NEW YORK; 
 

 IN MEMORIAM 
 FLOR1AN CAJORI 
 
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MENTAL ARITHMETIC; 
 
 OB, 
 
 ORAL EXERCISES IN ABSTRACT 
 
 COMMERCIAL ARITHMETIC, 
 
 WITH 
 
 FIRST LESSONS IS WRITTEN ARITHMETIC.) 
 
 FOR TUB 
 
 USE OF SCHOOLS. 
 BY CHAS. S. TENABLE, 
 
 / 
 
 PROFESSOR OP MATHEMATICS IN THE UNIVERSITY OP VIRGINIA*, FORMERLY PROFESSOR 
 OF MATHEMATICS AND ASTRONOMY, UNIVERSITY OF SOUTH CAROLINA; FOR- 
 MERLY PROFESSOR OF NATURAL PHILOSOPHY AND CHEMISTRY, UNI- 
 VERSITY OF GEORGIA; FORMERLY PROFESSOR OF MATHEMA- 
 TICS IN HAMPDEN SIDNEY COLLEGE, VIRGINIA. 
 
 NEW YORK: 
 
 RICHARDSON & COMPANY, 
 
 14 BOND STREET. 
 
 1867. 
 
MATHEMATICAL SERIES. 
 
 1. First Lessons in Numbers. An illustrated, easy, and attractive 
 
 Mental Arithmetic, for the use of Families and Primary Schools. 
 Adapted to the OBJECT METHOD. 30 cents. 
 
 2. Mental Arithmetic. After the inductive method of Pestalozzi, 
 
 with first lessons in Written Arithmetic. For Common Schools and 
 Academies. 45 cents. 
 
 3. Elements of Written Arithmetic. In which the fundamental 
 
 operations on Abstract and Denominate Numbers, and their various 
 applications, are thoroughly explained and illustrated by numerous 
 examples adapted to the business of practical life. For the use of 
 Schools. $1.00. 
 
 4. High School Arithmetic. FOR THE PSE OF SCHOOLS AND COL- 
 
 LEGES. In this work the reasoning on the fundamental principles 
 will be more fully developed, and illustrated by a greater variety 
 and number of applications and examples. In addition to the sub- 
 jects usually treated in Text-books of Arithmetic, it will contain 
 several chapters on Notation, Theory of Numbers, Progressions, 
 Logarithms, and the use of Logarithmic Tables. 
 
 5. Algebra. FOR THE USE OF SCHOOLS ANI> COLLEGES. Containing the 
 
 latest improvements. To a thorough discussion of the subjects 
 usually treated in Textbooks on this subject will be added chapters 
 on the Theory of Determinants, and other lessons from the modern 
 higher Algebra, for the use of University Students. 
 
 6. Legendre's Geometry* To the complete translation of the latest 
 
 edition of the standard work of Lespendre will be appended many 
 exercises in Geometrical Analysis, with hints for the solution of the 
 same. Also, notes on the application of the method of Limits to 
 the demonstration of theorems in Elementary Geometry. 
 
 7. Analytic Geometry, Part First Plane Co-ordinate Geometry, 
 
 The Point, Straight Line, Conic Sections, some of the hisrher Plane 
 Curves, Algebraic and Transcendental. Also, the Method of 
 Abridged Notation. 
 
 Part Second The Point and Straight Line in Space, the Plane, 
 Surfaces of the Second Order, Cones, Cylinders, and Surfaces of 
 Revolution, Conoids, Helicoids, Curves of Double Curvature, Ana- 
 lytical Theory of Projections. 
 
 RICHARDSON & CO., PUBLISHERS, NEW YORK. 
 
 Entered according to Act of Congress, in the year 1866, 
 
 BY RICHARDSOX & CO., 
 
 In the Clerk's Office of the District Court of the United States for the 
 Southern District of New York. 
 
A 
 
 CONTENTS. 
 
 CHAPTER I. 
 
 PAGE 
 
 Formation of Numbers by Counting Addition Use of the Arabic 
 
 Figures 7 
 
 CHAPTER II. 
 Subtraction 20 
 
 CHTAPTER III. 
 Multiplication 30 
 
 C3APTER IT. 
 Division 40 
 
 CHAPTER V. 
 Miscellaneous Questions 59 
 
 CHAPTER VI. 
 Multiples, Divisors, and Prime Factors 67 
 
 CHAPTER VII. 
 Fractions Proportion 72 
 
IV CONTENTS. 
 
 CHAPTER VIII. 
 
