260 dollars, and receive 
 for my yearly dividend 86 dollars. What is that 
 per cent.? 
 
214 PER CENTAGC. 
 
 9. I purchase flour at $4.75 per barrel. What 
 must I sell it for to gain 12 per cent.? 
 
 10. A merchant bought a ship for 11475 dollars, and 
 sold her for $13680. What did he gain per cent. ? 
 
 11. The population of the state of New York, in 
 1810, was 959949. In 1820, it was 1372812. What 
 was the gain per cent, in that term of 10 years? 
 
 12. In 1830, it was 1918604. What was the gain 
 per cent, from 1820 to 1830? 
 
 13. In 1840, it was 2428921. What was the gain 
 per cent, from 1830 to 1840? 
 
 14. The population of Ohio, in 1810, was 230760. 
 In 1820, it was 581434. What was the gain per 
 cent. ? 
 
 15. The population of Ohio, in 1830, was 937903. 
 What was the gain per cent, from 1820 to 1830? 
 
 16. In 1840, it was 1519467. What was the gain 
 per cent, from 1830 to 1840? 
 
 17. Massachusetts had, in 1810, 472040 inhabitants. 
 In 1820, it had 523287. What was the gain per cent, 
 in 10 years? 
 
 18. Massachusetts had, in 1830, 610408 inhabitants. 
 What was her gain per cent, from 1820 to 1830? 
 
 19. In 1840, Massachusetts had 737699 inhabitants. 
 What was the gain per cent, from 1830 to 1840? 
 
 20. An agent sells 12000 dollars' worth of cloth for 
 a factory, charging 2| per cent, commission. What 
 will be his remuneration? 
 
 21. If I buy for a merchant, at a commission of 4 
 per cent., 500 barrels of flour, at $4.40 per barrel, 
 what am I entitled to for my commission ? 
 
 22. What is 3 per cent, on $674.54? 
 
 23. What is 2 per cent, on $781.50? 
 
 24. What is the value of five 100 dollar shares in a 
 bank, at 4£ per cent, advance? 
 
 25. What is the value of seven 100 dollar shares, 
 at 6 per cent, discount ? 
 
PER CENTAGE. 215 
 
 26. What is the value of 18 shares bank stock, 60 
 dollars a share, at 4 per cent, discount ? 
 
 27. What is the duty on a quantity of broadcloth, 
 whose value is 1735 dollars, at 15 per cent. ? 
 
 28. What is the duty on a quantity of iron, whose 
 value is 3456 dollars, at 18 per cent.? 
 
 29. What is the commission on the sale of 1246 
 dollars' worth of cloth, at 3 per cent. ? 
 
 30. A man bought a lot of hay for 13 dollars a ton. 
 He sold it for $ 14.25 a ton. What did he gain per 
 cent. ? 
 
 31. Bought tea for 46 cents a pound. What must 
 I sell it for a pound to gain 12 per cent. ? 
 
 32. What is the worth of 750 dollars, bank stock, 
 at 7£ per cent, advance ? 
 
 33. What is the worth of 8500 dollars, bank stock, 
 at 9 per cent, discount? 
 
 34. I sell flour at $5.32 per barrel, and thereby 
 gain 12 per cent, on my outlay. What did the flour 
 cost? 
 
 Every $1.00 laid out in the purchase has brought 
 me a return of $1.12. The number of dollars I paid 
 out on a barrel must therefore equal the number of 
 times $1.12 will go in $5.32. 
 
 35. A merchant sells a ship for 13680 dollars, gain- 
 ing thereby 14-& per cent, on what she cost him. 
 What did the ship cost ? 
 
 36. 300 dollars is 2£ per cent, on what sum? 
 
 37. $15.63 is 2 per cent, on what sum? 
 
 38. Bought 12 barrels of flour, each containing 196 
 pounds, at $5.42 per barrel, and sold it at 26 cents for 
 7 pounds. How much gain in the whole, and how 
 much gain per cent.? 
 
 39. Bought 43 dozen pairs of shoes, at $4.30 per 
 dozen, and sold them at 62 cents per pair. What gain 
 in all? What gain per cent.? 
 
216 ALLIGATION. 
 
 40. Bought 20 barrels of apples, each containing 
 2f bushels, at $2.10 per barrel, and sold them at 
 $1.25 per bushel. What gain in all? What gain 
 per cent.? 
 
 41. Bought 375 barrels of flour, at $5.20 per 
 barrel, and sold 200 barrels at $6.10; the remainder 
 at $6.42 per barrel. What gain in all? What gain 
 per cent. ? 
 
 42. Bought 34 acres of land, at 41 dollars per acre. 
 Sold it for $1700.00. How much gain in all? What 
 gain per cent. ? 
 
 SECTION XXXIII. 
 
 ALLIGATION.* 
 
 The operations under this rule show the method of 
 rinding the value of a mixture, when the price and 
 quantity of each of its ingredients are given ; also, to 
 find the quantity of each ingredient, when its price is 
 given, and it is required to unite them so as to form a 
 mixture of a given value. 
 
 Case 1. — To find the value of the mixture, when 
 the quantity and -price of each of the ingredients 
 are given. 
 
 1. Mix 15 bushels of oats, at 40 cents per bushel ; 
 12 bushels of barley, at 60 cents; and 24 bushels of 
 
 * The word alligation signifies a tying together, and has reference 
 to a particular way of linking numbers together, by means of which 
 operations of this kind have been performed. The name is retained 
 as a matter of convenience ; but I have thought it best for the prog- 
 ress of the pupil, that he should pursue a strictly analytical method 
 in all the operations. 
 
ALLIGATION. 217 
 
 corn, at 83 cents. What will the mixture be worth 
 per bushel? 
 
 It is evident that, if you find the value of the whole, 
 and divide the sum by the number of bushels, the 
 quotient will be the value per bushel. 
 
 r 2. Mix 20 pounds of tea, at 43 cents per pound ; 
 18 lbs. at 61 cents ; and 11 lbs. at 74 cents per pound. 
 What will the mixture be worth? 
 
 3. If 41 lbs. of coffee, at 13 cents per lb., be mixed 
 with 45 lbs. at 9£ cents, and 27 lbs. at 15 cents, 
 what will the mixture be worth per pound? 
 
 Case 2. — To find the quantity of each ingredient, 
 when its price and that of the required mixture are 
 given. 
 
 4. If I mix oats, worth 2 s. per bushel, with rye, 
 worth 5 s., so as to make the mixture worth 3 s. per 
 bushel, in what proportion must I mix them ? 
 
 It is evident, that, if I put in 1 bushel of oats, I 
 gain 1 shilling. Now, I must put in rye enough with 
 this bushel of oats to lose 1 shilling. On every bushel 
 of rye put in, I lose 2 shillings ; therefore, in order to 
 lose 1 shilling, I must put in £ a bushel. I must 
 therefore put in 1 bushel of oats to £ a bushel of rye. 
 It is evident that, if I double the quantity thus found 
 of each ingredient, the value of the mixture will be 
 the same ; or I may take any equal multiples of the 
 quantities, as 4 bushels of oats and 2 bushels of rye, 
 6 bushels of oats and 3 bushels of rye, 20 bushels of 
 oats and 10 bushels of rye, &c. 
 
 5. If I mix oats, worth 2 s. per bushel, with rye, 
 worth 6 s., so as to make the mixture worth 3 s. per 
 bushel, in what proportion must they be mixed ? 
 
 6. Mix oats, worth 3 s. per bushel, with wheat, 
 worth 7 s., so as to make the mixture worth 5 s. per 
 bushel. In what proportion must they be mixed ? 
 
 19 
 
218 ALLIGATION. 
 
 7. Mix the same ingredients, at the same price, so 
 as to make the mixture worth 6 s. per bushel. In 
 what proportion must they be mixed? 
 
 8. In what proportion must oats, worth 2 s., and 
 wheat, worth 8 s., be mixed, to make the mixture 
 worth 4s. per bushel? 
 
 9. How can you mix corn, worth 80 cents per bushel, 
 and rye, worth 85 cents, with barley, worth 46 cents, 
 so as to make a mixture worth 60 cents per bushel ? 
 
 Here you have three ingredients. First, mix barley 
 with one of the dearer ingredients, so as to make a 
 mixture of the required value. Then mix barley 
 with the other ingredient, and see how much you 
 have taken of each. 
 
 10. Mix 3 sorts of tea, at 25 cents, 33 cents, and 40 
 cents, per pound, so as to make a mixture worth 30 
 cents per pound. 
 
 11. Mix tea at 20 cents, with tea at 45 cents, and 
 tea at 54 cents, per pound, so as to make a mixture 
 worth 38 cents per pound. 
 
 12. If you mix sugar, at 6 cents, 8 cents, 10 cents, 
 and 11 cents, per pound, in what quantities may they 
 be taken so as to make a mixture worth 9 cents per 
 pound ? 
 
 First, take two of the ingredients, one cheaper and 
 one dearer than the mixture. Form a mixture of 
 these. Then take the two remaining ingredients in 
 the same way. 
 
 13. If three sorts of spirit, worth 60 cents, 75 cents, 
 and 80 cents, per gallon, are mixed with water, costing 
 nothing, what must be the proportion to make a mix- 
 ture worth 70 cents per gallon ? 
 
 It is immaterial in what way you select the pairs of 
 ingredients, provided, in each pair, one of the ingre- 
 dients be cheaper and the other dearer than the 
 
ALLIGATION. 219 
 
 required mixture. Thus a great variety of answers 
 may be obtained whenever there is more than one 
 pair of ingredients. In all cases, however, the cor- 
 rectness of the operation may be proved in the follow- 
 ing way : — 
 
 Find the total value of all the ingredients. If this 
 is equal to the value of the whole mixture at the 
 required price, the work is right. 
 
 14. Mix 5 sorts of grain, at 25 cents, 30 cents, 33 
 cents, 45 cents, and 50 cents, so as to make a mixture 
 worth 40 cents, per bushel. 
 
 Case 3. — When the quantity of one ingredient is 
 given. 
 
 15. Mix brandy, at 74 cents per gallon, with 24 
 gallons of brandy, at 1 dollar per gallon, so that the 
 mixture may be worth 80 cents per gallon. 
 
 Here you observe that the quantity of one of the 
 ingredients is given. We will first make a mixture of 
 the two, without regard to this circumstance. If I put 
 in 1 gallon at 1 dollar, I lose 20 cents. For every 
 gallon at 74 cents, which is put in, I gain 6 cents. In 
 order to gain 20 cents, I must, therefore, put in 3-£ 
 gallons. The quantities stand, then, 1 gallon at 1 
 dollar, 3£ gallons at 74 cents. But I wish to put in 
 24 gallons at 1 dollar. To balance this, I must there- 
 fore put in 24 times 3£ gallons at 74 cents ; that is, 80 
 gallons. 
 
 16. Mix sugar, at 8 cents, 11 cents, and 12 cents, 
 with 100 lbs. of sugar at 7 cents, so as to make the 
 mixture worth 10 cents, per pound. 
 
 Case £. — When the quantity of the required mix- 
 ture is given. 
 
 17. Mix oats, at 40 cents, with corn, at 60 cents, so 
 
220 EQUATION OF PAYMENTS. 
 
 as to form a mixture of 100 bushels, worth 48 cents 
 per bushel. 
 
 If I put in 1 bushel at 40 cents, I gain 8 cents ; if I 
 put in 1 bushel at 60 cents, I lose 12 cents. To lose 
 8 cents, therefore, I must put in only f of a bushel. 
 The quantities are, then, 1 bushel at 40 cents, § of a 
 bushel at 60 cents ; making, when added, If bushels. 
 But 100 bushels is the quantity required. 100-r-$ = 
 60. Each ingredient, therefore, must be multiplied by 
 60. 60X1 = 60; 60Xf = 40. The quantities, then, 
 are 60 bushels at 40 cents, and 40 bushels at 60 cents. 
 
 SECTION XXXIV. 
 
 EQUATION OF PAYMENTS. 
 
 If A owes B several sums of money, to be paid at 
 different times, he may desire to pay the whole at 
 once, and consequently to know at what time the 
 whole becomes due. This time is found by making 
 an equation of the payments, multiplied by the time, 
 as follows : — 
 
 1. A owes B 200 dollars; 100 due Jan. 1, 1844; 
 100 due Jan. 1, 1846. He wishes to pay it all at 
 once. At what time should he pay it ? 
 
 Now, on Jan. 1, 1844, A is entitled to the use of 
 100 dollars for 2 years longer; 100X2 = 200; equal 
 to the use of 1 dollar for 200 years. If he is to pay 
 the whole together, he must keep the 200 dollars long 
 enough to balance that claim. 200 ) 200 ( I year, the 
 answer. The whole should be paid one year from 
 Jan. 1, 1844. 
 
EQUATION OF PAYMENT. 221 
 
 2. A owes B 100 dollars due hi 6 monLhi,, 200 dol- 
 lars due in 12 months. In how many montns should 
 the whole be paid together ? 
 
 100 X 6= 600 
 200X12^2400 
 
 300: 300)3000 
 
 10 months, — the answer. 
 
 The above is the method usually employed, and is 
 sufficiently exact for the necessities of business ; but it 
 gives a result a little in favor of the debtor ; that is, it 
 makes the equated time a little later than it should be. 
 To find the exact equated time, is a problem far too 
 difficult to be used in ordinary business. 
 
 Rule. — Multiply each payment by the length of 
 time before it becomes due. Divide the sum of the 
 products by the sum of all the payments. The quo- 
 tient will express the length of time in which the 
 whole is due. 
 
 3. A owes B several sums, due at different times, as 
 follows: $600 in 2 months, $150 in 3 months, $75 
 in 6 months. What is the equated time for the 
 whole ? 
 
 4. A man owes $1000; of which 200 are to be 
 paid in 3 months, 400 in 6 months, and the remainder 
 in 8 months. What is the equated time for the pay- 
 ment of the whole ? 
 
 5. If I owe $1200, one half to be paid in 3 
 months, one third in 6 months, and the remainder in 
 9 months, in what time should the whole be paid ? 
 
 6. A owes B $640; 150 due in 30 days, 200 due 
 in 60 days, and the remainder in 90 days. What is 
 the equated time for the whole ? 
 
 7. A merchant buys goods to the amount of $1800; 
 one third to be paid in 30 days, one third in 45 days, 
 
 19* 
 
222 SQUARE MEASURE. 
 
 and the remainder in 90 days. What is the equated 
 time for the whole ? 
 
 8. If I owe $1000, half to be paid in 60 days, and 
 half in 120 days, and if I pay $ 100 down, what will 
 be the equated time for the remainder? 
 
 SECTION XXXV. 
 
 SQUARE MEASURE. 
 
 [See Section XV., Part L] 
 
 1. There is a field in the form of a square, 15 rods 
 on a side. How many square rods does it contain ? 
 
 2. If the square be 15£ rods on a side, how many 
 square rods will it contain ? 
 
 3. How many square rods are there in a square field 
 measuring 17 rods on a side ? 
 
 4. If the field measure 17£ rods on a side, how 
 many square rods will it contain ? 
 
 5. What is the contents of a square field measuring 
 2l£ rods on a side ? 
 
 6. What is the area of a rectangular field, its length 
 being 64 rods, and its breadth 12£ rods ? 
 
 7. There is a rectangular field, its dimensions being 
 24^ rods and 76£ rods. What is the area ? 
 
 8. How many acres are there in a rectangular field, 
 its dimensions being 94 rods and 76£ rods ? 
 
 9. There is a rectangular field containing 7 acres. 
 Its length is 35 rods. What is its breadth ? 
 
 10. There is a rectangular farm, its length being 
 132 rods, its breadth 86£. How many acres does it 
 contain ? 
 
 11. There is a rectangular lot of land containing 
 325 acres. It measures on one side 176 rods. What 
 will it measure on the other ? 
 
SQUARE MEASURE. 
 
 223 
 
 12. There is a board containing 12 square feet. It 
 is 13 inches wide. How long is it ? 
 
 13. A table contains 15 square feet. It is 4 feet 
 long. How wide is it ? • 
 
 14. A certain room contains 30 square yards. It is 
 ,16 feet wide. How long is it? 
 
 * 15. A piece of cloth is If yards wide. How much 
 in length will it require to make 8 square yards? 
 
 16. There is a room 15 feet by 18. How many 
 yards of carpeting, £ of a yard wide, will it require to 
 cover it? 
 
 17. How many feet of boards will it require to 
 cover the sides and ends of a barn, as high as to the 
 eaves, (its length is 42 feet, width 34, and height 18,) 
 allowing one fifth of the boards to be wasted in 
 cutting ? 
 
 18. What will the above-named amount of boards 
 cost at $11.50 a thousand feet? 
 
 19. A road, 3£ rods wide, passes through a man's 
 land 1 mile. How much of his land does it take? 
 
 20. To what damages will he be entitled, allowing 
 him 28 dollars an acre ? 
 
 21. There is a right-angled triangle. Its base is 64 
 rods, and perpendicular 20 rods. How many acres 
 does it contain? (See Sec. XVII., Pt. I.) 
 
 22. There is a right-angled triangle. Its base is 84 
 rods, and perpendicular 26 rods. How many acres 
 does it contain? 
 
 23. There is a right-angled triangle. Its base is 49 
 r rods, perpendicular 34 rods. How many acres does it 
 
 contain ? 
 
 24. There is a right-angled triangle. Its area is 
 640 rods; the base is 64 rods. What is the per- 
 pendicular ? 
 
 25. A right-angled triangle has an area of 1092 
 rods. Its base is 60 rods. What is the perpendic- 
 ular ? 
 
224 
 
 DUODECIMALS. 
 
 SECTION XXXVI. 
 
 DUODECIMALS. 
 
 In measuring wood and lumber, the dimensions are 
 taken in feet and inches. As one inch is T V of a foot, 
 the multiplication of feet and inches by feet and 
 inches is the same as multiplying integers and twelfths 
 by integers and twelfths. Take the following ex- 
 ample : — 
 
 1. What is the contents of a 
 board 3 feet 7 inches long, and 
 2 feet 4 inches wide ? 
 
 Operation. 
 
 A 
 
 A 
 
 6 if 
 
 28 
 TI1> 
 
 . Ans. 6 ff T 2 ¥ V 
 
 This answer may be reduced to more simple terms. 
 t^ — ^ + i^t] adding T 2 2- + ff = ff, and this again 
 3= 2 feet + T V ; adding the 2 feet to the 6 feet, the 
 answer stands, 8 feet-f-y^-f-Tfr- 
 
 As the fractions decrease in value at a twelve-fold 
 rate, whenever the numerator exceeds 12, the excess 
 may be set down, and the 1 or more carried to the 
 next higher fraction. 
 
 2. Multiply 5 feet 2 inches 
 by 11 feet 9 inches. 
 
 5 
 
 A 
 
 
 11 
 
 A 
 
 
 3 
 
 « 
 
 T fj. 
 
 56 
 
 « 
 
 
 tIws. 60 A T | ¥ . 
 
 To render the operation more simple, call the 12ths 
 or inches, primes, (marked',) and the 144ths or frac- 
 tions of the jsecond order, seco?ids, (marked";) then 
 begin with the lowest order and multiply, setting each 
 
DUODECIMALS. 225 
 
 product in^its own place, with the mark appropriate 
 to express its value. 
 
 3. Multiply 13 ft. 5 in. by 2 
 ft. 11 in. 
 
 ft. 
 13 5 
 2 11 
 
 12 3 
 26 10 
 
 I Ans. 39 1' 7" 
 
 4. Multiply 3 ft. 9 in. by 7 ft. 4 in. 
 
 5. Multiply 9 ft. 8 m. by 4 ft. 9 in. 
 
 6. Multiply 15 ft. 2 in. by 9 ft. 1 in. 
 
 7. Multiply 8 ft. 6 in. by 2 ft. 4 in. 
 
 8. What is the contents of a board 14 ft. 5 in. long, 
 and 1 ft. 1 in. wide? 
 
 9. How many feet in a load of wood 8 ft. 6 in. 
 long, 4 ft. 2 in. wide, and 3 ft. 7 in. high ? 
 
 Multiply two of the dimensions together, and that 
 product by the third dimension. 
 
 10. How much wood in a load 11 ft. 3 in. long, 4 
 ft. 4 in. wide, 3 ft. 11 in. high? 
 
 Divide the cubic feet by 128 for cords, and the re- 
 mainder by 16 for cord feet, or eighths of a cord. 
 
 11. How much wood in a pile 38 ft. 6 in. long, 4 
 ft. 2 in. wide, and 4 ft. high? 
 
 12. How much wood in a load 9 ft. 4 in. long, 4 ft. 
 3 in. wide, 3 ft. 8 in. high? 
 
 13. How much wood in a load 7 ft. 8 in. long, 4 ft. 
 
 2 in. wide, 3 ft. 4 in. high ? 
 
 14. How much wood in a load 8 ft. 2 in. long, 4 ft. 
 wide, 4 ft. 3 in. high? 
 
 15. How many cords of wood will a shed contain, 
 whose dimensions inside are 22 ft. 6 in. long, 10 ft. 6 
 in. wide, 7 ft. 8 in. high? 
 
 16. Three men own equal shares in a lot of wood 
 lying in two piles. One pile is 13 ft. 4 in. long, 4 ft. 
 
 3 in. wide, 4 ft. 4 in. high. The other pile is 17 ft. 
 
 F 
 
226 
 
 EXTRACTION OF THE SQUARE ROOT. 
 
 long, 4 ft. wide, 3 ft. 10 in. high. How much wood 
 is each man's share? 
 See note on page 105. 
 
 SECTION XXXVII 
 
 EXTRACTION OF THE SQUARE ROOT. 
 
 [See Section XVI., Part I.] 
 
 This operation will be best understood by taking 
 first the simplest case, where the number is an exact 
 square, and the root containing only two figures. 
 
 What is the square 
 root of 196? 
 
 Operation. 
 
 i% 
 
 100 
 
 20 
 
 96 
 
 80 
 16 
 
 96 
 
 00 
 
 10, 1st part of the root. 
 4, 2d part of the root. 
 
 14, Ans. 
 
 Place a period over the unit figure ; another over 
 that of hundreds. This will show how many fig- 
 ures there will be in the root; for the square of a 
 number has always either twice as many figures as 
 the number, or one less than twice as many. Find 
 the greatest square of tens in the first period, (in the 
 given example, 100,) and set its root (10) in the 
 quotient. This will be the first part of the root. 
 Square the root. Subtract the square from the first 
 period, and bring down the figures of the next period 
 for a dividend. To the left hand, place double the 
 
EXTRACTION OF THE SQUARE ROOT. 227 
 
 part of the root already found for a trial divisor. Find 
 by trial what the next figure of the root must he, and 
 set it down under the first part of the root. This is 
 the second part, or unit figure of the root. [In try- 
 ing for this figure, remember that it must be so small, 
 that, when the divisor shall be multiplied by it, and 
 the square of itself shall be added to the product, the 
 sum shall not exceed the dividend.] Multiply the 
 divisor by the new figure of the root; to this add the 
 square of the same figure, and subtract the sum from 
 the dividend. If the number is an exact square of 
 two periods, as in the above example, there will 
 be no remainder; and the two parts of the root thus 
 found, when added together, will give the whole root. 
 
 2. What is the square root of 225? Of 324? 
 
 3. What is the square root of 289? Of 529? 
 
 4. What is the square root of 361? Of 729? 
 
 5. What is the square root of 625? Of 1024? 
 
 6. What is the square root of 784? Of 1296? 
 
 7. What is the square root of 841? Of 1849? 
 
 8. What is the square root of 961? Of 2601? * 
 
 If there are more than two periods, first consider 
 only the two left-hand periods, and find their root as 
 above directed ; then consider the part of the root ex- 
 pressed by these two figures as the first part with 
 reference to the next figure, (to indicate this, you 
 must annex a cipher,) and work for the next; and 
 so on. 
 
 9. W^hat is the square root of 15625 ? 
 
 10. What is the square root of 60516 ? Of 104976? 
 Of 213444? 
 
 Square Root of a Decimal. 
 
 If there are decimals in the number, point off each 
 way from the place of units; adding a cipher, if ne- 
 cessary, to make the right-hand period complete. 
 
228 EXTRACTION OF THE SQUARE ROOT. 
 
 11. What is the square root of 2.56 ? Of 12.25? 
 
 12. What is the square root of 2.25? Of 20.25? 
 
 13. What is the square root of 156.25? Of 132.25 ? 
 
 14. What is the square root of 13.69? Of 21.16? 
 
 15. What is the square root of 88.36 ? Of 53.29 ? 
 
 16. What is the square root of 1.69? Of 1.44? 
 
 17. What is the square root of .81 ? Of .64? 
 
 18. What is the square root of .01 ? Of 6.25? 
 
 Square Root of a Vulgar Fraction. 
 
 To obtain the square root of a vulgar fraction, find 
 the square root of the numerator, and of the denomi- 
 nator, and write the -former over the latter. 
 
 19. What is the square root of | ? Of £f ? 
 
 20. What is the square root of T \ ? Of A ? 
 
 21. What is the square root of £ff ? Of ff ? 
 
 22. What is the square root of fff ? Of fff ? 
 The correctness of the answer may always be tested 
 
 by multiplying the answer found, by itself. If cor- 
 rect, it will reproduce the original square. 
 
 23. What is the square root of i? Of *V? 
 
 24. What is the square root of £f ? Of t Vt ? 
 
 Another Method of finding the Root of a Fraction. 
 — Reduce the fraction to a decimal, and proceed as 
 already directed in the case of decimals. 
 
 25. What is the square root of i ? The square root 
 of 1 is 1 ; the square root of 4 is 2. Ans. £. Or re- 
 duce i to a decimal, — .25; square root, .5, Ans. 
 
 If the number is not a complete square, annex 
 periods of ciphers, as decimals, and carry the opera- 
 tion as far as desired. 
 
 26. What is the square root of 70 ? Of 80 ? 
 
 27. What is the square root of 90 ? Of 45 ? 
 
 28. What is the square root of 60 ? Of 84 ? 
 
 29. What is the square root of 200? Of 120? 
 
EXTRACTION OF THE SQUARE ROOT. 229 
 
 30. There is a field in the form of a square, contain- 
 ing 1 acre. How many rods does it measure on a side ? 
 
 31. There is a right-angled triangle, its hypote- 
 nuse measuring 60 rods. What is the sum of the 
 squares of the two legs? (See Sec. XVII., Part I.) 
 
 32. There is a right-angled triangle. The squares 
 of its legs added together are 81 rods. What is the 
 length of the hypotenuse ? 
 
 33. There is a right-angled triangle. Its legs meas- 
 ure, one 25, the other 30 rods. How long is the hy- 
 potenuse ? 
 
 34. Two men start from the same place. One 
 travels 8 miles east j the other, 15 miles north. How 
 far are they then apart ? 
 
 35. A ladder 40 feet long stands against a house, 
 the foot resting on the ground, on a level with the 
 foundation of the house, and 20 feet distant from it. 
 How far up will it reach? 
 
 36. The floor of a room measures 16 feet in length, 
 and 14 feet in width. How long a line will reach 
 diagonally from corner to corner ? 
 
 37. The two parts of a carpenter's square, one 12, 
 the other 24 inches long, may be regarded as the legs 
 of a right-angled triangle. How long would be the 
 hypotenuse connecting their extremities ? 
 
 38. There is a room 16 feet long, 14 feet wide, and 
 10 feet high. How long must a straight line be, 
 reaching from a corner of the room at the bottom to 
 the diagonal corner at the top ? 
 
 39. There is a room, the length, breadth, and 
 height of which are, each, 10 feet. How far is it 
 from a corner of the room at the bottom to the 
 diagonal corner at the top ? 
 
 40. There is a room, the length, breadth, and 
 height of which are equal. The distance from a 
 corner at the bottom to the diagonal corner at the 
 top is 18 feet. What is the size of the *room ? 
 
 20 
 
230 
 
 EXTRACTION OF THE CUBE ROOT. 
 
 41. I have a cubic block measuring 4 inches each 
 way. How far apart are its diagonal corners ? 
 
 42. How large a cube can be cut from a sphere 
 which is 1 foot in diameter ? 
 
 SECTION XXXVIII. 
 
 EXTRACTION OF THE CUBE ROOT. 
 
 [See Section XIX., Part I.] 
 
 We will first consider those numbers the cube root 
 of which is expressed by a single figure. Every exact 
 cube, of not more than three figures, will have for its 
 root some number less than 10, and, consequently, it 
 will be expressed by a single figure. This root can 
 be found by successive trials. 
 
 Examples. 
 
 1. What is the cube root of 125? 
 
 2. What is the cube root of 216? 
 
 3. What is the cube root of 512? 
 
 4. What is the cube root of 729? 
 
 We will next take perfect cubes, the root of which 
 consists of two figures. 
 
 5. What is the cube 
 root of 4096 ? 
 
 300 
 30 
 
 Operation. 
 
 4096 
 1000 
 
 330 
 
 3096 
 
 10, 1st part of the root. 
 6, 2d part of the root. 
 
 16, Ans. 
 
 1800 
 
 1080 
 
 216 
 
 3096 
 
 0000 
 
EXTRACTION OF THE CUBE ROOT. 231 
 
 Rule. — Place a period over the unit figure, and 
 another over that of thousands. Tind the greatest 
 cuhe in the first period, whose root is expressible in 
 tens. Set down this root as a quotient in division. 
 Find the cube of the root, and subtract it from the 
 first period, and bring down the second period as a 
 dividend. At the left hand of this set down three 
 times the square of the root, and under this three 
 times the root ; add these together, for a trial divisor. 
 Find, by trial, what the next figure of the root will 
 be, and set it under the first part, already found. 
 Multiply by this figure three times the square of the 
 first part of the root, setting the product under the 
 dividend. Multiply by the square of this figure three 
 times the first part of the root, setting the product 
 underneath the other. Under these set the cube of 
 the root figure last found. Add these three numbers 
 together, and subtract their sum from the dividend. 
 If the work be correct, there will be no remainder. 
 Add together the two parts of the root for the answer. 
 
 6. What is the cube root of 2744 ? 
 
 7. What is the cube root of 
 
 8. What is the cube root of 
 
 9. What is the cube root of 
 
 10. What is the cube root of 
 
 11. What is the cube root of 
 
 We will next consider the case where there are 
 more than two figures in the root. The number of 
 figures in the root can always be determined by the 
 number of periods placed over the sum, beginning 
 with units, and placing a period over every third 
 place. If there are more than three periods in the 
 cube, regard, first, only the two left-hand periods, 
 obtaining the first and second figures of the root, just 
 as if they constituted the whole root. Then, after 
 bringing down the figures of another period, add the 
 
 2744? 
 
 Of 205379 ? 
 
 3375? 
 
 Of 5832 ? 
 
 4913? 
 
 Of 10648? 
 
 9261? 
 
 Of 15625? 
 
 13824? 
 
 Of 19683? 
 
 46656 ? 
 
 Of 39304 ? 
 
232 EXTRACTION OF THE CUBE ROOT. 
 
 two parts of the root, and consider their sum as the 
 first part of the root, and proceed to find the next 
 part. To indicate this, you must annex a cipher to 
 the figures of the root already found. 
 
 12. What is the cube root of 1953125? 
 « 13. What is the cube root of 2406104? 
 
 14. What is the cube root of 3796416? 
 
 If there are decimals in the given sum, point off 
 both ways from the units' place, adding ciphers, if 
 necessary, to the decimal, in order to make the period 
 complete. 
 
 15. What is the cube root of 15.625? 
 
 16. What is the cube root of 35.937? 
 
 If the number given is not a perfect cube, add 
 periods of ciphers, and carry out the root in decimals 
 as far as may be desired. 
 
 17. What is the cube root of 10? 
 
 18. What is the cube root of 20? 
 
 19. What is the cube root of 50? 
 
 20. What is the cube root of 100? 
 
 21. A bushel, even measure, contains 2152 solid 
 inches. What would be the inside measure of a 
 cubic box containing 12 bushels? 
 
 22. A gallon, wine measure, contains 231 cubic 
 inches. What must be the inside measure of a cubic 
 cistern containing 10 barrels ? 
 
 23. What would be the measure of a cubic pile of 
 wood containing one cord ? 
 
PROPORTION. 233 
 
 SECTION XXXIX. 
 
 PROPORTION. 
 
 [See Section XX., Part I.] 
 
 Several changes that may be made in the terms of a 
 proportion are exhibited in page 131. In continuing 
 the subject, we will first state some further changes 
 that may be made in the terms without destroying the 
 proportion. 
 
 1. Multiply all the terms by the same number. 
 
 2. Divide all the terms by the same number. 
 
 3. Add the terms of the first ratio for the first an- 
 tecedent, and the terms of the second ratio for the 
 second antecedent. 
 
 4. Add the terms of the first ratio for the first con- 
 sequent, and the terms of the second ratio for the 
 second consequent. 
 
 5. Instead of the sum of the terms in the third case 
 above, take the difference of the terms. 
 
 6. Instead of the sum of the terms in the fourth 
 case above, take the difference of the terms. 
 
 7. Raise each term to the same power, as second or 
 third power. 
 
 8. Extract of each term the same root. 
 
 The result, after each of these operations, will still 
 be a proportion, and may be proved to be so, by mul- 
 tiplying the extremes together, and finding the product, 
 equal to that of the means. 
 
 Take the proportion, 4 : 16 : : 9 : 36, and perform on 
 it the first change, using any number you please for a 
 multiplier, and then prove the proportion. 
 
 Perform on the same proportion the second change. 
 
 Perform the third change. 
 
 Perform the fourth change. 
 20* 
 
234 
 
 PROPORTION. 
 
 Perform the fifth change. 
 
 Perform the sixth change. 
 
 Perform the seventh change, raising to the second 
 power. 
 
 Perform the eighth change, extracting the square 
 root. 
 
 Finally, you may, in any case, invert the whole 
 proportion ; or, invert the terms of each ratio ; or, 
 invert the means or the extremes. 
 
 Practical Questions. 
 
 1. If 7 lbs. of flour cost 31 cents, what will 196 
 lbs. cost? 
 
 As the smaller quantity is to the larger quantity, so 
 is the price of the smaller quantity to the price of the 
 larger. 
 
 2. If 3 cwt. of hay cost 2 dollars, what will 35 cwt. 
 cost ? 
 
 3. If 4 qts. of molasses cost 38 cents, what will 10 
 qts. cost? 
 
 4. If a horse travels 19 miles in 3 hours, how far 
 will he travel in 11 hours? 
 
 5. If the freight of 7 cwt. cost 2 dollars, what will 
 the freight of 20 cwt. cost ? 
 
 6. If 11 dollars buy 3 cords of wood, how many 
 cords will 50 dollars buy ? 
 
 7. If 7 bushels of oats last a horse 2 months, how 
 long will 23 bushels last him, at the same rate ? 
 
 8. A man bought a horse for 84 dollars, and sold 
 him for $93. What did he gain per cent.? 
 
 As the whole outlay is to 1 dollar, so is the whole 
 gain to the gain on a dollar. 
 
 9. A merchant buys flour at $4.35 a barrel, and sells 
 it for $4.63. What is his gain per cent. ? 
 
 1.0. A and B form a partnership in trade. A puts in 
 
PROPORTION. 235 
 
 $500, and B $300, for the same time. They gain 
 $ 180. What ought each to share ? 
 
 As the whole stock is to each one's share, so is the 
 whole gain to each one's gain. 
 
 11. C and D trade in company. C puts in 750 dol- 
 lars, and D $450, for the same time. They gain 240 
 dollars. How much gain ought each to receive ? 
 
 12. Two men buy a lot of wood in company for 
 340 dollars. One takes away 42 cords ; the other, the 
 remainder, which was 34 cords. What ought each to 
 pay? 
 
 13. Two men hire a sheep-pasture in company for 
 20 dollars. One keeps 30 sheep in it 14 weeks ; the 
 other, 24 sheep, 16 weeks. What ought each to pay ? 
 
 Find how many weeks' pasturing for a single sheep 
 each one had. 
 
 14. Two men purchase a lot of standing grass for 
 $36.50. One takes 34 tons ; the other, If tons. 
 What ought each to pay? 
 
 Reduce the quantity of hay to fourths of a ton, and 
 then state the proportion. 
 
 15. 'There is a circular piece of ground, whose di- 
 ameter is 14 rods. What will be the diameter of a 
 circle containing twice as much ? 
 
 16. There is a circular piece of ground containing 
 2.5 acres. What will be the area of a circle, the 
 diameter of which is 3 times as great ? 
 
 17. There are two similar triangular fields. The 
 smaller contains 3 acres, the larger 4. The base of 
 the smaller is 44 rods. How long is the base of the 
 larger ? 
 
 18. There are two similar rectangular fields. The 
 smaller is 34 rods wide, and 60 rods long. The other 
 has twice as great an area. What are its dimensions? 
 
 19. There is a grindstone 4 feet in diameter. What 
 will be its diameter after half of it is ground off ? 
 
236 PROPORTION. 
 
 20. There are two similar triangular pieces of land. 
 The base of one measures 44 rods. The other piece 
 has an area 7 times as large as the first. What is the 
 length of its base ? 
 
 21. There are two cisterns of the same shape. One 
 is 5 feet deep. The other has a capacity three times 
 as great. How deep is it ? 
 
 22. If a ball 5 inches in diameter weighs 14 lbs., 
 what will be the weight of one of the same material 
 6 inches in diameter ? 
 
 23. What, on the same supposition, will be the 
 weight of a ball of 7 inches diameter ? 
 
 24. There are two marble statues of the same form, 
 but differing in size. One is 5 feet high, and weighs 
 740 lbs. The other is 7 feet high. What will it 
 weigh ? 
 
 25. If a tree, 2£ feet in diameter at the ground, 
 contains 3 cords of wood, how much will there be in 
 a tree of the same shape, 3£ feet in diameter? 
 
 26. There are two similar stacks of hay. The 
 smaller is ll£ feet high, and contains 4£ tons of hay. 
 The larger is 14 feet high. How much hay does it 
 contain, supposing both to be of the same solidity ? 
 
 27. If an iron field-piece, 5J- feet long, weighs 1140 
 lbs., how many lbs. will an iron cannon of the same 
 shape weigh, that is lOf feet long ? 
 
 28. There are two anchors of similar form. The 
 smaller weighs 1100 lbs. The larger is 2f times as 
 long. What is its weight ? 
 
 When a cause and an effect are connected together, 
 the increase of the one is always coifnected with an 
 increase of the other. If 6 horses eat 20 bushels of 
 oats, we may regard the horses as the cause, and the 
 consumption of the oats the effect ; or, if we please, 
 we may regard the oats as the cause, and the support 
 of the horses as the effect. But, in either case, an 
 
PROPORTION'. 237 
 
 increase of one would require an increase of the other. 
 When numbers are connected in this way, in a pro- 
 portion, having the relation of cause and effect to each 
 other, the proportion is said to be direct. 
 
 But it often happens that quantities are connected 
 together, not as cause and effect, but as limitations of 
 each other ; where an increase of one quantity re- 
 quires a diminution of the other. 
 
 Thus, if the provisions of a ship's company are suf- 
 ficient to last 17 weeks, at the rate of 13 oz. of bread 
 per day for each man, it is evident that these quanti- 
 ties, 17 and 13, are not cause and effect, but limitations 
 of each other. If one is increased, the other must be 
 diminished. So, if, with a speed of 11 miles per hour, 
 a journey be performed in 31 hours, it is evident that 
 an increase of one term must diminish the other. 
 When quantities mutually limiting each other enter 
 into a proportion, it is called indirect proportion. No 
 special rule, however, is needed for the statement of 
 such questions; for you can always determine, by 
 strict attention, whether the statement you make is 
 reasonable. 
 
 29. If, with a speed of 11 miles per hour, a journey 
 is performed in 37 hours, how long will it take to per- 
 form the same journey with a speed of 15 miles per 
 hour ? 
 
 30. If a stable-keeper has grain for the supply of 29 
 horses 43 days, how long will the supply last, if he 
 buys 6 horses more ? 
 
 31. If a barrel of flour last a family of 7 persons 6 
 weeks, how long will it last 15 persons? 
 
 32. If. 42 men can do a job of work in 60 days, 
 how long will it take 53 men to perform the same 
 work ? 
 
 33. A ship-builder employs 50 men to complete a 
 ship, which they can do in 45 days. If 7 of the men 
 
238 COMPOUND PROPORTION. 
 
 fail to engage in the work, how long will it take the 
 others to perform it ? 
 
 34. If 8 yds. of cloth, 7 qrs. wide, cost 54 dollars, 
 what will be the cost of 15 yds. of cloth, of the same 
 quality, 9 qrs. wide? 
 
 In this example, the length of the two pieces of 
 cloth will not represent the ratio of their values, for 
 they are of different widths. The answer can be 
 found by two statements. First, regarding the two 
 pieces as of the same width, 
 
 8 : 15 : : 54 : 10l£ dollars, first answer. 
 
 Next, taking the width into view, 
 
 7 : 9 : : 10 1£ : 130^8 dollars, final answer: 
 
 If we examine the above question, we shall see 
 that in neither of them is the quantity of cloth ex- 
 pressed; but, in the first statement, its length, and in 
 the second, its width. Now, the quantity of cloth is 
 expressed by the length multiplied by the breadth. In 
 the smaller piece, it is 7X8 = 56 qrs. of a square yard; 
 in the larger piece, it is 15X9=135 qrs. of a square 
 yard. These numbers, 56 and 135, express the quan- 
 ties of cloth ; and taking these, instead of the dimen- 
 sions, a single proportion gives us the answer; — 
 
 56: 135::54:'130 2 V 
 
 As the question is first stated, you observe that, in- 
 stead of the numbers which form the ratio, 56 : 135, 
 you have only the factors of those numbers given. 
 This is called a Compound Proportion. 
 