 PAGB 
 
 Denominate Numbers 119 
 
 CHAPTER IX. 
 The Metric System of Weights and Measures 132 
 
 CHAPTER X. 
 Percentage Interest 1ST 
 
 CHAPTER XI. 
 Miscellaneous Problems 151 
 
 9 
 
 CHAPTER XII. 
 Elementary Lessons in Written Arithmetic 1GO 
 
PREFACE. 
 
 THE advantages of the study of Mental, or Oral, Arithmetic 
 are very generally acknowledged, and yet it is much neglected 
 in the schools. The present work is prepared as a companion 
 to those treatises on written Arithmetic which are not mere 
 collections of examples, but whose aim is to place the study in 
 its proper position among other branches of a rational education. 
 
 Great care has been taken to arrange the subjects and to 
 select and frame the questions in such a manner that the pupil 
 may not only acquire readiness and accuracy in the solution of 
 numerical problems, but may gain a thorough knowledge of all 
 the simpler relations among numbers, and learn the reason for 
 each step taken in the various operations performed on them. 
 
 The method pursued is that identified with the name of Pes- 
 talozzi. The clearest exposition of this method in abstract 
 numbers is to be found, perhaps, in Reiner's Lessons in a Pesta- 
 lozzian school in Surrey, England. Use has been made of this, 
 and of Menzel's valuable G-erman collection of examples for 
 " Head-reckoning." 
 
 The attention of teachers is called to the following points in 
 this little work : Its order and method ; the chapters on mul- 
 tiples, divisors, and prime factors ; the treatment of vulgar frac- 
 tions ; the mode of teaching proportional numbers ; the brief and 
 direct analysis of the various solutions of questions given ; the 
 tables of French decimal weights and measures (now authorized 
 in this country by the United States Government), with appro- 
 priate questions, illustrating the superiority of these tables over 
 tho ordinary ones for ready-reckoning. 
 
Yl PREFACE. 
 
 A few pages of written Arithmetic are appended, for the 
 convenience of those who may wish to acquire some familiarity 
 with the operations' on higher abstract numbers before entering 
 upon the next book of the series. These pages may also be 
 used to advantage concurrently with the rest of the book. 
 
 SUGGESTIONS TO TEACHERS. 
 
 1. Require the pupil to commit to memory the necessary 
 tables of Addition, Subtraction, Multiplication, and Division. 
 
 2. Practice the pupil frequently in counting forwards and 
 backwards by twos, threes, tens, &c., &c. This will cause him 
 to understand thoroughly the basis of all the operations on 
 whole numbers, and that Multiplication and Division are 
 abridged forms of Addition and Subtraction. 
 
 3. Do not enter upon the subject of fractions until the pupil 
 is very familiar with the questions on factors, multiples, and 
 divisors. 
 
 4. Add examples and illustrations to those in the book, and 
 require your pupils in each lesson to frame questions and give 
 their solutions. This will be found a most useful exercise, 
 especially in the miscellaneous questions on Proportion and its 
 applications. 
 
 5. Pupils in written Arithmetic should be called on to solve, 
 orally, questions similar to those in each .set of examples which 
 
 . they work on the slate. 
 
MENTAL ARITHMETIC, 
 
 CHAPTER I. 
 
 ADDITION. 
 LESSON I. 
 
 1. John has one knife ; if his father gives him 
 another, how many will he have ? 
 
 2. Frank has two oranges, and Charles one ; how 
 many have they both together ? 
 
 3. James is in school three hours every morning, 
 and one hour each evening ; how many hours is he in 
 school each day ? 
 
 4. A boy took four eggs from one bird's nest, and 
 one from another ; how many birds' eggs did he take ? 
 
 5. Mary has five tulips in bloom on one bed in the 
 
 farden, and one upon another ; how many tulips 
 loom on both beds ? 
 
 6. One and one are How many ? Two and one ? 
 Three and one ? Four and one ? Five and one ? Six 
 and one? Seven and one ^ Eight and one? Nine 
 and one ? Ten and one ? 
 
 7. Count from one to ten. Count the birds you 
 know. Count the flowers you know. Count from ten 
 to twenty. Count the number of words in the first two 
 questions of this lesson. Count from twenty to thirty. 
 Count from thirty to forty. 
 
 8. Begin at twenty and add successively the letters 
 of the sentence, " Labor overcomes the greatest diffi- 
 
8 ADDITION. 
 
 culties." Count from fifty to sixty. From sixty to 
 seventy, eighty, ninety, one hundred. Commence at 
 the first word of this lesson, and count the words as 
 far as one hundred. 
 
 9. How many tens have you counted in counting 
 twenty ? in counting thirty ? forty ? fifty ? sixty ? 
 seventy ? eighty ? ninety ? one hundred ? 
 