 A compound proportion, then, is one in which two 
 or more of the terms of the simple proportion are ex- 
 pressed in the form of their factors. 
 
 Every question containing a compound proportion 
 may be solved by means of two or more simple pro- 
 portions : or, it may be reduced to one simple propor- 
 
COMPOUND PROPORTION. 239 
 
 tion, as is seen in example 34, above. This method, 
 however, often requires calculations in large numbers. 
 It may therefore be desirable to have a method given 
 by which the process may be made less tedious. 
 
 The following rule is offered, as applicable to all 
 cases of proportion, simple or compound, — direct or 
 inverse. It is short, and the attention required in 
 applying it will afford a good discipline for the rea- 
 soning powers. 
 
 Rule of Proportion. 
 
 Draw a horizontal line. Then examine the condi- 
 tions of the question, and consider, in the case of 
 each, whether its increase would make the answer 
 greater or smaller. If it would make it greater, set it 
 above the line ; if smaller, set it below. 
 
 Regard the numbers, thus set down, as the terms of 
 a compound fraction. Cancel common factors. Mul- 
 tiply together the terms that remain, for the answer. 
 
 Example. 
 
 35. If 8 men build a wall 36 ft. long, 12 ft. high, 
 and 4 ft. thick, in 72 days, when the days are 9 hours 
 long, how many men will build a wall 100 ft. long, 
 10 ft. high, and 3 ft. thick, in 24 days, when the days 
 are 10 hours long? 
 
 Cancelling like factors above and below the line, 
 and multiplying the remaining terms, 
 
 Operation. 
 
 8 72 9 100 10 3 
 36 12 4 24 10 =a * 2 = 37 * men > Answer - 
 
 Explanation. 
 
 The question is, "How many men?" "If 8 men 
 will build," &c. Now, if it took 80 men to build the 
 
240 
 
 PROPORTION. 
 
 first wall, instead of 8, it would require more men to 
 build the second; then put 8 above the line. "Build 
 a wall 36 ft. long." If it were 360 feet long, instead 
 of 36, it would take fewer men to build the second 
 wall; therefore put 36 below the line. Pursue the 
 same reasoning with all the other conditions. 
 
 36. If 15 horses consume 40 tons of hay in 30 
 weeks, how many horses will consume 56 tons of hay 
 in 32 weeks? 
 
 37. If 1 dollar gain .06 of a dollar interest, in 12 
 months, how much will 740 dollars gain in 8 months? 
 
 38. If a crew of 75 men have provisions for 5 
 months, allowing each man 30 oz. per day, what must 
 be the allowance per day, to make the provisions last 
 6^ months? 
 
 39. If 18 bricklayers, in 12 days, of 9 hours each, 
 build a wall 175 feet long, 2 feet thick, and 18 feet 
 high, in how many days will 6 men, working 10 hours 
 each day, build a wall 100 feet long, 1J feet thick, and 
 16 feet high ? 
 
 40. If 10 masons lay 160 thousand of bricks in 12 
 days, working 8 hours per day, how many men will 
 lay 224 thousand in 15 days of 10 hours each? 
 
 Partnership. 
 
 41. Two men trade in company. One puts in 1000 
 dollars, the other 2000, for the same length of time. 
 They gain 600 dollars. What is each one's share 
 of the gain? 
 
 It is evident that each man's share ought to be in 
 proportion to the sum he put in. 
 
 As the whole investment is to each partner's invest- 
 ment, so is the whole gain or loss to each partner's 
 gain or loss. 
 
 42. Two men trade in company. One puts in 
 
PROPORTION'. 
 
 241 
 
 3500 dollars ; the other, 4000. They gain 600 dol- 
 lars. What is each one's share of the gain ? 
 
 43. Three men trade in company. The first puts 
 in 3400 dollars ; the second, 800 ; and the third, 1200 
 dollars. They gain 475 dollars. What is each part- 
 ner's share ? 
 
 44. Two men purchase a ship for 11000 dollars. 
 One pays 8000 dollars ; the other, 3000. They sell 
 the ship for 9500 dollars. What is each one's loss? 
 
 45. Two men trade in company. One puts in 1000 
 dollars for 6 months ; the other puts in 1000 for 18 
 months. They gain 600 dollars. What is each one's 
 share of the gain? 
 
 Here, though the money was equal, it is evident 
 the gain of one ought to be three times as great as the 
 other, because his money was in three times as long. 
 
 Where the investments are made for different times, 
 each partner's interest will be expressed by multiply- 
 ing his money by the time it was in trade. Then, as 
 the sum of all the interests is to each partner's interest, 
 so is the whole gain or loss to each partner's gain or 
 loss. 
 
 46. Three men trade in company. The first puts 
 in 400 dollars for 8 months; the second, 1100 dollars 
 for 6 months; the third, 1000 dollars for 7 months. 
 They gain 840 dollars. What is each man's share? 
 
 47. Four men trade in company. The first puts in 
 1200 for 2 years; the second, 1500 dollars for 18 
 months; the third, 600 for 8 months; the fourth, 900 
 dollars for 6£ months. They gain 1340 dollars. What 
 is each man's share ? 
 
 General Rule for Cancelling. 
 
 In all operations involving simply multiplication and 
 division, set the multipliers above the line, in the form 
 
 21 Q 
 
242 
 
 PROPORTION. 
 
 of their factors, and the divisors below. Cancel com- 
 mon factors, and the resulting compound fraction will 
 be the answer. 
 
 Examples. 
 
 48. How many blocks of granite, 16 inches long, 
 12 inches wide, and 8 inches thick, will it take to 
 make a solid pile 128 feet long, 48 feet wide, and 36 
 feet high ? 
 
 Here, the dimensions of the pile constitute the fac- 
 tors, which are to be multiplied together to form the 
 dividend. The dimensions of a block are the factors 
 which are to be multiplied together to form the di- 
 visor. The operation, then, is as follows: — 
 
 Length. m Width. Height. Blocks. 
 
 X%X£X$$ l ] 12X^X12; J2X3X 12 = 248832, Ans. 
 X$XMX$ 
 
 49. How many cubic yards are there in a cellar 60 
 feet long, 45 feet wide, and 9 feet deep? 
 
 50. What are the solid contents, in yards, of a stone 
 wall 42 feet long, 15 feet high, and 44 feet thick? 
 
 As 4J == f , the 9 goes with the multipliers, and the 
 2 with the divisors ; thus, 
 
 0X7X3X5X0=105 cubic yards, Ans. 
 
 $X#X#X2 
 
 In all cases, Reduce mixed numbers to improper 
 fractions ; and if they are multipliers, write the terms 
 in their natural order, the numerator above, and the 
 denominator below, the line. If they are divisors, in- 
 vert them. 
 
 51. What are the solid contents, in yards, of a wall 
 48 feet long, 7£ feet high, and 2£ feet wide ? 
 
 52. What is the cost of digging a cellar 63 feet 
 ong, 18 feet wide, and 7£ feet deep, at 20 cents a 
 
 cubic yard ? 
 
PROGRESSION. 243 
 
 As 20 cents = | of a dollar, this fraction is a multi- 
 plier, giving the answer in dollars. If 20 is used, it 
 will give the answer in cents. 
 
 53. How many cords of wood are there in a pile 
 164 feet long, 4 feet wide, and 12 feet high? 
 
 54. How many cords of wood are there in a pile 80 
 feet long, 32 feet wide, and 7 feet high? 
 
 55. What is the value of a pile of wood, at $4.00 a 
 cord, the dimensions of which are 90 feet, 8 feet, and 
 6 feet ? 
 
 56. What is the cost of cutting a pile of wood, at 
 60 cents a cord, the dimensions of which are 44 feet, 
 8 feet, and 5i feet? 
 
 One eighth of a cord, or a pile, 1 ft. X4 ft.X4 ft. = 
 16 feet, is called a cord foot. 
 
 57. How many cord feet are there in a load of 
 wood 8 feet long, 4£ feet wide, and 3^ feet high? 
 
 $X9X7 = 63 
 
 - — - =7|- cord feet, Ans. 
 
 4X4X^X2=8 
 
 58. How many cord feet in a load of wood, 8 feet 
 long, 4£ feet high, and 5£ feet wide ? 
 
 59. How mauy cord feet in a load of wood, 7 feet 6 
 inches long, 4 feet 3 inches wide, and 3 feet 4 inches 
 high? 
 
 60. What is the value of a load of wood, 8 feet 
 long, 4$ feet wide, and 2 feet high, at $ of a dollar a 
 cord foot ? 
 
 SECTION XL. 
 
 PROGRESSION. 
 
 When a series of numbers is given, each one of 
 which has the same ratio to the number which fol- 
 lows it, the series is called a progression. 
 
244 PROGRESSION. 
 
 Progression is arithmetical or geometrical. Arith- 
 metical progression is made by the successive addition 
 or subtraction of a common difference. When the 
 common difference is added to each term, in order to 
 make the succeeding one, the series is called an 
 ascending series; as, 1, 3, 5, 7, 9, 11, &c. 
 
 When the common difference is subtracted, the 
 series is called a descending series; as, 11, 9, 7, 5, 3, 1. 
 
 If you know the first term and the common dif- 
 ference of an arithmetrical progression, you can write 
 the whole series, for to do this, you have only to add 
 or subtract the common difference for each succeeding 
 term. If the whole series is written out, it is evident 
 you can find by inspection any particular term, as the 
 7th, the 15th, the 20th, &c. But, if the series be a 
 long one, this may be a very tedious operation. 
 
 Suppose the series given above, 1, 3, 5, 7, &c, 
 were continued to 87 terms, and you were required 
 to find what was the last term. 
 
 By examining the series, you will see that the 2d 
 term = the 1st + the common difference ; the 3d = 
 1st + twice the common difference ; the 4th = 1st +3 
 times the common difference; the 5th = 1st + 4 times 
 the common difference ; &c. Any term whatever equals 
 the 1st term + the common difference, multiplied by 
 a number one less than that which expresses the place 
 of the term. The 87th term in the above series, there- 
 fore, is 1 + 2X86=173. 
 
 1. What is the 38th term of the series 1, 3, 5, 7, &c. ? 
 
 2. What is the 53d term of the same series? The 
 91st term? The 89th term? The 107th term? 
 
 3. In an arithmetical series, the first term of which 
 is 1, and the common difference 3, what is the 64th 
 term? The 75th term? The 81st term? 
 
 4. In the series 1, 5, 9, 13, &c, what is the 40th 
 term? The 67th term? The 80th term? 
 
ARITHMETICAL PROGRESSION. 245 
 
 5. In the series 2, 4, 6, &c, what is the 45th term ? 
 What is the 100th term ? What is the 200th term ? 
 
 Hence, if you know the number and place of any 
 term, and the common difference, you may find the first 
 or any other term. 
 
 6. If the 5th term of an arithmetical series is 13, 
 and the common difference 3, what is the 1st term? 
 What is the 24th term ? What is the 191st term ? 
 
 7. If the 6th term of a series is 77, and the com- 
 mon difference 15, what is the 2d term ? What is the 
 14th term? 
 
 8. If the 22d term in a series is 89, and the com- 
 mon difference 4, what is the 10th term ? What is 
 the 43d term ? 
 
 By knowing the number and the place of any two 
 terms, we may find the common difference. 
 
 9. In a certain series, the 4th term is 10, and the 7th 
 term is 19. What is the common difference ? 
 
 19 — 10 = 9. Now, this difference, 9, is made by 
 the addition of the common difference three times ; 
 for 7 — 4 = 3. The common difference, therefore, is 
 9^3 = 3. 
 
 10. In a certain series, the 5th term is 9, and the 
 11th term is 21. What is the common difference ? 
 
 11. In a certain series, the 4th term is 13, and the 
 9th term is 33. What is the common difference ? 
 
 If we know the 1st term, the common difference, 
 and the number of terms, we can find the sum of all 
 the terms. 
 
 12. How many strokes does a clock strike in 24 
 hours, from noon to noon? 
 
 We might write down the series 1, 2, 3, &c, up 
 
 to 12, which would express the number of strokes in 
 
 12 hours, from noon till midnight; we might write 
 
 the same series again, for the time from midnight till 
 
 21* 
 
246 ARITHMETICAL PROGRESSION. 
 
 noon ; and, by adding these numbers together, might 
 obtain the answer. But a much shorter way may be 
 found. To exhibit it, we will write the two series 
 thus : — 
 
 1st series, 1 2 3 4 5 6 7 8 9101112, from noon till midnight. 
 2d series, 121110 9 8 7 6 5 4 3 2 1, from midnight till noou. 
 
 13 13 13 13 13 13 13 13 13 13 13 13 
 
 Sum of both series equal to 12 times 13. But 13 
 is the sum of the first and last terms, and 12 is the 
 number of terms. Therefore, The sum of the first 
 and last terms, multiplied by the number of terms, 
 gives the sum of all the terms of both series. Half 
 this number will be the sum of one series. 
 
 13. What is the sum of the series 1, 4, 7, to 20 
 terms ? 
 
 First find the 20th term. 
 
 14. What is the sum of 50 terms of the series 2, 
 6, 10? 
 
 15. A farmer instructed his boy to carry fencing- 
 posts from a pile to the holes in the ground where 
 they were to be inserted, taking one post at a time. 
 The holes are 12 feet apart, in a straight line, and the 
 pile of posts 30 feet from the first hole. How far 
 must he travel, in carrying to their places 100 posts ? 
 
 16. If the hours in a whole week were numbered 
 in regular progression, and were struck in this way by 
 the clock, how many strokes would the clock strike 
 for the last hour of the week? 
 
 What would be the whole number of strokes in the 
 week ? 
 
 If we know the first and last terms and the com- 
 mon difference, we can find the number of terms. 
 
 17. The first term of a series is 4 ; the last term is 
 19 ; the common difference is 3. What is the number 
 of terms? 
 
ARITHMETICAL PROGRESSION. 247 
 
 The difference between the extremes, 19 — 4, is 15. 
 This, you know, is the common difference, 3, taken a 
 certain number of times ; 15-1-3 = 5. There are then 
 5 additions of the common difference. Now, the num- 
 ber of terms is 1 more than the number of times the 
 common difference has been added. To find the num- 
 ber of terms, then, 
 
 Find the difference of the extremes ; divide it by the 
 common difference ; increase the quotient by 1 for the 
 number of terms. 
 
 DESCENT OF FALLING BODIES. 
 
 18. A body falling through the air falls, in the 1st 
 second, 16.1 feet ;* in the 2d second, 48.3 feet ; in the 
 3d second, 80.5 feet. How many feet farther does it 
 fall each second than it fell the second before ? 
 
 19. Taking the answer to the preceding question as 
 the common difference, and 16.1 as the first term of a 
 series, how far will a body fall in 4 seconds ? 
 
 20. How far will a body fall in 5 seconds ? 
 
 21. How far will a body fall in 6 seconds? 
 
 22. A stone, in falling to the ground, falls the last 
 second 209.3 feet. How many seconds has it fallen, 
 and from what height ? 
 
 23. A stone, in falling to the ground, falls the last 
 second 241.5 feet. How many seconds has it fallen, 
 and from what height ? 
 
 24. If a stone, dropped into a well, strikes the water 
 in 3 seconds, how far is it to the surface of the water ? 
 
 * This is a more exact statement than that made in Part I. (See 
 Olmsted's Natural Philosophy.) It should be remarked, also, that 
 no allowance, in these examples, is made for the resistance of the 
 atmosphere, which always diminishes the speed somewhat, and be- 
 comes greater and greater as the speed increases. 
 
248 GEOMETRICAL PROGRESSION. 
 
 SECTION XLI. 
 
 GEOMETRICAL PROGRESSION. 
 
 A series of numbers, such that each is the same part 
 or the same multiple of the number that follows it, is 
 called a geometrical series. The ascending series 1, 
 3, 9, 27, is of this kind, for each term is one third of 
 that which succeeds it. So, in the descending series 
 64, 16, 4, 1, each term is 4 times the following term. 
 
 The number obtained by dividing any term by the 
 term before it, is called the ratio of the progression. 
 Thus, in the first of the above examples, the ratio is 
 3 ; in the second example, it is £. 
 
 Let us take the series 2, 6, 18, 54, and observe by 
 what law it is formed. The ratio is 3 ; the first term, 
 2. The second term is 2X3, or the first term X the 
 ratio. The third term is 2X3 2 , or the first term X the 
 second power of the ratio. The fourth term is 2X3 3 , 
 or the first term X the third power of the ratio. 
 
 Thus each term consists of the first term multiplied 
 by the ratio raised to a power whose index is one less 
 than the number expressing the place of the term. 
 
 1. What is the 7th term in the series 1., 4, 16, &c. ? 4 
 
 2. What is the 10th term in the series 3, 6, 12, &c. ? 
 
 3. A glazier agrees to insert a window of 16 lights 
 for what the last light would come to, allowing 1 cent 
 for the first light, 2 for the second, and so on. What 
 will the window cost ? 
 
 4. If, in the year 1850, the population of the United 
 States shall be 20000000, and if it shall thenceforward 
 double once in every thirty years, what will be the 
 population in 1970? 
 
GEOMETRICAL PROGRESSION. 249 
 
 To obtain the sum of the terms, when the first and 
 last terms are given, and the ratio, 
 
 Take the following question: — What is the sum of 
 a series whose first term is 1, last term 64, and ratio 4? 
 
 We can write the series in full, and then obtain the 
 • answer by adding the terms together; thus, 1 + 4+16 
 + 64=85. In this way we might obtain the sum of 
 any given series ; but the operation becomes tedious if 
 the series be a long one, and hence a shorter method 
 is devised. 
 
 What is the sum of a series whose first term is 2, 
 last term 162, and ratio 3 ? 
 
 To show how a general rule is obtained, we will 
 write this series in full ; 
 
 thus, 2, 2X3, 2X3 2 , 2X3 3 , 2X3* ; 
 or, 2, 6 , 18 , 54 , 162 . 
 
 If, now, we multiply this series by the ratio, and set 
 the terms of the product one degree to the right, over 
 the terms of the series, we shall have this ; 
 
 6 +18 + 54 +162 + 486 — 3 S. or 3 times the sum. 
 2+6+18 + 54+162 =S. or once the sum. 
 
 If, now, we subtract the lower line of terms from 
 the upper, the remainder, it is clear, will be twice the 
 whole sum of the series j for it will be once the sum 
 subtracted from 3 times the sum. But, in this sub- 
 traction, all the intermediate terms cancel each other, 
 leaving only the first term, 2, to be subtracted from 
 the last term, 486. Now, this last term, 486, was 
 obtained by multiplying the last term given in the 
 question by the ratio. Therefore we might have 
 saved all our work, and simply multiplied the last 
 term by the ratio, and subtracted the first term from 
 the product. This would have given us, at once, 
 twice the sum of the series, or, in general terms, the 
 series multiplied by a number one less than the ratio. 
 Hence we have the following 
 
250 
 
 MENSURATION OF SURFACES. 
 
 Rule. — Multiply the last term by the ratio, sub- 
 tract the first term from this product, and divide the 
 remainder by the ratio diminished by one. 
 
 5. A gentleman promises his son. 11 years old, one 
 mill when he shall be 12 years old, and, on each suc- 
 ceeding birthday, till he is 21 years old, ten times as 
 much as on the preceding birthday. What will the 
 son's fortune be, without interest, when he is 21 years 
 old? 
 
 6. What is the sum of 14 terms of the series 2, 6, 
 18, &c? 
 
 First obtain the last term. 
 
 7. What is the sum of 16 terms of the series 5, 10, 
 20, &c? 
 
 8. What is the sum of 18 terms of the series 1, 2, 
 4, &c. ? 
 
 9. A builder offers to build a church for 16000 dol- 
 lars ; or, if the people prefer it, he will take for his 
 pay 1 cent for the first pew, 2 cents for the second, 4 
 for the third, 8 for the fourth, &c, till he shall have 
 received pay for forty pews. What would the meet- 
 ing-house have cost on these last-named terms? 
 
 SECTION XLII. 
 
 MENSURATION OF SURFACES. 
 
 \ 
 
 For the mensuration of the triangle and the paral- 
 lelogram, when the base and height are known, see 
 Section XV1L, Part I. 
 
 To find the area of an equilateral triangle, when the 
 sides only are known, 
 
 Square one side; multiply that product by the 
 decimal .433. 
 
MENSURATION OF SURFACES. 251 
 
 To find the circumference of a circle, when the 
 diameter is known, 
 
 Multiply the diameter by 3.1416. 
 
 1. What is the circumference of a circle the diam- 
 eter of which is 36 feet ? 
 
 2. What is the circumference of a circular race- 
 course whose diameter is l£ mile ? 
 
 3. What is the circumference of a wheel the diam- 
 eter of which is 24 feet 6 inches ? 
 
 4. What is the circumference of the earth on the 
 line of the equator, its diameter being 7925.65 miles? 
 
 To find the diameter of a circle, when the circum- 
 ference is known, 
 
 Divide the circumference by 3.1416. 
 
 5. How far is it across a circular pond, the circum- 
 ference of which is 231 rods ? 
 
 To find the area of a circle, when the diameter and 
 circumference are known, 
 
 Multiply the circumference by one fourth of the 
 diameter ; or, multiply the square of the diameter by 
 the decimal .7854. 
 
 6. What is the area of a circle whose diameter is 
 34 rods ? 
 
 7. What is the area of a circle whose diameter is 
 24 feet ? 
 
 When a circle is given, to find a square which shall 
 have an equal area, 
 
 Find the area, of the circle, extract the square root, 
 which will be one side of the square. 
 
 8. There is a circular piece of land, 40 rods in di- 
 ameter. What will be the side of a square of equal 
 area? 
 
 9. There is a circular green, containing 8 acres. 
 What will be the side of a square of equal area ? 
 
252 
 
 MENSURATION OF SOLIDS. 
 
 10. There is a circle 35 rods in diameter, and a 
 square 31 rods. Which is the greater ; and how 
 much? 
 
 SECTION XLIII. 
 
 MENSURATION OF SOLIDS. 
 
 A plane solid is bounded by flat surfaces ; a round 
 solid is bounded by curved surfaces. 
 
 To find the surface of a solid bounded by plane 
 surfaces, 
 
 Find the area of each plane surface, and add the 
 sums together for the whole surface. 
 
 A prism is a solid, whose ends are any equal, paral- 
 lel, and similar rectilineal figures, and whose sides are 
 parallelograms. 
 
 To find the solidity of a prism, 
 
 Multiply the area of the base by the height. 
 
 1. What is the solidity of a prism whose ends are 
 equilateral triangles, 14 inches on a side, and whose 
 height is 8 feet ? 
 
 A cylinder is a round solid, whose 
 ends are equal and parallel circles. 
 
 To find the solidity — the same rule 
 as for a prism. 
 
 2. What is the solid contents of a cylinder whose 
 ends are circles 18 inches in diameter, and whose 
 height is 12 feet ? 
 
 A regular pyramid is a solid whose sides are equal 
 and similar triangles, meeting in a point at the top. 
 The slant height is the distance from the point, at the 
 top, to the middle of the base of one of the triangles. 
 
MKXSURATION OF SOLIDS. 253 
 
 To find the solid contents of a pyramid, 
 Multiply the area of the base by one third the per- 
 pendicular height. 
 
 3. What is the solid contents of a four-sided pyra- 
 mid, whose base measures 40 feet on a side, and whose 
 height is 42 feet ? f 
 
 A cone is a round solid, standing on a 
 circular base, and terminating in a point 
 at the top. 
 
 To find the solidity of a cone — the 
 same rule as for a pyramid. 
 
 4. What is the solidity of a cone whose 
 base is 13 feet in diameter, and whose 
 height is 22 feet ? 
 
 5. What is the solidity of a cone whose base is 4 
 feet in diameter, and height 18 feet ? 
 
 To find the surface of a sphere, 
 
 Multiply the diameter by the circumference. 
 
 6. How many square miles of surface has the earth, 
 regarding it as a sphere the diameter of which is 
 7925.65 miles? 
 
 To find the solidity of a sphere, 
 Multiply the surface by £ of the diameter ; or, mul- 
 tiply the cube of the diameter by the decimal .5236. 
 
 To find the measure of a sphere, when the solidity 
 is given, 
 
 Divide the solidity by the decimal .5236 ; extract the 
 cube root of the quotient, which will be the diameter of 
 the sphere. 
 
 7. What is the solid contents of a sphere the diam- 
 eter of which is 14 inches ? 
 
 8. What will be the solidity of the largest sphere 
 that can be cut from a cubic block 1 foot on a side ? 
 
 22 
 
254 MISCELLANEOUS THEOREMS AND QUESTIONS. 
 
 SECTION XLIV 
 
 MISCELLANEOUS THEOREMS AND QUESTIONS. 
 
 Given the sum and the difference of two numbers, 
 to find the larger and the smaller numbers, 
 
 Add half the difference to half the sum, for the 
 larger ; subtract half the difference from half the sum, 
 for the smaller. 
 
 1. The sum of two numbers is 140, the difference 
 32. What are the numbers ? 
 
 2. The sum of two numbers is 572, the difference 
 94. What are the numbers ? 
 
 3. The sum of two numbers is 187, the difference 
 44. What are the numbers ? 
 
 4. The sum of two numbers is 190, the difference 
 57. What are the numbers? 
 
 Given the sum and the product of two numbers, to 
 find the larger and the smaller number. 
 
 5. There are two numbers ; their sum is 80, and 
 their product 1551. What are the numbers? 
 
 The theorem on which the solution of this question 
 depends, is this: — If a number be divided into two 
 equal parts, and also into two unequal parts, the 
 product of the two equal parts, that is, the square of 
 half the number, will equal the product of the two 
 unequal parts, plus the square of the difference between 
 one of the equal and one of the unequal parts. 
 
 Take 16: divide it equally, 8 + 8, and unequally, 
 9 + 7; the difference between an equal and an unequal 
 part, 1; 8 2 = 64: 9X7 = 63 ; loss, 1, which is the 
 square of the difference. 
 
 Divide unequally into 10 + 6; difference, 2; 8 2 = 64: 
 
MISCELLANEOUS THEOREMS AND QUESTIONS. 255 
 
 10X6 = 60; loss, 4, which is the square of the dif- 
 ference. 
 
 Divide unequally into 11 + 5; difference, 3 ; 8 2 = 
 64:11X5 = 55; loss, 9, which is the square of the 
 difference. 
 
 Hence, to solve the question, — Subtract the product 
 of the unequal parts from the square of half the num- 
 ber ; find the square root of the difference ; add it to 
 half the number for the greater, subtract it from half 
 the number for the less. 
 
 6. There are two numbers the sum of which is 
 100. Their product is 2419. What are the num- 
 bers? 
 
 7. There is a rectangular piece of land, the two. 
 contiguous boundaries of which measure together 120 
 rods. The area of the piece is 2975 square rods. 
 What is its length ? What is its width ? 
 
 8. A rectangular piece of land is surrounded by 480 
 rods offence. The area is 13104 square rods. What 
 is its length and breadth ? 
 
 Given the sum of two numbers, and the difference 
 of their squares, to find the greater and the less 
 number, 
 
 The product of the sum and the difference of two 
 numbers is equal to the difference of their squares. 
 
 Take the two numbers 6 and 9; their sum is 15; 
 their difference, 3; 15X3 = 45. The square of 6 is 
 36; the square of 9 is 81 ; 81 — 36 = 45. 
 
 Hence, if we divide the difference of the squares by 
 the sum, the quotient will be the difference, and from 
 this we may find the greater and the less. 
 
 9. There is a triangle, the hypotenuse and one leg 
 of which measure together 90 feet ; the other side 
 measures 40 feet. What are the lengths of the two 
 first-named sides respectively ? 
 
£56 MISC KLLANEOIS QUESTIONS. 
 
 10. There is a triangle ; the hypotenuse and base 
 measure together 120 feet ; the perpendicular measures 
 64 feet. What is the length of each of the first- 
 named sides? 
 
 This principle enables you to multiply readily any 
 two numbers, one of which exceeds a certain number 
 of tens by as many units as the other falls short of 
 it ; as, 63X57. The first exceeds 60 by 3 ; the second 
 falls short of it by 3. 
 
 Square the tens — 3600; subtract from this the 
 square of the units — 9. 3591, answer. 
 
 11. Multiply 64X56. 3600 — 16 = 3584, answer. 
 
 12. Multiply 82X78; 47X33; 92X88. 
 
 Theorem of Parallel Sections. 
 
 If a line is drawn in a triangle, parallel to one of 
 the sides, and meeting the other two sides, it divides 
 those sides proportionally ; and the small triangle cut 
 off is similar to the whole undivided triangle. 
 
 If a plane pass through a pyramid or cone, parallel 
 to the base, it divides all the lines it meets proportion- 
 ally; and the small solid cut off at the top is similar 
 to the whole undivided solid. 
 
 13. There is a triangular field, containing 7 acres. 
 A line is drawn through it, parallel to one side, cutting 
 the other two sides § of the distance from the apex to 
 the base. How much land does it cut off? 
 
 14. There is a board, in the form of a right-angled 
 triangle, 8 feet in perpendicular height. How far 
 from the top must a line be drawn parallel to the base 
 to cut off f of the board ? 
 
 15. A man had a field of 3 acres, in the form of a 
 right-angled triangle, with the base equal to the per- 
 pendicular. He sells one acre, to be cut off by a line 
 running parallel to the base. He then sells another 
 
MISCELLANEOUS QUESTIONS. 257 
 
 acre, to be cut off by another line parallel to the base. 
 How far from the base must the first line be ? How 
 far from the base must the second line be ? 
 
 16. There was a cone 20 feet high ; but, the upper 
 part being defective, 11 feet in height of the top was 
 taken down. How much of the cone has been re- 
 moved? 
 
 17. There was a square pyramid, the base of which 
 measured 48 feet on a side. When it was partly com- 
 pleted, its slant height, measuring from the middle of 
 a side at the bottom to the middle of the same side 
 at the top, was 40 feet, and the width of a side at 
 the top was 12 feet. How high was the apex of the 
 pyramid when completed? And what part of the 
 pyramid remained to be built ? 
 
 18. In a right-angled triangle, whose base and per- 
 pendicular are equal, what is the ratio of the square of 
 the hypotenuse to the square of the base ? What is 
 the ratio of the hypotenuse to the base ? 
 
 19. If a man travel on Monday 6 miles due north, 
 and on Tuesday 8 miles due east, how far is he then 
 from where he set out on Monday ? 
 
 If, on Wednesday, he travels 12 miles due south, 
 how far will he then be from where he was on Mon- 
 day morning? How far from where he was on Mon- 
 day night? 
 
 20. In repairing a meeting-house, it was thought 
 desirable to alter the form of the posts, which were 
 one foot square. It was proposed to cut away the 
 corners, so as to make them regular eight-sided prisms. 
 How wide must each face be, so as to have all the 
 eight faces of exactly the same width ? 
 
 21. Sound moves through the air at the rate of 
 1090 feet in a second. How long would it be in 
 passing 100 miles ? 
 
 22. At the above rate, how long would it require 
 for a wave of sound to compass one half the circuit of 
 
 22* R 
 
258 SPECIFIC GRAVITY. 
 
 the globe, on the line of the equator, the circumfer- 
 ence being 24899 miles? 
 
 23. Two men purchase, in equal shares, a stick of 
 hewn timber, 40 feet long, 2 feet square at the larger 
 end, and 1 foot square at the smaller end. How far 
 from the larger end shall they cut it in two, so that 
 each may have exactly one half? 
 
 24. A surveyor, in laying out a lot of land, first runs 
 a line due north, to a certain tree. From the tree he 
 runs between south and west till he comes to a point 
 due west from the place he started from. The whole 
 of these two lines is 212 rods; but those who measured 
 it neglected to note how far the tree was from the 
 starting point. On measuring a third line, connecting 
 the extremities of the two first lines, they find it 98 
 rods. How many acres does the triangle contain ? 
 
 Specific Gravity. 
 
 The specific gravity of a body is its weight com- 
 pared with the weight of an equal bulk of water. To 
 find the specific gravity of a body heavier than water, 
 
 Weigh the body in water, and out of water, and 
 fi,nd the difference in the weight. Then, as the dif- 
 ference in the weight is to the weight out of water, so 
 is 1 to the specific gravity. 
 
 The weight of a cubic foot of water is 62J- lbs. 
 avoirdupois. The specific gravity of the most im- 
 portant of the metals is as follows : — 
 
 Iron, 7.78; Tin, 7.2; Copper, 8.895; 
 
 Lead, 11.325; Mercury, 13.568; Silver, 10.51. 
 Gold, 19.257; Platinum, 21.25. 
 
 From the above table, we may find the weight of 
 any mass of one of the above metals the magnitude 
 of which is known. 
 
 25. What is the weight of a cubic foot of iron ? 
 
MECHANICAL POWERS. 259 
 
 26. What is the weight of an iron ball six inches in 
 diameter ? 
 
 27. What is the diameter of a 24 lb. cannon ball ? 
 
 28. What is the diameter of a 48 lb. cannon ball ? 
 
 29. What is the weight of a cannon ball one foot in 
 diameter ? 
 
 30. If the column of mercury in the barometer be 
 29J- inches high, what would be the weight of a col- 
 umn of mercury of that height one inch square ? 
 
 31. As the weight of mercury in the barometer 
 equals the weight of the atmosphere on the same base, 
 what is the pressure of the atmosphere on a square foot, 
 when the mercury in the barometer is 29 inches high? 
 
 The height to which water will rise in a suction 
 pump, and the height of the mercury in the barometer, 
 are in inverse proportion to the specific weight of 
 those two bodies ; that is, the water is as much higher 
 than the mercury, as mercury is heavier than water. 
 
 32. How high will water rise in a suction pump, 
 when the mercury in the barometer is 291 inches 
 high? 
 
 33. What is the weight of a copper prism, its base 
 being an equilateral triangle, 3 inches on a side, and 
 its height 15 inches? 
 
 Mechanical Powers. 
 
 The object to be gained by the application of me- 
 chanical powers, is to overcome a large weight or 
 resistance by means of a comparatively small power. 
 
 In doing this, however, the power must move 
 through a space as much larger than the space 
 which the weight moves through, as the weight is 
 heavier than the power ; 
 
 Or, the distance X tv eight of the power = distance 
 X weight of the weight or mass to be moved. 
 
260 THE LEVER. 
 
 This is the great law of mechanical powers, and 
 applies to them all, without exception. In the practi- 
 cal application of them, a certain allowance must be 
 made on account of the friction in the machine. The 
 amount of friction differs in different powers. No ac- 
 count of this will be taken in the examples which 
 follow, unless it is particularly mentioned; nor will 
 any difference be made between the power when in 
 motion, and when in equilibrium, or at rest. 
 
 The Lever. 
 
 The lever is a straight bar, used to support or raise 
 heavy weights. It is supported by a prop or fulcrum, 
 placed near the weight ; and the power is applied at 
 the other end of the lever. The distance from the 
 fulcrum to the weight, is called the shorter arm, the 
 distance from the fulcrum to the power, the longer 
 arm, of the lever. If the lever were to turn over 
 the fulcrum as a centre, the longer arm would de- 
 scribe a larger circle, and the shorter arm would de- 
 scribe a smaller circle. The circumferences of these 
 two circles, or an arc of the same number of degrees 
 in both, would be the distances passed through by the 
 power and the weight respectively. But we may take 
 the arms themselves as representing these distances, 
 for they are the radii of the two circles; and the radii 
 of different circles have the same ratio to each other 
 as the circumferences. 
 
 We have, therefore, this proportion: — 
 
 The longer arm is to the shorter arm as the weight 
 to the power. Or, let I- a. stand for the longer arm ; 
 s- a., for the shorter ; w., for the weight ; and p., 
 for the power. 
 
 1. a. : s. a. : : w. : p. ; and any change admissi- 
 ble in the terms of a proportion, may be made in 
 these terms. 
 
THE WHEEL AND AXLE. 261 
 
 34. If a lever 10 feet long have its fulcrum one foot 
 from the weight, how great must the power be, to 
 raise a weight of 1640 lbs. ? 
 
 35. If a lever 10 feet long have its fulcrum 18 
 inches from the weight, how great a weight will, 
 be raised by a power of 160 lbs. ? 
 
 36. A lever 18 feet long rests on a fulcrum 2 feet 
 from the end. How large a weight can two men 
 raise, (one weighing 164 lbs., the other 172 lbs.,) by- 
 applying their weight at the longer arm ? 
 
 37. If a lever 7£ feet long rest on a fulcrum 15 
 inches from the end, how heavy must the power be to 
 support a ton gross weight ? 
 
 38. If the weight be 3600 lbs., and the power 140 
 lbs., how far from the weight must the fulcrum be 
 placed, under a lever 12 feet long, so as to have the 
 weight and power balance? 
 
 39. If the weight be 6480 lbs., the power 312, and 
 the lever 16£ feet long, how far from the weight must 
 the fulcrum be, to have the weight and power bal- 
 ance ? 
 
 40. In a certain machine, it is necessary to adjust a 
 lever 3 feet long, so that a power of l± lbs. shall bal- 
 ance 13£ lbs. How far from the weight must the ful- 
 crum be placed? 
 
 The Wheel and Axle. 
 
 In this case, the power is applied at the circumfer- 
 ence of the wheel, and the weight is drawn up by a 
 rope passing round the axle, which is a smaller wheel. 
 The principle, therefore, is the same as in the lever. 
 The semi-diameter of the wheel is the longer arm ; 
 the semi-diameter of the axle, the shorter arm. 
 
 41. In a grocery store, the wheel and axle used in 
 raising heavy articles^ are of the following dimensions, 
 
262 
 
 THE WHEEL AND AXLE. THE SCREW. 
 
 viz. : the wheel 5 feet in diameter, the axle 7 inches 
 in diameter. What power must be applied to the rope 
 passing over the wheel, to balance a barrel of flour, 
 weighing 205 lbs., suspended by a rope passing over 
 the axle ? 
 
 42. With the same wheel and axle, what power 
 will raise a box of sugar, weighing 431 lbs., adding £ 
 to the power, to overcome the friction ? 
 
 43. In digging a well, the wheel employed in rais- 
 ing stones and earth, is 6 feet in diameter ; the axle, 
 6£ inches in diameter. What power will raise a rock 
 weighing 640 lbs., adding £ to the power, to overcome 
 the friction ? 
 
 44. If a wheel is 14 feet in diameter, what must be 
 the diameter of the axle, in order that a power of 140 
 lbs. may balance 5760 lbs. ? 
 
 45. If an axle is 16} inches in diameter, what must 
 be the diameter of the wheel, in order that a power of 
 56 lbs. may balance a weight of 1344 lbs. ? 
 
 The Screw. 
 
 In this case, the distance passed through by the 
 power in one revolution, is equal to the circumference 
 of the circle described by the lever which turns the 
 screw. The distance passed by the weight is the dis- 
 tance between two threads of the screw, measured in 
 the direction of its axis. 
 
 In the practical application of this power, a large 
 allowance must be made to compensate for the friction. 
 
 46. If the lever of a screw is 11 feet in length, and 
 the distance of the threads 1| inches, what power will 
 raise a weight of 6431 lbs., making no allowance for 
 friction ? 
 
 47. With the same conditions as in the last exam- 
 ple, what weight will be raised by a power of 124 
 lbs.? 
 
STRENGTH OF BEAxMS TO RESIST FRACTURE. 263 
 
 48. What must be the length of the lever of a 
 screw, the threads of which are 1 inch asunder, in 
 order that a power of 3 lbs. may balance a weight of 
 1640 lbs., making no allowance for friction? 
 
 49. How far asunder must the threads of a screw 
 be, so that, with a lever of 8£ feet in length, 26 lbs. 
 will balance 6590 lbs. 
 
 Strength of Beams to resist Fracture. 
 
 [See Section XX., Part I.] 
 
 In addition to the principles that have already been 
 stated in estimating the strength of timbers, the fol- 
 lowing are among the most important. It is under- 
 stood, in all cases, when timbers are compared, that 
 they are of the same wood, and equally good in 
 quality. 
 
 When the depth of two beams is the same, and the 
 thickness the same, the strength is inversely as the 
 length. 
 
 50. There are two beams, of the same depth and 
 thickness ; one, 18 feet in length ; the other, 13. 
 The longer beam will sustain a weight of 68 cwt. 
 What weight will the shorter beam sustain ? 
 
 51. Two beams, of the same size, measure in length 
 22 and 17£ feet. The shorter beam will sustain 76 
 cwt. How much will the longer beam sustain ? 
 
 52. Two beams, of equal thickness, have a depth 
 of 14 and 16 inches respectively. The deeper beam 
 is 20 feet long, and will sustain 84 cwt. The other 
 is 17 feet in length. What weight will it sustain ? 
 
 First take into view the length; then, in a second 
 proportion, the depth. 
 
 53. If a beam, 25 feet in length and 9 inches in 
 depth, will sustain a weight of 12 cwt.. what weight 
 
264 STRENGTH OF BEAMS TO RESIST FRACTURE. 
 
 will be sustained by a beam of the same thickness 18 
 feet long, and 10 inches in depth ? 
 
 When beams are of the same length and depth, the 
 strength varies directly as the width. 
 
 54. There are two beams of equal length and 
 depth ; one, 9 inches in width ; the other, 7£ inches. 
 The wider beam will sustain 47 cwt. What weight 
 will the narrower beam sustain? 
 
 55. There are two beams of equal depth. One 
 measures 20 feet in length, and 11 inches in width, 
 and will sustain 94 cwt. The other beam is 14 feet 
 in length, and 10 inches in width. What weight 
 will it sustain ? 
 