 10. Count from one hundred to two hundred. 
 
 11. How would you count from two hundred to 
 three hundred ? 
 
 ANS. By counting from one to one hundred^ and 
 repeating two hundred before each number counted. 
 
 12. How would you count from three hundred to 
 four hundred ? to five hundred ? six hundred ? seven 
 hundred? eight hundred? nine hundred? ten hun- 
 dred? 
 
 13. What do we call ten hundred ? 
 ANS. One thousand. 
 
 LESSON II. 
 
 1. A man has fifty-two white sheep and one black 
 one; how many has he altogether? 
 
 2. One company of soldiers has sixty-five men in 
 it, a second company has one man more ; how many 
 men in the second company ? 
 
 3. Of a flock of blackbirds, forty-five are on a tree, 
 and one on the fence : how many blackbirds in the 
 flock? 
 
 4. Ten and one are how many? Twenty and one ? 
 Thirty and one ? Forty and one ? Fifty and one ? 
 Seventy and one ? Ninety and one ? 
 
 5. Eleven and one are how many ? Twenty-one 
 and one ? Thirty-one and one ? Fifty-one and one ? 
 Sixty-one and one ? Eighty -one and one ? One hun- 
 dred and one and one ? 
 
 6. Twelve and one are how many ? Twenty-two 
 
ADDITION. 9 
 
 and one ? Thirty-two and one ? Forty-two and one ? 
 Fifty-two and one? Sixty-two and one? Seventy- 
 two and one ? Eighty -two and one ? Ninety-two 
 and one ? 
 
 7. Fourteen and one? Twenty-four and one? 
 Thirty- four and one ? Forty-four and one ? Fifty-four 
 and one? Sixty-four and one ? Seventy-four and one ? 
 Eighty-four and one ? Ninety-four and one ? 
 
 8. Fifteen and one are how many ? Twenty-five 
 alid one ? &c. 
 
 9. Nineteen and one are how many ? Twenty-nine 
 and one? &c. 
 
 10. One ten and one ten are how many ? Two tens 
 and one ten ? Three tens and one ten ? Seven tens 
 and one ten? Eight tens and one ten? Ten tens and 
 one ten ? 
 
 11. One hundred and one hundred are how many? 
 Two hundred and one hundred ? Three hundred and 
 one hundred? Four hundred and one hundred? 
 Nine hundred and one hundred ? 
 
 LESSON III. 
 
 1. Two oranges and two oranges are how many 
 oranges ? 
 
 2. A planter made three bales of cotton on one acre 
 of land, and two on another acre ; how many bales 
 did he make on both ? . 
 
 3. Charles has three marbles, and James has three ; 
 how many have they together ? 
 
 4. A traveller walked three miles the first hour, and 
 four miles the second ; how many miles did he walk 
 in the two hours ? 
 
 5. A planter makes five hogsheads of tobacco one 
 year, and four the next ; how many hogsheads in the 
 two years ? 
 
 i* 
 
10 ADDITION. 
 
 6. One and two are how many ? One and three ? 
 One and four ? 
 
 7. Two and two ? Two and three ? Two and four ? 
 
 8. Three and two? Three and three? Three and 
 four? 
 
 9. Four and two ? Four and three ? Four and 
 four? 
 
 10. Five and two ? Five and three ? Five and 
 four? 
 
 11. Six and two? Six and three? Six and four? 
 
 12. Seven and two ? Seven and three ? Seven and 
 four? 
 
 13. Eight and two ? Eight and three ? Eight and 
 four ? 
 
 14. Nine and two ? Nine and three ? Nine and 
 four ? 
 
 15. Ten and two ? Ten and three ? Ten and four ? 
 
 16. Twenty and two are how many ? Thirty and 
 two? Forty and two? Fifty and two? Sixty and 
 two ? Seventy and two ? Eighty and two ? Ninety 
 and two ? 
 
 17. Twenty and three are how many ? &c. 
 
 18. Twenty and four are how many? &c. 
 
 The teacher will cause the pupil to go through the same addition as 
 above with three and four. 
 
 i 
 
 19. Eleven and two are how many? Twenty-one 
 and two ? Thirty-one and two ? Forty-one and two ? 
 Fifty-one and two ? Sixty-one and two ? Seventy- 
 one and two ? Eighty-one and two ? Ninety-one and 
 two? 
 