 56. There are two beams, of the same width. One 
 measures 16 feet in length, and 10 inches in depth, 
 and will sustain 66 cwt. The other is 18 feet long, 
 and 12 inches in depth. What weight will it sustain ? 
 
 It is sometimes desirable to know how the strength 
 of a beam will vary by removing the point on which 
 the pressure is made, as in the following example : — 
 
 57. A beam 20 feet in length will sustain, at its 
 centre, a weight of 44 cwt. What weight will it sus- 
 tain applied 7 feet from one end ? 
 
 The following formula will give the variation in the 
 strength : — 
 
 As the product of the two unequal sections of the 
 beam (in this case, 13X7) is to the square of half the 
 length, so is the weight which the beam will sustain 
 at the centre to the weight it will sustain at the other 
 given point. 
 
 58. A beam 24 feet in length will sustain at its 
 centre 56 cwt. What weight will it sustain at the 
 distance of 9 feet from one end ? 
 
STIFFNESS OF BEAMS TO RESIST FLEXURE, 
 
 265 
 
 59. A beam 28 feet in length will sustain at its 
 centre 33 cwt. What weight will a beam of the same 
 width and length, and of f the depth of the former, 
 sustain at the distance of 10 feet from the end ? 
 
 Stiffness of Beams to resist Flexure. 
 
 The stiffness of beams of the same length and 
 width varies as the cube of the depth. If the depth 
 is the same, the stiffness varies as the width. 
 
 60. There are two beams, of equal length and 
 width. One is 8 inches in depth; the other, 11 
 inches. If it require 30 cwt. to bend the former 1 
 inch, what weight will it require to bend the latter 1 
 inch ? 
 
 61. There is a stick of timber 8 inches by 6. If it 
 require 24 cwt. to bend it 2 inches when lying flat, 
 what weight will bend it 2 inches when turned up on 
 the edge? 
 
 62. If 10 cwt. will bend the stick just described l£ 
 inches when it lies flat, what weight will be requisite 
 to bend it 1£ inches when turned up on the edge ? 
 
 63. There is a board 12 inches wide, and 1 inch in 
 thickness. What is the ratio of its strength when 
 lying flat, supported at the ends, to its strength when 
 turned edgewise ? 
 
 64. If it require 12 lbs. to bend the same board £ an 
 inch, when lying flat, how much will it require to 
 bend it £ an inch when turned edgewise ? 
 
 23 
 
266 BUSINESS FORMS AND INSTRUMENTS. 
 
 SECTION XLV. 
 
 BUSINESS FORMS AND INSTRUMENTS. 
 Promissory Notes. 
 
 1. — On Demand, with Interest. 
 
 $500. — Boston, March 1, 1846. For value re- 
 ceived, I promise A. B. to pay him, or his order, five 
 hundred dollars, on demand, with interest. T. M. 
 
 2. — On Time, with Interest. 
 
 $200. — Boston, March 1, 1846. For value re- 
 ceived, I promise A. B. to pay him, or his order, two 
 hundred dollars, in three months, with interest. 
 
 T. M. 
 3. — On Time, without Interest. 
 
 $400. — Boston, March 1, 1846. For value re- 
 ceived, I promise A. B. to pay him, or his order, four 
 hundred dollars, in sixty days from date. I. M. 
 
 4. — Payable by Instalments, with Periodical Interest 
 
 $1000. — Boston, March 1, 1846. For value re- 
 ceived, I promise A. B. to pay him, or his order, one 
 thousand dollars, as follows, viz. ; — two hundred dol- 
 lars in one year, two hundred dollars in two years, and 
 six hundred dollars in three years, from this date, with 
 interest semi-annually. I. M. 
 
 Remarks on Promissory Notes. 
 
 When the words "or order" are inserted in a note, 
 the holder of the note may endorse it, that is, write 
 
BUSINESS FORMS AND INSTRUMENTS. 267 
 
 his name on the back of it, and pass it to a third per- 
 son, who can collect it in the same manner as if he 
 were the original holder. If the maker of the note 
 neglects to pay, the holder may collect it of the en- 
 dorser. 
 
 If the words "or bearer" are inserted instead of 
 "or order," any person who has possession of the note 
 may collect it of the maker. Such a note would be 
 like a bank note, which passes from hand to hand 
 without endorsement. 
 
 A note, in order to be legal in the first holder's 
 hands, must be for value received. A note, therefore, 
 given to pay a debt incurred in gambling or betting, 
 cannot be collected by law, unless it has passed into 
 the hands of an innocent holder. 
 
 When a note contains the promise to pay interest 
 annually, and the interest is not collected annually, the 
 law does not permit the holder to draw compound in- 
 terest. The holder may compel the payment of the 
 interest when it becomes due ; but if he neglect to do 
 this, he can recover only simple interest. 
 
 When a note is given to pay in a certain commodity, 
 as wood, grain, &c, if the note is not paid when due, 
 the holder may compel the payment of the equitable 
 value of the commodity in money. The reason of 
 this is, that it is supposed that the commodity may 
 have a value to the holder at the time when it is 
 promised, which it will lose if not paid then. 
 
 Receipts. 
 
 1. — A general Form. 
 
 $500. — Boston, March 1, 1846. Received of 
 O. P. the sum of five hundred dollars, in full of all 
 demands against him. A. B. 
 
268 BUSINESS FORMS AND INSTRUMENTS. 
 
 2. — For Money paid by another Person. 
 
 $300. — Boston, March 1, 1846. Received of 
 O. P., by the hand of Y. Z., three hundred dollars, in 
 full payment for a chaise by me sold and delivered to 
 the said O. P. A. B. 
 
 3. — For Money received for Another. 
 
 $700. —Boston, March 1, 1846. Received of 
 O. P. seven hundred dollars, it being for the balance 
 of account due from said O. P. to Y. Z. A. B. 
 
 4. — In Part of a Bond. 
 
 $3000. — Boston, March 1, 1846. Received of 
 O. P. the sum of three thousand dollars, being a part 
 of the sum of five thousand dollars due from said O. P. 
 to me on the day of . A. B. 
 
 5. — For Interest due on a Pond. 
 
 $600. —Boston, March 1, 1846. Received of 
 O. P. six hundred dollars, due this, day from him to 
 me, as the annual interest on a bond, given by said O. 
 P. to me on the 1st of May, 1831, for the payment to 
 me of ten thousand dollars in three years, with inter- 
 est annually. A. B. 
 
 6. — On Account. 
 
 $50. — Boston, March 1, 1846. Received of O. P. 
 fifty dollars, for which I promise to account to him on 
 a settlement between us. A. B. 
 
 7. — Of Papers. 
 
 Boston, March 1, 1846. Received of O. P. several 
 contracts and papers, which are described as follows: — 
 [describe the papers ;] which I promise to return to the 
 said O. P. on demand. A. B. 
 
business forms and instruments. 269 
 
 Order at Sight. 
 
 Boston, April 18, 1846. At sight, pay to the order 
 of John Brown, one thousand dollars, value received, 
 which place to account of 
 
 Your obedienf servants, A. W. & Co. 
 
 Jacob Smith, Esq., New York. 
 
 Order on Time. 
 
 Boston, April 18, 1846. Six months after date, pay 
 to the order of John Brown, one thousand dollars, value 
 received, which place to the account of 
 
 Your obedient servants, A. W. & Co. 
 
 Jacob Smith, Esq., New York. 
 
 Remarks. — If J. B. present this order to J. S., and J. S. write his 
 name across the face of it, it becomes what is called an acceptance. 
 J. S. agrees to pay it at the date named. 
 
 If J. B. writes his name on the back of the acceptance, it becomes 
 negotiable. He may pass it to a third person, who may endorse it, 
 and pass it to a fourth. All those whose signatures are on the order 
 are bound for its payment ; — the acceptor to the drawer ; the acceptor 
 and drawer to the first endorser ; and they and each endorser to the 
 one succeeding him ; and the last endorser, and all previous parties, 
 to the holder. 
 
 Award by Referees. 
 
 We, the undersigned, appointed by agreement of the 
 parties herein named, having met the parties, and heard 
 their several allegations, arguments, and proofs, and 
 duly considered the same, do award and determine that 
 A. B. shall recover of C. D. the sum of , to- 
 gether with all the costs of this reference, which are 
 
 to the amount of ; and that, this shall be final 
 
 and in full of all claims and dues of the parties on 
 matters herein referred to us. I. M. 
 
 R. N. 
 L. S. 
 
 23 
 
270 business forms and instruments. 
 
 Letter of Credit for Goods. 
 
 Boston, March 1, 1846. 
 Messrs. Y. & Z., Merchants, Baltimore. 
 
 Gentlemen, — Please to deliver Mr. C. D., 
 
 of , or to his order, gt>ods and merchandise to an 
 
 amount not exceeding in value, in the whole, one hun- 
 dred dollars ; and, on your so doing, I hereby hold my- 
 self accountable to you for the payment of the same, 
 in case Mr. C. D. should not be able so to do, or should 
 make default, of which default you are required to give 
 me reasonable and proper notice. 
 
 Your obedient servant, A. B. 
 
 A letter of credit for money may be given in the general form of 
 the above ; specifying, in the letter, the amount of credit granted. 
 
 Power of Attorney. 
 Know all men by these presents, that I, A. B., of 
 -, do hereby appoint C. D., of , to be my 
 
 sufficient and lawful attorney, to act for me, and in my 
 name, [here slate the ohjects for which he is to act.] And 
 for the purposes aforesaid, I hereby grant unto my said 
 attorney full power to execute all needed legal instru- 
 ments, to institute and prosecute all claims in my be- 
 half, to defend all suits against me, to submit to arbi- 
 tration, or settle all matters in dispute, and to do all 
 such acts as he shall think expedient for the full ac- 
 complishment of the objects for which he is appointed 
 my attorney, as fully as I might myself do them if 
 present ; and all acts done by the said C. D., my attor- 
 ney, under authority of this appointment, I will ratify 
 and confirm. 
 
 In testimony whereof, I hereby set my hand and 
 
 seal, this day of , in the year . 
 
 A. B. [l. s.] 
 
 Signed, sealed, and delivered, 
 in the presence of S. N. 
 W. F. 
 
THE STANDARD OF WEIGHTS AND MEASURES. 271 
 
 SECTION XLVI. 
 
 ON THE STANDARD OF WEIGHTS AND MEASURES. 
 
 In the earlier states of society, the standard of 
 weights and measures was, of necessity, very indefi- 
 nite and fluctuating. In one nation it was one thing; 
 in another nation, another j and in no case was it de- 
 serving of a very high degree of confidence. 
 
 Sometimes the length of the king's foot was the 
 standard for all measures of length ; again, the length 
 of the king's arm, from the elbow to the extremity of 
 the fingers, was made the standard. 
 
 The length of journeys was measured by the hours 
 or days employed in performing them, or by the num- 
 ber of steps taken. 
 
 In land measure, the standard was, what a yoke of 
 oxen could plough in a day, when, in fact, one yoke 
 might plough twice as much as another. 
 
 In dry measure, it was as much as a man could 
 conveniently carry, without first deciding how strong 
 the man should be. 
 
 In weight, the standard was, what a man could hold 
 and swing in his hand. 
 
 Sometimes vegetables were taken as measures; as, 
 "three barley corns make one inch." But barley 
 corns do not all grow of exactly equal length, any 
 more than the feet and arms of kings. 
 . As science advanced, and commerce became far- 
 Uher extended among different nations, the mischiefs 
 of these vague and fluctuating methods of measure- 
 ment became more and more deeply felt. 
 
 But it was far easier to see the faults of the old sys- 
 tem, than to devise a new one that should be perfect. 
 What object could be selected as an ultimate standard 
 for all weights and measures? 
 
272 THE STANDARD OF WEIGHTS AND MEASURES. 
 
 We have seen that the parts of animals or of vege- 
 tables are too liable to change to deserve any confi- 
 dence. If some arbitrary standard should be adopted, 
 as a foot, or yard, and this measure should be kept as 
 the standard, by which all others should be tried, what 
 security could there be that it would never be altered 
 by fraud or destroyed by accident ? Or, if some natu- 
 ral distance were taken, as the distance between two 
 points of some well-known rock or cliff, this distance 
 might vary with a change of temperature or be altered 
 by some convulsion of nature. 
 
 We will proceed to give a short account of the 
 English system of weights and measures, adopted by 
 their Act of Uniformity, which took effect Jan. 1, 1826. 
 To begin with measures of capacity ; all English 
 measures of capacity, whether for liquors or grain, 
 are referred to the standard imperial gallon. This 
 gallon contains 277i cubic inches. From this gallon, 
 quarts, pints, and gills, are obtained by subdivision ; 
 and pecks and bushels by multiplication. Hence you 
 can find the number of cubic inches in an English 
 quart, pint, peck, or bushel. Thus the adoption of the 
 imperial gallon introduces entire uniformity into all 
 English measures of capacity. It refers them all 
 ultimately to the cubic inch. We must now inquire, 
 what has been done to fix the measure of the inch ; 
 for, if there is any error or variation here, it will ren- 
 der false all the measures of capacity which depend 
 upon it. 
 
 To determine the measure of the inch, it is made 
 by law -gV of the standard yard. That standard yard* 
 is a straight brass rod in the custody of the clerk of 
 the House of Commons. The yard is the distance 
 on that rod between the centres of the points in the 
 two gold studs or pins in the rod. And as heat would 
 make the yard longer, and cold would make it shorter, 
 the law requires that it shall be used w r hen it is of the 
 
THE STANDARD OF WEIGHTS AND MEASURES. 273 
 
 temperature of 62° (Fahrenheit.) This standard yard, 
 however, may be destroyed by accident. We must 
 then inquire for a still more permanent standard. To 
 effect this, the law declares that the standard yard, if 
 destroyed, may be restored, by making it ttt8M °f tne 
 length of a pendulum, that vibrates seconds in the lati- 
 tude of London, in a vacuum, at the level of the sea. : 
 If all these conditions are fulfilled, a pendulum that 
 vibrates seconds must have an absolutely invariable 
 length. 
 
 Thus we have brought the whole system of meas- 
 ures back to seconds, as the standard. The whole 
 scheme now depends upon seconds being of an in- 
 variable length. 
 
 Seconds are parts of a year. The year is not made 
 up by multiplying seconds, but seconds are obtained 
 by dividing the year. If, then, the year is of a fixed 
 length, seconds are so. Now, the year is the time of 
 the revolution of the earth round the sun. It is the 
 same, without change, from one year to another, and 
 from century to century. 
 
 Thus the whole system of measures has been 
 brought, for its ultimate standard, to the unalterable 
 period of the earth's revolution round the sun. 
 
 We will now retrace the steps of this investiga- 
 tion, beginning with the primary standard, the earth's 
 yearly revolution. 
 
 The time of the earth's revolution round the sun is 
 always the same. Therefore, a second, which is a 
 certain part of this time, is an exact measure. If the 
 second is a fixed measure, then the pendulum which, 
 under the same circumstances, vibrates seconds, is of 
 a fixed length. If the length of the pendulum vibra- 
 ting seconds is fixed, the length of the standard yard 
 is fixed, for it is $f$$$$ of the pendulum. If the 
 standard yard is fixed, the inch is fixed ; consequently 
 the cubic inch, the gallon, quart, pint, gill, and bushel. 
 
 s 
 
274 THE STANDARD OF WEIGHTS AND MEASURES. 
 
 In the preceding investigation, no mention has been 
 made of the standard of weight. It is obtained by 
 making a cubic inch of distilled water equal to 
 252.458 grs., of which 5760 make a pound Troy, 
 and 7000 make a pound avoirdupois. 
 
 Thus weights and measures are alike brought to an 
 unalterable standard. 
 
 The imperial gallon contains 277^ cubic inches. 
 
 The Winchester* gal., wine measure, 231 " " 
 
 " " " beer measure, 282 " " 
 
 The imperial gallon of water weighs 1 /6s. avoirdupois. 
 
 The system of weights and measures established 
 by law in the United States, is very nearly the same 
 as the English. The gallon, United States measure, 
 contains 9 lbs. 14 oz. of water. This is the legal 
 standard for all measures, dry and liquid. In many 
 parts of the country, however, especially in the interior, 
 the legal standard has not supplanted the system de- 
 rived in earlier times from the English. 
 
 In France, where th§ system of weights has been 
 carried to greater perfection than in any other country, 
 the decimal ratio is* adopted in all denominations. In 
 some cases, however, there is still retained some part 
 of the old system, combined, with the decimal. 
 
 In obtaining an ultimate standard of measure, the 
 French measured one quarter of a meridian line of 
 longitude. 
 
 One ten millionth part of this arc they made the 
 basis of their system of measures. This standard, 
 the metre, is 3.28 feet. The lower denominations are 
 made by successive divisions of this, by 10, 100, &c, 
 and the -higher by multiplication. 
 
 * Winchester ; so called because the standard measures were kept 
 *at Winchester. 
 
THE STANDARD OF WEIGHTS AND MEASURES. 275 
 
 The following table presents the French decimal 
 weights and measures, with the English equivalents. 
 
 French Long Measure. 
 
 feet. 
 
 10 millimetres make .... 1 centimetre, 0328 
 
 10 centimetres, 1 decimetre, 328 
 
 10 decimetres, 1 metre, 3.28 
 
 10 metres, 1 decametre, 32.8 
 
 10 decametres, 1 hectometre, 328 
 
 10 hectometres, 1 kilometre, 3280 
 
 10 kilometres, 1 myreametre, .... 32800 
 
 French Square Measure. 
 
 The unit square measure is the are, which is the 
 square of the decametre ; consequently, it is the square 
 of 32.8 feet, — a little less than 4 square rods. 
 
 This unit is multiplied for the higher denomina- 
 tions, and divided for 'the lower, in the same way 
 as the metre. 
 
 French Decimal Weight. 
 
 10 milligrammes make . 1 
 10 centigrammes, ...... 1 
 
 10 decigrammes, 1 
 
 10 grammes, 1 
 
 10 decagrammes, 1 
 
 10 hectogrammes, . . . . 1 
 
 10 kilogrammes, 1 
 
 10 myriagrammes, .... 1 
 10 quintals, 1 
 
 decigramme, . , 
 gramme, . . . 
 decagramme, , 
 hectogramme, 
 kilogramme, . , 
 myriagramme, 
 quintal, .... 
 million , 
 
 grs. Trov. 
 
 .1543402 
 1.543402 
 15.43402 
 154.3402 
 1543.402 
 15434.02 
 154340.2 
 1543402. 
 15434020. 
 
 
276 MISCELLANEOUS EXAMPLES. 
 
 MISCELLANEOUS EXAMPLES. 
 
 The examples that follow are designed to carry 
 still farther the practice in Written Arithmetic. 
 
 1. James Ball bought of Amos Sewall three pieces 
 broadcloth, measuring 12^, 13, and 24 yards, at 
 $4.87^ per yard; five pieces kerseymere, measuring 
 24£, 25, 27, 26i, and 26 yards, at 67 cts. per yard ; 
 eight pieces cotton sheeting, measuring 33 yards each, 
 at 9£ cents per yard ; £ per cent. off. What was the 
 amount of the bill ? 
 
 2. Bought of John Jones, on six months' credit, 47 
 yards broadcloth, at $4.31 per yard; 16 yards vest- 
 ings, at $1.15 per yard ; 63^ yards satinet, at 62^ cents 
 per yard ; 5J- pieces sheeting, containing 33 yards a 
 piece, at 8f cents per yard. John Jones agrees, if paid 
 in cash, to deduct 4 per cent, from the bill. What is 
 the cash amount of the bill ? 
 
 3. Bought of Asa Wood, on six months' credit, 45 
 barrels flour, at $5.37 per barrel; four hhds. molasses, 
 containing 124^ 131, 134, and 136 gallons, at 27£ 
 cents per gallon ; five bags coffee, containing 54£, 56, 
 49£, 62, and 65^- lbs., at 8£ cents per pound. Gave, in 
 payment of the above, a note payable in six months. 
 What was the cash value of the note when it was 
 given, reckoning interest at 6 per cent. ? 
 
 4. A bought of B, on six months' credit, goods with 
 the amount as follows : — 
 
 April 3, 1845, $254.75. 
 
 June 8, " 135.00. 
 
 Aug. 1, " v . . . . 200.00. 
 
 ♦Sept. 14, " 168.25. 
 
 At what date shall A make an equated payment of the 
 whole amount ? 
 
MISCELLANEOUS EXAMPLES. 277 
 
 5. A gives B a note for $600, payable in six months. 
 What is the cash value of the note, two months after 
 date, reckoning the interest at 6 per cent. ? 
 
 6. A man agrees to dig and stone a well on the fol- 
 lowing terms: — $1.00 per foot for the first 10 feet, 
 $2.50 per foot for the second 10 feet, and $4.00 per 
 foot for the remainder, till he finds a supply of water. 
 For every foot of rock through which he digs, he 
 shall receive double pay. He digs 42 feet in all, and 
 through rock from 17 to 31£ feet from the surface. 
 What pay is he entitled to for the whole ? 
 