 20. Eleven and three are how many ? Twenty -one 
 and three ? &c. 
 
 21. Eleven and four are how many? Twenty-one 
 and four? &c. 
 
 22. Twelve and two are how many ? .Twenty-two 
 and two ? &c. 
 
ADDITION. 11 
 
 23. Twelve and three are how many ? Twenty-two 
 and three ? &c. 
 
 24. Twelve and four are how many ? Twenty -two 
 and four ? &c. 
 
 25. Thirteen and two ? Twenty-three and two ? &c. 
 
 26. Thirteen and three ? Twenty -three and three? 
 &c. 
 
 27. Thirteen and four? . Twenty-three and four? 
 &c. 
 
 28. Fourteen and two ? Twenty-four and two ? &c. 
 
 29. Fifteen and two ? Twenty-five and two ? &c. 
 
 30. Sixteen and two ? Twenty-six and two ? &c. 
 
 31. Seventeen and two ? Twenty-six and two ? &c. 
 
 32. Eighteen and two ? Twenty-eight and two ? &c. 
 
 33. Nineteen and two ? Twenty-nine and two ? &c. 
 
 The teacher should extend these questions as far as ninety-three in, 
 the same manner with those which are completed, and make the 
 same series of additions with three and four. 
 
 34. Count by twos to one hundred and one, begin- 
 ning with one. 
 
 35. Count by twos to one hundred, beginning with 
 two. 
 
 36. Count by threes to one hundred, -beginning with 
 one. 
 
 37. Count by threes to one hundred and one, begin- 
 ning with two. 
 
 38. Count by threes to ninety-nine, beginning with 
 three. 
 
 39. Eighty-seven and one, and two, and three, are 
 how many ? 
 
 40. Ninety two and two, and three, and four, are 
 ow many? 
 
 41. Thirty-five and two, and three? Forty-three 
 ,nd four, and two? Fifty-four and four, and 
 
 three ? 
 
12 
 
 ADDITION. 
 
 LESSON IV. 
 
 We have thus far written out the names of the 
 different numbers. For the sake of abbreviation in 
 the succeeding lessons, we will use the figures of the 
 Arabic notation. 
 
 These are 
 
 one two three four five six seven eight nine zero or naught 
 1234567 89 
 
 NOTE. For a more detailed explanation of the use of figures, see 
 first lessons in Numbers, page 31. 
 
 We have seen, in counting, there are two tens in 
 twenty, three in thirty, four in forty, &c., and ten 
 tens in one hundred. If now we wish to express 
 numbers greater than nine by those figures, we may 
 do so readily by the use of columns and the 
 (zero). 
 
ADDITION. 13 
 
 Hundreds. Tens. Units or ones. 
 
 5 6 
 
 4 
 
 123 
 100 
 
 Thus 5 in the column of tens and 6 in the column 
 of ones or units, means five tens and six units, or fifty- 
 six ; 4 in the column of tens and in the column of units 
 indicates four tens and no units, of forty ; 1 in the 
 column of hundreds, and 2 in the column of tens, and 3 
 in that of units, indicates one hundred, two tens, and 
 three units, or one hundred and twenty-three ; 1 in 
 the hundreds column, in tens, and in units, indi- 
 cates one hundred. We leave out the columns in the 
 actual writing of numbers and write them thus : 56, 
 40, 123, 100, and simply recollect the local values of 
 the figures. 
 
 1. What is the local value of r the figure on the right 
 of a number ? 
 
 ANS. Units. 
 
 2. What is the local value of the figure next to the 
 right-hand figure ? 
 
 3. The third figure from the right ? the fourth ? 
 
 4. What is the meaning and use of the 0? 
 
 NOTE. The pupil should now be required to read the following 
 table, and be drilled in the use of figures, until he can readily write any 
 number from one to one thousand in figures: 
 
 10 20 40 30 88 
 
 11 21 41 31 90 
 
 12 22 42 32 93 
 
 13 23 43 60 99 
 
 14 24 44 61 100 
 
 15 25 45 62 110 
 
 16 26 50 70 101 
 
 17 27 51 78 255 
 
 18 28 52 80 999 
 
 19 29 53 81 1000 
 
14 
 
 ADDITION. 
 
 LESSON V. 
 
 1. John bought a lemon for five cents, and an apple 
 for five cents ; how much did they both cost him ? 
 
 2. One man rides six miles in one hour, and another 
 rides three miles more than the first ; how many miles 
 does the latter ride in one hour? 
 
 3. Edward begins to work at six o'clock in the 
 morning and works five hours ; at what o'clock does 
 he stop, work ? 
 
 4. Harry has seven peaches, and Frank has five ; 
 how many peaches have both ? 
 
 5. A farmer has seven cows in one pasture, and nine 
 in another ; how many has he in both ? 
 
 1 and 7 \ 
 
 6. 
 7. 
 8. 
 9. 
 10. 
 11. 
 12. 
 13. 
 14. 
 