 7. A man engages to build 160 rods of road, one half 
 for $1.42 per rod, the other half for $1.83 per rod. 
 He hires 93 days of men's labor at 84 cents per day ; 
 pays for board of the same at $1.50 per week of six 
 days; pays for tools and repairs, $11.60. He works 
 himself, with 4 oxen and his son, 34 days. What 
 wages will he receive per day for himself, for his oxen, 
 and for his son, allowing for the 4 oxen as much as for 
 himself, and for his son half as much ? 
 
 8. A bought a lot of standing wood for 105 dollars, 
 and agreed with B to cut and haul it to market for 
 three fifths of the proceeds. There were 54 cords of 
 pine, which was sold for $2.84 per cord, and 61 cords 
 of hard wood, which sold for $4.75 per cord. Did A 
 gain, or lose, and how much? B labored himself 35 
 days, employed one yoke of oxen 24 days, and hired 
 68 days of men's labor at 85 cents per day. What 
 pay does he receive per day for himself and for his 
 oxen, allowing for his oxen, per day, two thirds as 
 much as for himself? 
 
 9. A and B bought a quantity of grass, ready for 
 cutting, for 26 dollars, for which they paid 13 dollars 
 apiece. In cutting and curing it, A furnished 3 days 
 of hired men's labor, and worked himself 2J days; B 
 hired 7 days of men's labor, a team 1J- days, and 
 worked himself 24- days. There were 6 \ tons of hay. 
 
 24 
 
278 MISCELLANEOUS EXAMPLES. 
 
 What is each one's share of the hay, allowing for the 
 labor $1.00 per clay, and for the whole work of the 
 team $1.50? 
 
 10. A agrees to dig and stone a cellar 7 feet deep. 
 It is to be 13 feet wide and 16 feet long inside the 
 walls, which are to be 2 feet in thickness. He is to 
 receive for digging 22 cents a cubic yard for earth, and 
 $1.55 a cnbic yard for rock, and for stoning S7 cents 
 for every perch of 25 cubic feet for which the stone is 
 found in digging the cellar, and $1.50 a perch when 
 he has to bring the stone from another place. In dig- 
 ging the cellar, he digs 7^ cubic yards of rock, which 
 furnishes stone for 7£ cubic yards of wall. What is 
 he to receive for the whole job ? 
 
 11. A man agrees to dig a canal 10 rods long, 6 feet 
 in depth, 33 feet wide at the top, and 24 feet wide at 
 the bottom, for 10 cents per cubic yard. What sum 
 will the work amount to ? 
 
 12. What is the value of six loads of wood, at 
 $4.67 per cord, measuring as follows: — first, 8 ft., 4 
 ft. 3 in., 3 ft. 10 in. ; second, 8 ft. 4 in., 4 ft. 1 in., 
 4 ft. ; third, 8 ft. 2 in., 4 ft. 7 in., 3 ft. 11 in. ; fourth, 
 8 ft., 4 ft. 2 in., 4 ft. ; fifth, 8 ft. 2 in., 3 ft. 10 in., 
 3 ft. 9 in. ; sixth, 8 ft. 6 in., 4 ft., 4 ft. 4 in. ? 
 
 13. What will be the dimension, at the two ends, of 
 the largest square stick of timber that can be hewn 
 from a round log 3 feet in diameter at the larger end, 
 and 2 feet in diameter at the smaller? 
 
 14. What will be the solid contents of such a stick, 
 if it is 16 feet in length ? 
 
 15. What will be the width at the two ends of the 
 largest stick of timber that can be hewn from a log 
 3 feet in diameter at the larger end, and 2 feet in 
 diameter at the smaller, if the stick is hewn 10 inches 
 in thickness through its whole length ? 
 
 16. What will be the solid contents, of such a stick, 
 if it is 20 feet in length ? 
 
MISCELLANEOUS EXAMPLES. 279 
 
 17. There are three pieces of cloth. The length of 
 the first is to that of the second as 3 to 2, the length 
 of the second is to that of the third as 4 to 15, and 
 the length of the three added together is 50 yards. 
 What is the length of each piece ? 
 
 18. A man bought a chaise, and paid 20 dollars for 
 repairing it. He then sold it for one fifth more than 
 he gave, and found that, allowing one dollar for his 
 own trouble in the business, he had lost thirteen dol- 
 lars. What did he give for the chaise ? 
 
 19. A gentleman began his preparation for college 
 at a certain age, and spent in school and at college 
 half as many years as he had lived before. He then 
 went to Europe, and, after spending there one ninth 
 as many years as his age amounted to when he left 
 Europe, spent in his profession one third as many 
 years as he had lived when he entered it, and was then 
 36 years old. At what age did he begin his prepara- 
 tion for college ? 
 
 20. One half of three fourths of A's age equals one 
 sixth of B's age, and the sum of their ages is 78. 
 What is the age of each ? 
 
 21. Three fourths of the liquor in a cask equals five 
 sixths of what has leaked out ; and the whole, before 
 any leaked out, was sixty gallons. How much is 
 there in the cask? 
 
 22. Reduce 5 pence to the decimal of a shilling, 
 carrying the decimal to the sixth figure. 
 
 23. Reduce 9£ gallons to the decimal of a barrel, 
 wine measure, carrying the decimal to the tenth 
 figure. 
 
 24. Reduce 7% quarts to the decimal of a bushel. 
 
 25. What is the least common multiple of 784, 
 1386, and 1235? 
 
 26. What are the prime numbers between 1010 and 
 1020? 
 
 27. What are the prime factors of 2326? 
 
280 MISCELLANEOUS EXAMPLES. 
 
 1 T 1 " 1 91 
 
 28. Reduce — — , — — f and — — , to simple fractions, 
 
 6&-$ 19 30^- 
 
 with a common denominator. 
 
 29. What is the value, in shillings and pence, of 
 the following decimals, when added together : — 
 £.0431; .67142s.; .73462d.? 
 
 30. What is the interest of $642.25 for 1 year and 
 3 months, at 7i per cent. ? 
 
 31. What is the interest of $954.30 for 2 years, 4 
 months, and 21 days, at 5 per cent. ? 
 
 32. What is the present worth of a note of $640.50 
 due in 3 months ? 
 
 33. What is the present worth of a note of $ 1263.00 
 due in 3£ months, at 5 per cent, interest ? 
 
 34. Boston, June 14, 1836. For value received, I 
 promise to pay John Ball, or order, three hundred and 
 sixty-five dollars in four years with interest annually. 
 
 James Frost. 
 If no interest is paid, and the note is renewed an- 
 nually, what will be the amount of the note four years 
 after date ? 
 
 35. New York, July 1, 1840. For value received, 
 I promise to pay Abel Jones, or order, five hundred and 
 forty dollars, on demand, with interest. 
 
 John Frost. 
 
 Endorsements, — Oct. 3, 1840, $63.00 
 
 Feb. 4, 1841, 120.00 
 
 June 1, 1841, 60.00 
 
 Sept. 15, 1841, 200.00 
 
 What will be due Feb. 14, 1842, at seven per cent. 
 interest ? 
 
 36. A owes B $400.00, due in 2 months; $320.00, 
 due in 3 months; $600.00, due in 4£ months. What 
 is the equated time for the payment of the whole ? 
 
 37. What is the present worth of a bank note for 
 800 dollars payable in three months ? 
 
MISCELLANEOUS EXAMPLES. 281 
 
 38. What is the present worth of a bank note for 
 346 dollars payable in six months? 
 
 39. For what sum must I give a note to a bank 
 payable in three months, in order to obtain 674 dol- 
 lars? 
 
 40. Bought seven 100 dollar shares of bank stock 
 at 6£ per cent, advance. Gave in payment nine 60 
 dollar shares of railroad stock at 3 per cent, discount, 
 and a bank note payable in 60 days. What was the 
 face of the note ? 
 
 41. How many square inches are there in 15^- square 
 rods ? 
 
 42. What is the cost of plastering the sides, ends, 
 and ceiling of a room 22J- feet long, 17f feet wide, and 
 11 feet 1 inch in height, at 18 cents per square yard, 
 making no deduction for windows, doors, or wood- 
 work ? 
 
 43. There is a house 40 feet in length, and 26 feet 
 in breadth. From the beam to the ridgepole is 11 feet. 
 The roof projects 7 inches beyond the walls at each 
 end, and the line of the eaves is 8 inches, measured 
 horizontally, from the side walls. How many square 
 feet are there in the roof? 
 
 44. A painter agrees to paint the outside of a house 
 which is 44 feet long, 28 feet wide, 22£ feet in height 
 to the top of the beam, and 11 feet 8 inches from the 
 beam to the ridgepole, for 44 cents per square yard. 
 What is he entitled to for the job, making no deduc- 
 tion for windows, and no addition for cornices or other 
 projections ? 
 
 45. What is the square root of 9743 to three places 
 of decimals? 
 
 46. What is the square root of 17431 to two places 
 of decimals? 
 
 47. The base of a right-angled triangle is 744 feet, 
 the hypotenuse 834 feet. What is the perpendicular, 
 to two places of decimals? 
 
 24* 
 
282 MISCELLANEOUS EXAMPLES. 
 
 48. The base of a triangle is 76 rods, the sum of 
 the hypotenuse and perpendicular is 186 rods. What 
 is the length of the hypotenuse ? 
 
 49. From a cylinder 12 inches in. diameter, it is de- 
 sired to cut the largest possible four-sided prism, whose 
 opposite sides shall be parallel, and whose width shall 
 be to its thickness as 2 to 1. What will be its width, 
 and what its thickness? 
 
 50. What are the dimensions of the largest prism, 
 with parallel sides, that can be cut from a cylinder 12 
 inches in diameter, making the width to the thickness 
 as 3 to 1 ? * 
 
 51. What are the dimensions of the largest prism, 
 with parallel sides, that can be cut from a sphere 15 
 inches in diameter, making the length and breadth 
 equal, and each of them double of the thickness ? 
 
 52. What is the cube root of 674 to three places 
 of decimals? 
 
 53. What is the cube root of 1736 to two places 
 of decimals ? 
 
 54. What is the cube root of 31 to three places of 
 decimals ? 
 
 55. Two men purchase a lot of land for 750 dollars. 
 One pays $406.50; the other, the remainder. They 
 expend $341 in equal shares on its improvement, and 
 sell the land for $1430.00. What is each one's share 
 of the gain? 
 
 56. 1841 are how many times four-fifths of 76£ ? 
 
 57. How many bottles, each containing 1$ pints, 
 can be filled from a hogshead containing 63 gallons, 
 allowing a loss of one eleventh in the process ? 
 
 58. What is the value, in Federal money, of £456 
 at 9J- per cent, advance ? 
 
 59. How many gallons, each containing 231 cubic 
 inches, will fill a cylindrical cistern 4 feet in diameter 
 and 5 feet deep? 
 
 60. There is a cylindrical cistern 6 feet deep, con- 
 
MISCELLANEOUS EXAMPLES. 283 
 
 taining 10 barrels of 31 J- gallons each, each gallon 
 containing 231 cubic inches. What is the diameter 
 of the cistern ? 
 
 61. There is a cistern, in the form of an inverted 
 cone, 8 feet deep, and of the same capacity as the cis- 
 tern last named. What is its diameter at the top? 
 
 62. Bought 74 barrels of flour at $4.56 per barrel, 
 and, after keeping it 35 days, sold it at $5.16 per bar- 
 rel. What per cent, did I gain, allowing 6 per cent, 
 interest on the money invested ? 
 
 63. The first and fourteenth terms of an arith- 
 metical series are 3 and 19. What is the common 
 increase ? 
 
 64. The fifth term of an arithmetical series is 18, 
 the 16th term is 39. What is the first term? 
 
 65. What is the sum of an arithmetical series, the 
 extremes of which are 9 and 164, and the number of 
 terms forty? 
 
 66. Find the ninth term of a geometrical series 
 whose first term is 2 and ratio f. 
 
 67. What is the fifth term of a geometrical series 
 whose second term is 4 and ratio f ? 
 
 6§. James Wildes bought of John Good, 
 
 45£ bushels Salt, at 39£ cents per bushel; 
 143| lbs. Rice, at 3f cts. per lb. ; 
 
 43J- lbs. Tea, at 39 cts. per lb. ; 
 
 94J lbs. Coffee, at 101 cts. per lb. ; 
 
 12 bbls. Flour, at $5.87 per bbl. 
 What is the amount of the bill ? 
 
 69. If he pays the above bill by a bank note, 
 discounted for 60 days, what must be the sum 
 named in the note ? 
 
 70. Add 23f + 18f+f + j|+| 
 
 71. Reduce to a common denominator, 23 and £ of 
 A of 121 and f of T V 
 
 72. Bought 3 boxes of sugar, containing 3 cwt. 2 
 
284 MISCELLANEOUS EXAMPLES. 
 
 qrs. 17 lbs.; 3 cwt. 3 qrs. 11^ lbs.; 3 cwt. 2 qrs. 22} 
 lbs. ; at 6& cts. per lb. Paid in corn at 57% cts. per 
 bushel. How many bushels did it take? 
 
 73. What is the solid contents of a wall 5 ft. high, 
 2 ft. 3 in. wide at the bottom, and 1 foot 10 in. wide 
 at the top, and 34 ft. in length? 
 
 74. Three men agreed to build a wall 6 ft. high, 3 
 ft. wide at the bottom, and 2 ft. wide at the top. 
 The first man built the wall to the height of 2 feet 
 from the ground, the second raised it 2 feet more, and 
 the third finished it. What proportional share of the 
 pay ought each to receive ? 
 
 75. There is a lever, 13 ft. in length, which is sup- 
 ported by a fulcrum 14 in. from the end. How many 
 pounds applied to the longer end will balance a weight 
 of 17 cwt. 2 qrs. at the shorter end ? 
 
 76. The axle of a wheel is 13 in. in diameter, the 
 wheel is 11£ ft. in diameter. What power, applied to 
 the circumference, will balance 3^ tons suspended at 
 the axle? 
 
 77. If the threads of a screw are l£ in. apart, what 
 power, applied at the end of a lever 9J- ft. in length, 
 will support 7 tons, allowing nothing for friction? 
 
 78. With the same screw, what power would sup- 
 port 7 tons, making an allowance of one fourth for 
 friction? Reflect whether the friction in this case is 
 in favor of the weight or in favor of the power. 
 
 79. With the same screw, what power would raise 
 7 tons, allowing one fourth for friction? Notice, in 
 this case, in which way the friction will operate. 
 
 80. There are two right-angled triangles upon a 
 base of 21 ft. in length ; the perpendicular of the larger 
 is 9 ft. in length, that of the smaller 8£ ft. What is 
 the difference in the length of the hypotenuse of the 
 two triangles? 
 
 81. Divide the number 78 in two such numbers 
 that the first shall be 6 more than one fifth part of the 
 second. 
 
MISCELLANEOUS EXAMPLES. 285 
 
 82. Divide the number 82 into two such parts that 
 the first diminished by 5 shall equal one sixth part of 
 the second. 
 
 S3. What is the value, in Federal money, of £194 
 16s., at 8£ per cent, advance? 
 
 84. How many Winchester gallons, wine measure, 
 would be contained in a cubical vat measuring 4 
 ft. each way? 
 
 85. Divide the number 1520 into three such parts 
 that twice the first shall be 40 less than the second, 
 and the second shall be half as great as the third. 
 
 86. How many yards of lining, $ of a yard wide, 
 will line 13£ yds. of cloth l T 7 e yds. wide ? 
 
 87. There is a rectangular field containing 7£ acres, 
 the width of which is one half as great as its length. 
 What is the length of a diagonal line connecting its 
 opposite corners? 
 
 88. Around a rectangular common, containing 18 
 acres, the length of which is to its breadth as 6 to 5, a 
 road runs 40 ft. in breadth. How many rods would 
 be saved in travel by crossing the common diag- 
 onally, rather than going round on two sides of it, 
 supposing the traveller to begin and end, in both 
 cases, in the middle of the road in range with the 
 diagonal line? 
 
 89. Whatsis the difference between the square root 
 of half of 4, and half the square root of 4, carried 
 to three decimal places? 
 
 90. What is tile difference between the square root 
 of one third of 12, and one third the square root of 
 12, carried to three places of decimals? 
 
 91. How many times £ of 17£ are equal to 13£ 
 times 15f? 
 
 92. When it is noon in Boston, Lon. 71° 4' W., 
 what time is it at Liverpool, Lon. 2° 59' W.; at 
 Greenwich, Lon. 0; at Havre, Lon. 16' E. ; and 
 at Paris, Lon. 2° 20' E. ? ^ 
 
286 MISCELLANEOUS EXAMPLES. 
 
 93. When it is noon at London, Lon. W W., 
 what time is it at New York, Lon. 74° T W. ; at 
 Washington, Lon. 77° 2' W. ; and at Cincinnati, 
 Lon. 84° 27' W. ? 
 
 94. A field, in the form of an equilateral triangle, 
 contains 8^ acres. What is the length of one of 
 its sides? 
 
 95. Reduce # of — to fifths. Reduce f of — to 
 fifteenths. 
 
 96. What is the amount, in Federal money, of 6 s. 
 7d., 5s. 3d., 14s. 9d., and lis. 6}d.? 
 
 97. What is the cube root of 144, to three places of 
 decimals ? 
 
 98. What is the weight of a cylinder of lead 3 ft. 
 long and 4 inches in diameter ? 
 
 13i 34A 
 
 99. Multiply 7J times ~ by i of ~. 
 
 100. How many square feet in a triangle whose 
 base measures" 45 feet, and whose height is 17 feet? 
 
 101. What is the solid contents of a triangular 
 pyramid whose base measures 18 feet on each side, 
 and whose height is 23 feet? 
 
 102. What is the interest of $546.25 for 23 mo. 13 
 days, at 7£ per cent. ? 
 
 103. What number is that of which i and i of it 
 added together exceed £ of it by 2.? 
 
 . 104. What is the cube root of 197 to two decimal 
 places ? 
 
 105. What is the cube root of 501 to four decimal 
 places ? 
 
 106. What is the 43d term of an arithmetical series 
 whose first term is 7| and common difference 3f ? 
 
 107. What is the sum of an arithmetical series of 
 57 terms whose 4th term is 15 and common dif- 
 ference 2£? 
 
 108. How many pounds of coffee at 11 cents per 
 
MISCELLANEOUS EXAMPLES. 287 
 
 pound can be mixed with 56 lbs. at 8 cents, and 96 
 lbs. at 9J- cents, so as to make the mixture worth 
 10i cents? 
 
 109. If a sphere of gold weigh 36 oz., how many 
 ounces will a sphere of silver weigh of equal size, the 
 specific gravity being as given on page 258? 
 
 110. What will be the weight of a ball of iron 6 
 inches in diameter? 
 
 111. What will be the weight of the largest cube 
 that can be cut from a ball of iron 6 inches in di- 
 ameter? 
 
 112. What is the value, in Federal money, of £13 
 6s., 2 guineas, 3d., 7^d., added together? 
 
 113. How many times will a wheel 4 ft. 3 in. in 
 diameter go round in traversing the circumference of 
 a circle containing 5 acres? 
 
 114. If a lever is 16 ft. in length, the weight 13 
 cwt. 3 qrs. 11 lbs., and the power 94 lbs., what must 
 be the distance of the fulcrum from the weight in 
 order that the weight and power may balance? 
 
 115. If the longer arm of a lever be 10 feet and 
 the shorter arm 2 feet in length, how must 480 
 pounds be divided so that one part shall be the 
 weight and the other the power that will balance 
 it on the lever? 
 
 116. Divide f of l of 16i by 18£ times f of ff 
 
 117. Reduce 1 pk. 3 qts. 1 pt. 1 gill, to the deci- 
 mal of a bushel. 
 
 118. How many shillings and pence in .4562 of 
 a £? 
 
 119. A general drew up his army in a square, with 
 the number in rank and file equal, and had 576 men 
 left. 'He then increased the square by placing two 
 lines of men in front, and two files on one side from 
 front to rear, when he found he wanted 12 men to 
 complete the square. How many men had he ? 
 
288 MISCELLANEOUS EXAMPLES. 
 
 120. What number is that one third of which ex- 
 ceeds two sevenths of it by 19? 
 
 54 9 
 
 121. What number is that -~ of which exceeds ™ 
 of it by 31? 28 
 
 122. How many tiles, each 8 inches square, will it 
 require to cover one acre ? 
 
 123. If the hypotenuse of a right-angled triangle 
 measure 34 feet, and the base 19^, what is the 
 measure of the perpendicular? 
 
 124. If the sum of the base and hypotenuse is 63 
 feet, and the perpendicular 14 feet, how long is the 
 base? 
 
 125. A man travels south 20 miles, then east 15 
 miles, then south 2J- miles, and east 7 miles. How 
 far is he then from where he set out ? 
 
 126. What is the dhTerence between the cube root 
 of one third of 12, and one third of the cube root 
 of 12? 
 
 127. What is the value, in dollars and cents, of 
 £94 16s. 3d. + £43 19s. 7d. + £14 13s. 9d.? 
 
 128. If a note of $1000, promising annual interest, 
 is renewed at the end of each year for five years, 
 without the payment of any interest, what is the 
 amount, principal, and interest, at the end of the 
 fifth year ? 
 
 129. What is the dfrTerence, in avoirdupois weight, 
 between a ball of silver and a ball of gold, each 3 
 inches in diameter? 
 
 130. There are three numbers; the first plus 4 is 
 equal to one sixth of the second, the second is one 
 half as great as the third, and the sum of the three is 
 186. What are the numbers? 
 
 131. There are two numbers; the first increased 
 by 4 equals one sixth of the second, and the second 
 diminished by 6 is eight times the first. What are 
 the numbers? 
 
MISCELLANEOUS EXAMPLES. 289 
 
 132. What is the value, in Federal money, of 13 
 shares of bank stock, par value $125 per share, and 
 sold at 11£ per cent, advance? 
 
 133. What is the square root of 14734? 
 
 134. What is the interest of $1974.36, for 3 yrs. 
 
 2 mo. 17 days, at 5f- per cent. ? 
 
 135. What is the 14th term of a geometrical se- 
 ries, the first term of which is 4 and the ratio f ? 
 
 136. What is the 16th term of a geometrical se- 
 ries, the first term of which is 7 and the ratio H? 
 
 137. A sells to B 5 loads of wood, measuring, first, 
 8 ft. 6 in., 4 ft., 3 ft. 9 in. ; second, 9 ft., 4 ft. 2 in., 
 
 3 ft. 11 in.; third, 9 ft. 2 in., 4 ft. 4 in., 4 ft. 1 in. ; 
 fourth, 8 ft. 2 in., 3 ft. 7 in., 4 ft. 2 in. ; "fifth, 7 ft. 
 11 in., 4 ft., 3 ft. 6 in.; at $4.75 per cord. He re- 
 ceives in payment 47 bushels of oats, at 38 cents per 
 bushel ; 56 lbs. of cheese, at 8} cents per pound ; 
 and the balance in butter, at 154- cents per pound. 
 How much butter does he receive? 
 
 138. What is the weight of one rod of lead pipe 
 one fourth of an inch in thickness, if the inner diame- 
 ter measures l£ inches ? 
 
 139. What is the weight of a plate of iron half an 
 inch in thickness, 4 feet long, and 2 feet 3 inches in 
 breadth ? 
 
 140. How many feet of silver wire, one tenth of an 
 inch in diameter, can be made from one pound avoir- 
 dupois of silver? 
 
 141. How •many gallons, imperial measure, will a 
 cylindrical cistern hold, 3 feet in diameter and 4^ 
 feet deep? 
 
 142. What is the cost of transporting 64 barrels of 
 flour, each containing 7 qrs. gross, 100 miles, at $3.12£ 
 per ton, allowing for the weight of each cask 16 lbs."? 
 
 143. If freight by railroad is $3.12^- per ton, for 
 100 miles, and freight by wagon road is $20 per ton 
 for 80 miles, how much is saved in, the freight of a 
 
 25 T 
 
290 MISCELLANEOUS EXAMPLES. 
 
 barrel of flour 100 miles by railroa^ allowing its 
 weight to be as in the preceding example? 
 
 144. At the rate named above, how far could a 
 barrel of flour be carried by wagon road, before the 
 freight should amount to as much as the flour was 
 worth, when the price is $5.62 per barrel? 
 
 145. If the distance from New York to Liverpool 
 is 3000 miles, what would be the cost of trans- 
 porting a barrel of flour that distance, at the rate 
 of $20 per ton for every 80 miles ? 
 
 146. If a barrel of flour can be transported from 
 New York to Liverpool for 65 cents, what would that 
 give for the transport of one ton 80 miles ? 
 
 147. The summit of the Rocky Mountains, visited 
 by Freemont, is in longitude 110° 8'. What time is 
 it at Greenwich when it is noon there ? 
 
 148. There is a field 20 rods long and 8 rods broad, 
 with a path 3£ feet wide running round it. How 
 many square feet are there in the path ? 
 
 149. What is the cost of excavating a cubical pit, 
 measuring 7-£ feet in each direction, at 31^ cents per 
 cubic yard ? 
 
 150. What is the weight of an iron cannon 94- feet 
 in length, 26 inches in diameter at the larger end, and 
 18 inches at the smaller, with a bore 9 feet long and 
 10 inches in diameter, allowing for no inequalities in 
 the surface ? 
 
 151. What will be the duties on an invoice of goods 
 amounting to $1156.80, at 30 per cent.?. 
 
 152. Bought an invoice of imported goods, amount- 
 ing to £564 15 s. I agree to give 18 per cent, in ad- 
 vance of the invoiced price. What is the amount 
 paid, in Federal money, reckoning $4,444 to the 
 pound ? 
 
 153. I buy for cash an invoice of imported goods, 
 amounting to £1146 16s. W r hat is the invoiced, price, 
 
MISCELLANEOUS EXAMPLES. 291 
 
 in Federal money, allowing Sterling money to be 9 
 per cent, in advance of the nominal par value ? 
 
 154. A merchant owes $16472.50. He fails, his 
 means of payment amounting to only $4345.62. 
 How much is a creditor entitled to, who holds a note 
 against him of $ 100, dated 7i months previous to the 
 final settlement, and promising interest 60 days after 
 date? 
 
 155. A road, 3£ rods wide, is laid out 1 mile and 
 14£ rods in length, for which damages are awarded to 
 the land-owners, as follows : — for 80 rods of the road, 
 at the rate of 37 dollars per acre; for 110 rods, at the 
 rate of 22 dollars per acre ; and for the remainder, at 
 the rate of 30£ dollars per acre. What is the whole 
 amount of the damages ? 
 
 156. What is due for the freight of 12 barrels of 
 flour 65 miles, at $ of a cent per pound, allowing each 
 barrel to contain 7 qrs. gross, of flour, and the cask to 
 weigh 18 pounds? 
 
 157. If 5 barrels of flour suffice for a family of 11 
 persons 7 months, how many barrels will suffice for a 
 family of 15 persons 4£ months ? 
 
 158. 13£ times 14| is 7£ times what number ? 
 
 159. 384 is -± of how many times 15} ? 
 
 160. Reduce 62£ cents to the decimal of a £, at 
 nominal par value. 
 
 161. How many seconds were there in the year 
 1844? 
 
 162. The report of a signal gun, fired on the equa- 
 tor, at 12 o'clock, is heard at a place due west distant 
 16 miles. At what time is the report heard at the 
 latter place, allowing 69£ miles to a degree of longi- 
 tude, and sound to move 1 mile and 10 rods in 5 
 seconds ? 
 
 163. A communication is made by the magnetic 
 telegraph from Boston to Washington, at 1 o'clock, 
 
292 MISCELLANEOUS EXAMPLES. 
 
 P. M. At what time will it be received at Washing- 
 ton, allowing no time for the transmission of the fluid; 
 longitude of Boston being 71° 4' 9" ; that of Wash- 
 ington, 77° T24"? 
 
 164. A, engaging in partnership with B and C, puts 
 in 1600 dollars for 9£ months ; B puts in 3100 dollars 
 for 14 months ; C puts in 2200 dollars for 12 months. 
 They gain 1046 dollars. What is each one's share of 
 the gain ? 
 
 165. If 100 dollars in one year gain 6£ dollars in- 
 terest, what will 467 dollars gain in 9J- months, at the 
 same rate ? 
 
 166. What is the value of 17 shares of bank stock, 
 par value 60 dollars per share, and sold at 6£ per cent, 
 advance ? 
 
 167. A man buys 640 barrels of flour at $5f per 
 barrel ; pays for freight, $42.75'; for storage, $11.17; 
 and sells it for 6 T \ per barrel, allowing a commission 
 of 2i per cent. What was his loss or gain per cent. ? 
 
 168. The pipe of an aqueduct, 12 inches in diame- 
 ter, is divided into two branches, such that their united 
 capacity is equal to that of the main pipe, and the /di- 
 ameter of one of the branches is 9 inches. What is 
 the diameter of the other ? 
 
 169. The pipe of an aqueduct is 3 feet in diameter. 
 Wliat number of pipes, each 2£ inches in diameter, 
 would have a capacity equal to that of the main pipe ? 
 
 170. If a tree measures, at the distance of 2 feet 
 from the ground, 12 feet in circumference ; and, at 14 
 feet from the ground, divides into two branches, meas- 
 uring respectively 9 feet and 7 feet in circumference ; 
 how many square inches more of surface would the 
 lower horizontal section of the tree contain than the 
 upper one ? 
 
 171. A certain tree measures, at the distance of 3 
 feet from the ground, 22| feet in circumference. At 
 some distauce above, its four branches measure, re- 
 
MISCELLANEOUS EXAMPLES. 293 
 
 spectively, 8 ft. 4 in., 9 ft., 7 ft. 6 in., and 6 ft. 5 in. in 
 circumference. What is the ratio of the magnitude 
 of the tree at the lower, compared with its magnitude 
 at the higher place" of measurement ? 
 
 172. The shadow of a certain tree, as cast upon the 
 ground, measures 102 feet in diameter. Allowing the 
 shadow to be circular, how many rods of ground does 
 it cover ? 
 
 173. How many cubic inches does a wine-glass 
 contain, measuring 3£ inches in depth, and 2 inches in 
 diameter at the top, the form being that of an inverted 
 cone ? 
 
 174. How many yards of lining, f of a yard wide, 
 will line 37£ yards of cloth 1^ yards wide ? 
 
 175. What is the 3d term of the square 900 + 420 
 
 + n? 
 
 176. What is the 3d term of the square 1600 + 480 
 
 + □? 
 
 177. Complete the square 4900 + CD+64. 
 
 178. What is the 4th term of the cube 27000 + 
 5400 + 360 + D? 
 
 179. What is the 4th term of the cube 64000 + 
 4800+120 + □? 
 
 180. What is the 4th term of the cube 125000 + 
 22500+1350 + □? 
 
 181. What is the cube root of 68437? 
 
 182. What is the cube root of 954326 ? 
 
 183. What is the solid contents of the largest sphere 
 that can be cut from a cubic block 13£ inches on a 
 side ? 
 
 184. From a sphere measuring 10 inches in diame- 
 ter, the largest possible cubic block has been cut ; and 
 from this block again the largest possible sphere has 
 been cut. What is the diameter of the last-named 
 sphere ? 
 
 185. From a sphere 20 inches in diameter what are 
 the dimensions of the largest parallel prism that can 
 
 25* 
 
294 ' MISCELLANEOUS EXAMPLES. 
 
 be cut, making the length to the breadth as 4 to 3, and 
 the breadth to the thickness as 3 to 2 ? 
 
 186. How many cubic feet are there in the Avails of 
 a brick house, the length and breadth of which are 
 each 44 feet outside, and the height 24 feet, supposing 
 the walls to be perpendicular outside, and the thickness 
 to be 2 feet at the bottom and 1 foot at the top, making 
 no deduction for doors or windows ? 
 
 187. What are the prime factors of 3746? Of 9862 ? 
 
 188. What is the least common multiple of 684, 
 963. and 8416? 
 
 189. What is the greatest common divisor of 94620, 
 3642, and 1646 ? 
 
 190. How many times will the wheel of a railroad 
 car, if it be 24- feet in diameter, revolve in going 40 
 miles ? 
 
 191. How many times will such a wheel revolve 
 in a minute, if the speed of the car be 20 miles an 
 hour ? 
 
 192. What is the cost, in Federal money, of the 
 freight of 940 bales of cotton, averaging 340 lbs. per 
 bale, at f of a penny per pound? 
 
 193. Sold 34 pieces cotton goods, averaging 31^ 
 yards each in length, at 9| cents per yard, l£ per cent, 
 off. What is the amount of the bill ? 
 
 194. Bought 840 barrels of flour, on six months' 
 credit, at 4& dollars per barrel. Sold the flour the 
 same day on three months' credit, at 4j| dollars per 
 barrel. Did I gain or lose, and how much, estimating 
 the present worth of the debts at the date of the 
 transaction ? 
 
 195. A merchant buys for me, on commission, 400 
 barrels of flour, for $4.34 per barrel, cash. He sells the 
 flour on the same day for cash, at $4.46 per barrel. 
 How much do I gain by the operation, allowing 1£ per 
 cent, commission on the purchases, and 2 per cent, on 
 the sales? 
 
MISCELLANEOUS EXAMPLES. 295 
 
 196. A sets out on a journey, travelling 24 miles a 
 day. B sets out 2 days after, travelling 3 If miles a 
 day. A, after travelling 3 days, goes 25 miles a day ; 
 and B, after travelling 4 days, goes 33£ miles a day. 
 In how many days after A sets out will B overtake 
 him ? 
 
 197. If a parallel prism is 4£ inches thick, 5 inches 
 wide, and 6 inches long, what must be the diameter 
 of the hollow sphere that would enclose it ? 
 
 198. If 18 horses, in 16 weeks, consume 204 bush- 
 els of oats, how many horses will it require to consume 
 413 bushels in 18 weeks ? 
 
 199. If 100 dollars gain 6 dollars interest in 12 
 months, what will be the interest of 840 dollars for 
 5} months ? 
 
 200. If a field containing 17 acres measures 81 rods 
 on one side, what must be the length of the corre- 
 sponding side of a similar field containing 26J- acres ? 
 
 201. The contents of two similar fields are as 4j- 
 to 7, and the smaller measures on one side 63 rods. 
 What must be the corresponding dimension of the 
 larger field ? 
 
 202. There are two circles ; their areas are as 14 to 
 19, and the diameter of the smaller is 16 rods. What 
 is the diameter of the larger ? 
 
 203. What is the area of a circle whose diameter is 
 44 feet ? 
 
 204. What is the circumference of a circle whose 
 diameter is 67 inches ? 
 
 205. Given the circumference of a circle 60 rods, to 
 find its area. 
 
 206. There are three equilateral triangles, whose 
 areas are to each other as the numbers 3, 4, and 7. A 
 side of the smallest measures 40 rods. What is the 
 sum of their areas ? 
 
 207. A grindstone is 3 feet in diameter. Allowing 
 the hole in the middle to be 2 inches in diameter, how 
 
296 MISCELLANEOUS EXAMPLES. 
 
 many inches must be ground off to grind away half 
 of the stone ? 
 
 208. The fore wheels of a wagon are 3 feet 10 
 inches in diameter, and the hind wheels 4 feet 2 
 inches in diameter. How many times more does one 
 of the fore wheels turn round than one of the hind 
 wheels in going one mile ? 
 
 209. What is the weight of a cast-iron cylinder 6 
 feet long and 4i inches in diameter, the specific grav- 
 ity being as already given? 
 
 210. The frustum of a cone 7 feet long is 14 inches 
 in diameter at the larger end, and 10 inches in diam- 
 eter at the smaller. How far from the base must it 
 be cut in two to divide it into equal parts ? 
 
 211. What is the first prime number above 901 ? 
 
 212. What is the first prime number below 10000 ? 
 
 213. What is the greatest common divisor of 1846, 
 3105, 684, and 1006? 
 
 214. What are all the prime factors of 801, of 3042, 
 of 586, of 908 ? 
 
 215. In what proportion may corn at 80 cents be 
 mixed with rye at 86 cents, and with oats at 43 cents, 
 per bushel, to make the mixture worth 50 cents per 
 bushel ? 
 
 216. Add the fractions v+^ + ^7- 
 
 217. What is the value of ? of ££, f of T \s., ex- 
 pressed in the decimal of a £ ? 
 
 218. What is the present value of a note of 584 
 dollars, payable in three months? 
 
 219. What is the bank discount on a note of 150 
 dollars, payable in three months? 
 
 220. What sum will be paid on a note of 240 dol- 
 lars, discounted at a bank, for 90 days ? 
 
 221. What is the interest of 1200 dollars for 10 
 days, at 7 per cent. ? 
 
MISCELLANEOUS EXAMPLES. 
 
 297 
 
 222. Divide the sum of the decimals 2016 + 9172 
 -f- 0064, by £ of f reduced to a decimal. 
 
 223. Divide 3 T. 17 cwt. 3 qrs. 19 lbs. by 6. 
 
 224. Divide 9 m. 3 fur. 21 r. 14 ft. by 8. 
 
 225. Multiply 31 d. 14 h. 37 m. 15 sec. by 19. 
 
 226. Multiply 83 A. 3 R. 22 r. by 12. 
 
 227. What is one fifth of 16 T. 11 cwt. 2 qrs. 
 20 lbs. ? 
 
 228. What is the value, at 4 dollars a cord, of three 
 loads of wood, measuring as follows : first, 8 feet 6 in., 
 4 ft. 2 in., 3 ft. 9 in. ; second, 9 ft., 4 ft. 1 in., 3 ft. 
 10 in. ; third, 8 ft. 1 in., 4 ft., 4 ft. 2 in. ? 
 
 229. What is the 15th term of an arithmetical 
 series, the 1st term of. which is 3 and the common 
 diri'erence § ? 
 
 230. If the 9th term of an arithmetical series is 23, 
 and the common difference |, what is the 3d term ? 
 
 231. What is the sum of an arithmetical series of 
 40 terms, if the 1st term is 2 and the common dif- 
 ference 3£? 
 
 232. What is the sum of an arithmetical series of 
 72 terms, if the 1st term is 1 and the common dif- 
 ference If ? 
 
 233. A man engages to walk 1000 miles in 1000 
 hours, on condition of receiving 1 cent for the first 
 mile, and for each mile after £ of a cent more than he 
 had for the mile preceding it.. What will he be enti- 
 tled to on the fulfilment of his contract ? 
 
 234. There are two grindstones, the thickness of 
 which is to the diameter as 2 to 11. The smaller 
 one is 2 feet in diameter ; the other is three times as 
 heavy. What is its diameter ? 
 
 235. A man has a triangular field containing 7 acres. 
 The vertex or point of the triangle is 54 rods from the 
 base. At what distance from the base must a line be 
 drawn parallel to it so as to cut off one half the field ? 
 
 236. The hypotenuse and perpendicular of a tri- 
 
298 MISCELLANEOUS EXAMPLES. 
 
 angle measure together 816 rods ; the base measures 
 61 rods. What is the length of the perpendicular? 
 
 237. A rope 100 feet long passes straight from the 
 ground, at the distance of 10 feet from a perpendicu- 
 lar pole, over the top of the pole, which is 25 feet in 
 height, and thence is drawn so as to reach the ground 
 at the farthest possible point. Allowing the ground 
 to be level, how far is the last-named point from the 
 foot of the pole ? 
 
 238. A stick of timber, in the form of a truncated 
 wedge, is 10 feet long, 2 feet wide through its whole 
 extent, 20 inches in thickness at one end, and 14 
 inches thick at the other. How far from the thicker 
 end must it be cut in two so as to divide it into two 
 equal parts ? . * 
 
RECOMMENDATIONS. 
 
 From Mr. George B. Emerson, Boston. 
 
 I have carefully examined the plan of Mr. Adams's work on Men- 
 tal Arithmetic, and have given some attention to its execution ; and I 
 am confident that it will prove a very valuable addition to the means 
 of instruction in Arithmetic. It is a successful extension of the ad- 
 mirable method of Colburn's First Lessons, with such modifications 
 as seemed to be required in a higher work on the same general 
 model. It occupies unappropriated ground ; and it deserves, and I 
 think it will take, a high place amongst the text-books. 
 
 GEO. B. EMERSON. 
 
 From Mr. Thomas Sherwin, Boston. 
 
 I have carefully examined, in manuscript, the work of Mr. Adams 
 on Mental Arithmetic, and am much pleased with it. His plan is 
 good, and well executed. I would, therefore, heartily recommend 
 his book to Teachers and School Committees, as one which will con- 
 tribute very materially to the attainment of that very important, but 
 much-neglected, branch of study, — Intellectual Arithmetic. 
 
 THOMAS SHERWIN, 
 Principal of the Boston English High School. 
 
 From Mr. Solomon Adams, Boston. 
 
 To the Publisher. 
 
 Dear Sir : — Having been favored with an opportunity of exam- 
 ining, in manuscript, a Treatise on Mental Arithmetic, by F. A. 
 Adams, A. M., I am most happy to find that our schools are about to 
 have a work of the kind, carried with much skill and judgment into 
 the higher departments of Arithmetic. 
 
 Very respectfully yours, 
 
 SOLOMON ADAMS, 
 Principal of the Young Ladies" School, Central Place. 
 
RECOMMENDATION'S. O 
 
 From Roger S. Howard, Esq., Newhuryport. 
 
 Mr. F. A. Adams. 
 
 Dear Sir: — I have looked over, with much care and pleasure, the 
 manuscript Arithmetic, which you put into my hands a few days since. 
 The plan of the work appears to me quite original, and many of the me- 
 thods you have adopted exceedingly ingenious, and, at the same time, 
 beautifully simple. Your rules and explanations are clear and concise ; 
 and the numerous examples for practice which you have inserted, are 
 judiciously selected and well arranged. The book, I think, is one which 
 will greatly facilitate the teaching of this important branch of education. 
 I am, sir, very respectfully yours, 
 
 ROGER S. HOWARD, 
 Principal of the Putnam High School. 
 
 From Mr. Rufus Putnayn, Salem. 
 
 Mr. F. A. Adams. 
 
 Dear Sir: — I have read with much satisfaction the manuscript copy 
 of the Mental Arithmetic you are intending to publish. The plan of the 
 work is, in many respects, different from its predecessors ; and, strange 
 as it may seem, to those who examine many of the new books in the 
 various departments of education, and who have not read yours, it occu- 
 pies much ground which has not been occupied by others. I think that, 
 in its arrangement, its definitions, its explanations, the examples for 
 practice, — indeed, in its whole matter, — it is happily adapted to its ob- 
 ject ; to release our youth from a part of their present bondage to slate 
 and pencil, and artificial rules, by qualifying them to perform correctly 
 and easily, in the mind, many of the operations which are almost univer- 
 sally performed on the slate. I commend it, with much confidence, to the 
 notice of all who are intrusted with the education of youth. 
 Yours, verv truly and respectfully, 
 
 R. PUTNAM, 
 Principal of the Bowditch (English High) School. 
 
 From Mr. Edwin Jocelyn, Salem. 
 Mr. F. A. Adams. 
 
 Dear Sir: — No one can hold "Colburn's First Lessons in Arithme- 
 tic" in higher estimation than I do; and I think, whoever undertakes to 
 furnish a substitute for that little book, which shall better answer the 
 purpose, will fail in his purpose. I am glad to see from your hand an ex- 
 tension of Mental Arithmetic on the plan of that inestimable school-book. 
 I have often felt the want of such a work, and have in practice extended 
 this course of teaching, somewhat : — and should have done it oftener 
 and farther, if I had had such a book at hand as you now propose to pub- 
 lish. The plan appears to me to be very happily carried out, and I feel 
 confident that it will meet with a wide appreciation and use. 
 Yours, with much esteem, 
 
 EDWIN JOCELYN, 
 Principal of the Female High School. 
 
4 RECOMMENDATIONS. 
 
 From Mr. Charles Norihend, Salem. 
 Mr. F. A. Adams. 
 
 Dear Sir: — Having examined, with some care, your manuscript en- 
 titled "Advanced Lessons in Mental Arithmetic," I feel no hesitation in 
 saying, that I consider the work very happy in design, and admirable in 
 execution. Wishing you much success, 
 
 I remain very truly yours, &c, 
 
 CHARLES NORTHEND, 
 Principal of Epes Grammar School. 
 
 From the North American Review. 
 
 To the late Warren Colburn belongs the high credit of first introducing 
 into our Schools, through his admirable First Lessons, the regular study 
 of Mental Arithmetic. Of this excellent little manual, the author of the 
 book before us justly observes, that, so completely has it performed the 
 work within its prescribed sphere, that there is little reason to desire or 
 to expect that it will ever be superseded. Mr. Colburn published also a 
 Sequel to Mental Arithmetic, in which the principles and rules of Written 
 Arithmetic were deduced from the solution and analysis of questions ac- 
 cording to the method adopted in the former treatise. This Sequel was 
 very well executed as far as it went ; but it was not full enough for all 
 the wants of the higher classes in our schools. It omitted Proportion and 
 Progression, the "Rule of Three," and the doctrine of Powers and Roots. 
 Mr. Adams has undertaken to supply these deficiencies, following mainly 
 in the track of Mr. Colburn, but appearing fully competent also to mark 
 out a path for himself. By this enlargement of plan, he has brought 
 many useful problems in Mensuration and Mechanics within the scope 
 of his work, and has extended the analysis and induction over much new 
 ground, though many of the new problems are still left to be performed 
 by arbitrary rules. 
 
 The First Part of Mr. Adams's book consists of exercises in Mental 
 Arithmetic, arranged under the different arithmetical rules. Where the 
 principles have not been taught in the First Lessons, they are here care- 
 fully deduced from an analysis of a number of simple questions, following 
 which are numerous and well-selected examples. These examples pass 
 gradually from simple to more complicated questions, so as to give the 
 pupil a thorough training. In the Second Part, the different processes 
 are arranged in the same order as before ; and when the operations are 
 complicated, distinct rules are given, illustrated by examples for practice 
 containing larger numbers than were suitable for the exclusively mental 
 operation. When the operations are simple, and sufficiently explained in 
 the analysis and induction contained in the First Part, a reference is 
 merely made to that Part, and the examples for practice follow, without 
 any enunciation of a rule. 
 
 The author's reasoning and explanations are very clear, simple, and 
 concise ; his disposition of the different parts judicious, and his selection 
 of examples well suited to exercise the mind of the pupil. As a whole, 
 we prefer this work to any Arithmetic we have seen in use. 
 
RECOMMENDATIONS. O 
 
 From Professor Chase, of Dartmouth College. 
 Mr. F. A. Adams. 
 
 My Dear Sir : — I have examined, with some care, your Treatise on 
 Arithmetic, and am much pleased with it. The practice and habit of 
 extending mental operations to large numbers, is of great utility. I have 
 occasion, very frequently, to see the inconvenience that young men suffer, 
 from the want of such a habit. Not less valuable than the habit of ope- 
 rating mentally upon large numbers, is the habit of performing the more 
 advanced operations of arithmetic without the aid of the pencil. 
 
 I like very much, also, the manner in which you have treated several 
 of the principles which you have developed ; as, for example, the subject 
 of the Common Divisor, the Least Common Multiple, the Roots, Ratio, 
 and Proportion. These are but few of the subjects, but I mention them 
 as examples. 
 
 I think the book will do much to promote the proper method of teach- 
 ing arithmetic, — by demonstration and explanation. 
 
 I am, dear sir, very truly yours, &c, 
 
 S. CHASE. 
 
 From Mr. Addison Brown, of Brattleboro, Vt. 
 Daniel Bixby, Esq. 
 
 Dear Sir : — I give you many thanks for the " First Book in Arithme- 
 tic," by F. A. Adams, which you had the kindness to send me, some time 
 since. To test its merits, I set my youngest child, a daughter, seven 
 years of age, to studying it. She has been about half through it ; and 
 having heard her lessons myself, and watched her progress, and the effect 
 the book has had in developing her powers of calculation, I am satisfied 
 that it is a very excellent work. I think, as a First Book in Arithmetic, 
 it is decidedly the best I have either used or examined, and I should be 
 glad to see it extensively introduced into our Schools. 
 
 Yours, with respect, 
 
 ADDISON BROWN. 
 
 From Mr. John Tatlock, Professor of Mathematics, and Mr. A. Hopkins, 
 Professor of Natural Philosophy. 
 
 I have examined a treatise on Arithmetic, by F. A. Adams, and am 
 much pleased with it. I think it well adapted to teach the science and 
 art of numbers, and at the same time to teach the art of thinking. I am 
 persuaded that a thorough training in this Arithmetic would prepare 
 students for the further study of mathematics better than nine-tenths are 
 now prepared. 
 
 I should be glad if every student who enters college was master of 
 this Arithmetic. 
 
 JOHN TATLOCK. 
 A. HOPKINS. 
 
D RECOMMENDATIONS. 
 
 From Mr. Roger B. Hildreth, Tyngshorough. 
 To the Publisher. 
 
 Dear Sir : — The " Advanced Lessons in Mental Arithmetic," upon 
 the plan adopted by Mr. F. A. Adams, will, I am persuaded, from a care- 
 ful examination of the work, be found very useful in instruction. It 
 peculiarly meets the wants both of the teacher and the pupil ; and as its 
 real merits become more generally known, I am confident it will be ex- 
 tensively used as a text-book. The work is arranged with admirable 
 clearness, — the rules and explanations are concise, and yet very simple. 
 I cheerfully commend it to the notice of all who wish for an excellent 
 treatise on Mental Arithmetic. 
 
 Very respectfully yours, 
 
 ROGER B. HILDRETH, 
 Principal of Tyngshorough High School. 
 
 From J. II. Purkitt, Esq., St. Louis. 
 
 Mr. Bixby. 
 
 Dear Sir : — I have used Frederic A. Adams's Arithmetic in my 
 School for many months ; and it gives me pleasure to say that it has met 
 my highest expectations. I regard it as b}' far the best Arithmetic in 
 the English language. Recommendations are extremely cheap, nor 
 -would I trouble you with an expression of my own opinion of its merits, 
 did I not believe it to be an act of common justice, when an author has 
 produced a really valuable book, that those who have thoroughly tested it, 
 and know and feel its value, should freely and cordially acknowledge it. 
 If it has a decided superiority over all other Arithmetics, in developing 
 power of thought, and training the mind to a stern and rigid analysis, 
 then the public have a right to know it. That this book, faithfully used, 
 will do this, I sincerely believe. No consideration whatever could induce 
 me to throw it out of my school, for the sake of introducing any other 
 Arithmetic yet published. I feel, therefore, that I am doing not only 
 what is just, but what, I hope, is a real good, by this expression of opi- 
 nion. Teachers may be slow in using it as a text-book in their schools ; 
 but that it will be ultimately adopted, and, when adopted, permanently 
 retained, I cannot, for a moment, doubt. To the la?y and unthinking, 
 and those who are bigotcdly attached to the common mode of teaching 
 Arithmetic, this book will not only not be acceptable, but, perchance, be 
 dogmatically condemned, and indignantly rejected. But those who love 
 to think themselves, and desire to develope the power of correct thinking 
 in others, will find in this book an exhibition of principles, beautiful as 
 they are true, and ingenious as they are rational. Wishing you great 
 success, 
 
 I am, dear sir, yours, with respect, 
 
 J. H. PURKITT, 
 Principal of the Young Ladies 1 High School. 
 
RECOMMENDATIONS. 1 
 
 From Mr. Charles C. Dame, Principal of the English High School, New- 
 buryport. 
 
 Mr. F. A. Adams. 
 
 Dear Sir : — I have examined somewhat carefully your " Mental and 
 Written Arithmetic," and find that it possesses peculiar merit. The 
 arrangement and exercises are such, I think, as will train the student to 
 a habit of thinking and reasoning for himself, and lead him readily to 
 apprehend the relations of numbers in their various combinations. The 
 book is not encumbered with "many rules," but the subjects of which it 
 treats appear to be taken up in a systematic manner; and the expla- 
 nations are clear and analytic. 
 
 The work throughout seems to have been planned and executed in a 
 way well calculated to fit those who may use it for the " active pursuits 
 of life," — to enable them to solve, in the most ready and natural way, 
 those arithmetical questions which may occur in business or otherwise. 
 The questions for exercise which it contains are, for the most part, prac- 
 tical, and so constructed as to afford encouragement in their solution. 
 
 I have formed such an opinion of the work, as to believe that no scholar 
 will hereafter leave school where it is known, and consider himself well 
 educated, unless he is familiar with its pages, or the principles which 
 they contain. 
 
 I am very truly and respectfully yours, 
 
 CHARLES C. DAME. 
 
 From E. Wyman, Esq., St. Louis. 
 Mr. F. A. Adams. 
 
 Dear Sir : — It is now nearly two years since I introduced your Arith- 
 metic into my School, and I have refrained from expressing any opinion 
 of its merits, until it should have been fairly and thoroughly tested. I 
 did not introduce it — nor do I any book — simply as a matter of experi- 
 ment. I believed it to be a good work, from examination of it ; and am 
 now prepared to add, that I know it to be a good one, from trial. 
 
 I cheerfully assent to the opinions I see expressed by others, on all the 
 valuable characteristics of the book. Beside these, I must mention the 
 fact, that upon examination of those students who have been carried 
 through the system, and made to comprehend it, I find them, in their 
 arithmetical processes, independent of arbitrary diction of rule, possessing 
 a strong analytical power, and arriving at results with great rapidity and 
 accuracy. The head-work takes precedence of the hand- work ; and they 
 have a why and wherefore for what they do. My commendation of the. 
 book is full and unequivocal. 
 
 Yours truly, 
 
 E. WYMAN, A. M., 
 
 Principal of the St. Louis English and Classical High School. 
 
a « ' 
 
 RECOMMENDAiTIOiNS. 
 
 From Elias Nusan, A. M., Principal of the Classical High School, 
 
 Newbimjperl. 
 
 Mr. Adams. 
 
 Dear Sir : — Having carefully examined your treatise on Arithmetic, I 
 would say that! have formed a very favorable opinion of it, and believe 
 it to be a valuable contribution to the department of instruction to which 
 it relates. Your method of performing mental operations on large num- 
 bers is philosophical, and, for the most part, original ; )^our rules and illus- 
 trations are written with clearness and precision, and your examples for 
 practice have been chosen with reference to the actual concerns of life. 
 You have not encumbered the book with arithmetical puzzles, which tend 
 to perplex the pupil, and occupy time which should be devoted to useful 
 problems. The practical and business-like character of the work is, in 
 my opinion, a great excellence. The examples are such as are of daily 
 occurrence in the family, the work-shop, and the counting-room ; and the 
 scholar who has become master of them will not need another course of 
 training, and a new set of rules, to meet the actual wants of trade and 
 business in the world. 
 
 Your, method of working Fractions, both Vulgar and Decimal, is 
 rational and easy ; and the doctrines of Proportion and of the Roots arc 
 plainly and distinctly unfolded. 
 
 The exposition of the laws of Mechanics, and the* sections on the Com- 
 parison of Similar Surfaces and Solids, on Per Centage, Mensuration, and 
 on Weights and Measures, are exceedingly valuable. 
 
 In short, I cannot but think that you have presented the whole science 
 of arithmetic in a very interesting and philosophical point of view; and 
 
 1 believe that those teachers who use your book wiilj find it admirably 
 fitted both to develop the intellectual powers of their pupils, and to make 
 them quick and accomplished arithmeticians. 
 
 With respect, I remain yours, 
 
 ELIAS NASON. 
 
 From Mr. William Smyth, Professor of Mathematics, Bitwdoin College. 
 
 I have examined the system of Arithmetic by the Rev. F. A. Adams, 
 Principal of Dummer Academy. The plan of the work, and the style of 
 its execution, appear to me well calculated to give to the learner clear 
 views of the general principles and operations of Arithmetic, and to 
 furnish the discipline requisite to a skilful and ready application of them. 
 The work, indeed, as should be the casern all works of the kind, appears 
 to have been composed in the recitation room, by one well conversant 
 with his subject, and possessing, in an eminent degree, the talents re- 
 quisite to a successful instructor ; and is therefore admirably adapted to 
 the wants both of the pupil and teacher. I should regard with much 
 pleasure its extensive introduction into our schools and academies. 
 
 WE SMYTH. - 
 
RECOMMENDATIONS. 9 
 
 From Mr. John D. Philbrick, Principal of the Mathematical Department 
 of the Mayhem School, Boston. 
 
 Mr. F. A. Adams. 
 
 My Dear Sir: — I am delighted with your Arithmetic. A careful 
 examination of every page of it has convinced me that it is a work of 
 icendcnt excellence. To say that it contains a great amount of mat- 
 ter well arranged ; that its rules and explanations are clear and logical, 
 and the examples well adapted to illustrate them, would be to accord to 
 it but a small part of its just meed of praise. 
 
 Its peculiar and crowning merit is, that it is calculated to emancipate 
 the learner from the bondage of rules, and even to give him dominion 
 over them, so that they shall be to him as clay in the hands of the potter. 
 I cannot but regard it as a superstructure worthy of its admirable basis, 
 Colburn's First Lessons; and if the one be a "faultless" school-book, the 
 other is not a whit less perfect. I am confident, therefore, that it needs 
 no other recommendations than its own merits, to insure it a hearty wel- 
 come every where among intelligent teachers. 
 
 Yours truly, 
 
 JOHN D. PHILBRICK. 
 
 From Mr. Stephen Holman, Principal of Fitchburg Academy. 
 
 Mr. F. A. Adams. 
 
 Dear Sir : — I have examined your work on Arithmetic, and find it 
 admirably adapted to teach the " science of numbers." It will be re- 
 ceived as a valuable assistant, by those teachers who aim to give their 
 pupils a thorough knowledge of the properties of numbers, as it enables 
 the learner to derive much from the text-book, which has heretofore been 
 communicated orally, if at all. The scholar who studies this book faith- 
 fully, will see the subject divested of its mystery, will learn to worship 
 rules less, and common sense more. 
 
 I have seen no Arithmetic so well adapted to the wants of our schools. 
 I am, sir, very respectfully yours, 
 
 STEPHEN HOLMAN. 
 
 From Mr. A. K. Hathaway, Principal of the Grammar School, Medford. 
 
 Mr. Adams. 4K 
 
 Dear Sir : — I have very carefully examined the " Advanced Lessons 
 in Mental Arithmetic," and with the highest satisfaction. The plan of 
 the work is admirable, and the execution shows great care and judg- 
 ment. 
 
 This work occupies new and unappropriated ground, and is just the 
 manual most needed in our schools, to give that high degree of mental 
 discipline, so necessary in training the young. It needs only to be known 
 to be appreciated ; and, when once known, I am fully confident it will be 
 very extensively used. 
 
 Yeurs very truly, 
 
 A. K. HATHAWAY. 
 
10 RECOMMENDATIONS. 
 
 From Rev. James Means, Principal of Lawrence Academy, Groton. 
 Daniel Bixby, Esa. 
 
 Dear. Sir : — I have carefully examined the " Mental and Written 
 Arithmetic " which you put into my hands, and am prepared to give it 
 my unqualified approval. 
 
 It would be inappropriate to detail all the excellences which it pos- 
 sesses. I will just mention, as points which gratify me, the method of 
 multiplying and dividing- large numbers, used by all good accountants, 
 and now introduced for the first time into a school-book ; the manner of 
 stating and illustrating the rules ; the nature of the questions proposed 
 for solution ; the introduction of matter quite new upon several points of 
 practical importance. I would refer to the rules for Compound Propor- 
 tion, Square and Cube Roots, several parts of Section XLIV., and all of 
 Section XLVI., as worthy of special notice. 
 
 I hope and believe that this book will be extensively used. I shall 
 commend it to the Committee of the Trustees here, who have such mat- 
 ters in charge, and make no doubt it will be permanently established as 
 a text-book. 
 
 Very respectfully yours, 
 
 JAMES MEANS. 
 
 From Teachers of the Public Schools in Lowell. 
 
 Having carefully examined the Mental and Written Arithmetic by F. 
 A. Adams, we do not hesitate to say, that, in its design " to continue and 
 extend the course of discipline in numbers," it is, in our opinion, far 
 superior to any thing that has fallen under our notice. 
 
 CHARLES MORILL, 8th Grammar School. 
 
 NASON H. MORSE, 4th do. do. 
 
 O. H. MORILL, 6th do. do. 
 
 JONA. KIMBALL, 3d do. do. 
 
 PERLEY BALCH, 1st do. do, 
 
 From the School Committee of Lowell. 
 
 At a meeting of the School Committee of Lowell, held March 15th, 
 1847, it was 
 
 Voted, That F. A. Adams's Arithmetic be adopted for the High School. 
 FREDERICK PARKER, Secretary. 
 
 At a meeting of the School Committee of Lowell, held May 2d, 1848, 
 the principals of several of the Grammar Schools having expressed a de- 
 sire to use F. A. Adams's Arithmetic in their schools, on motion, it was 
 
 Voted, That F. A. Adams's Arithmetic be adopted, to be used as a 
 text-book in any Grammar School the principal of which may be desirous 
 of using it. 
 
 FREDERICK PARKER, Secretary. 
 
RECOMMENDATIONS. 1 1 
 
 From Mr. A. Parish, Principal of Springfield High School. 
 
 Mr. F. A. Adams. 
 
 Dear Sir: — I have just completed the perusal of your Arithmetic, de- 
 signed to impart to pupils the power of solving - , mentally, problems in- 
 volving large numbers. The misgivings which I felt at the commence- 
 ment of my investigation vanished as I advanced. While the leading 
 feature of the work resembles, in some respects, Colburn's Unrivalled pro- 
 duction, in its processes and application of principles, it seems to open to 
 the pupil an entirely new field for mental calculations. The great facility 
 with which large numbers are rapidly disposed of without encumbering 
 the mind, is an element in the work which must contribute greatly to the 
 satisfaction as well as success of the scholar. 
 
 The very clear explanation and illustration of principles ; the original 
 and' varied character of the problems, happily graduated from the simple 
 to the more difficult ; the adaptation of the whole matter, both to produce 
 mental discipline and due preparation for business, must commend the 
 book forcibly to the attention of teachers who desire to employ the most 
 effectual aids in their profession. 
 
 A. PARISH. 
 
 
V 
 
YB 
 
 ^"7 j nr\ 
 
 
 
 911313 
 
 
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