 1 
 
 2 
 3 
 4 
 5 
 
 6 
 7 
 8 
 9 
 
 and 
 and 
 and 
 and 
 and 
 and 
 and 
 and 
 and 
 
 5 are how 
 5 ? 2 and 
 5 ? 3 and 
 5 ? 4 and 
 5 ? 5 and 
 5 ? 6 and 
 53 7 and 
 5 ? 8 and 
 5 ? 9 and 
 
 many \ 
 6? 2 
 6? 3 
 6? 4 
 6? 5 
 6? 6 
 6? 7 
 6? 8 
 6? 9 
 
 1 
 and 
 and 
 and 
 and 
 and 
 and 
 and 
 and 
 
 and 6? 
 
 7? 
 7? 
 7? 
 7? 
 7? 
 7? 
 7.1 
 7? 
 
 15. 10 and 5 are how many ? 20 and 5 ? 30 and 
 5 ? 40 and 5 ? 50 and 5 ? 60 and 5 ? 70 and 5 2 
 80 and 53 90 and 5 ? 
 
 16. 10 and 6 are how many ? 20 and 6 ? &c. 
 
 17. 10 and 7 are how many ? 20 and 7 ? &c. 
 
 18. 11 and 5 are how many ? 21 and 5 ? &c. 
 
 19. 11 and 6 are how many ? 21 and 6 ? &c. 
 
 20. 11 and 7 are how many ? 21 and 7 ? &c. 
 , 21. 12 and 5 are how many ? 22 and 5 ? &c. 
 
 22. 12 and 6 are how many ? 22 and 6 ? &c. . 
 
 23. 12 and 7 are how many ? 22 and 7 ? &c. 
 
 24. 13 and 5 are how many ? 23 and 5 ? &c. 
 
 25. 13 and 6 are how many ? 23 and 6 ? &c. 
 
ADDITION. 
 
 15 
 
 26. 13 and 7 are how many ? 23 and 7 ? &c. 
 
 27. 14 and 5 are how many ? 24 and 5 ? &c. 
 
 28. 14 and 6 are how many ? 24 and 6 ? &c. 
 
 29. 14 and 7 are how many ? 24 and 7 ? &c. 
 
 30. 15 and 5 are how many ? 25 and 5 ? &c. 
 
 31. 16 and 5 are how many ? 26 and 5 ? &c. 
 
 32. 17 and 5 are how many ? 27 and 5 ? &c. 
 
 33. 18 and 5 are how many ? 28 and 5 ? &c. 
 
 34. 19 and 5 are how many ? 29 and 5 ? &c. 
 
 To be carried out as far as 91, 92, 93, 94, &c., by the pupil, for 5, 6, 
 and 7. 
 
 LESSON VI C 
 
 I. 1 Count by 5's to 101, beginning with 1. To 102, 
 beginning with 2. To 103, beginning with 3. To 
 104, beginning with 4. To 100, beginning with 5. 
 
 2. Count by 6's, beginning with 1, 2, 3, 4, 5, 6, re- 
 spectively. Thus, 1, 7, 13, &c. 2, 8, 14, &c., &c. 
 
 3. Count by 7's, beginning with 1, 2, 3, 4, 5, 6, re- 
 spectively. Thus, 1, 8, 15, &c. 2, 9, 16, &c., &c. 
 
 4. A man had sixty-five sheep in one fold, five in 
 a second, six in a third, and seven in a fourth ; how 
 many had he altogether ? 
 
 5. A farmer gathered fifty-six barrels of apples 
 from his orchard, and seven barrels from two trees 
 which stood in the corner of his yard ; how many bar- 
 rels did he gather ? 
 
 6. There are three hundred soldiers in one regi- 
 ment, and six hundred in another ; how many in the 
 two regiments? 
 
 7. 2 tens and 5 tens are how many ? 2 tens and 6 
 tens ? 2 tens and 7 tens ? 
 
 8. 9 tens and 5 tens are how many ? 9 tens and 
 6 tens ? 9 tens and 7 tens ? 
 
 9. 3 hundred and 5 hundred are how many ? 3 
 hundred and 6 hundred ? 3 hundred and 7 hundred ? 
 
16 ADDITION. 
 
 10. 4 hundred and 5 hundred are how many ? 4 
 hundred and 6 hundred ? 4 hundred and 7 hundred ? 
 
 11. Add fifty-seven, seven, six, and five. 
 
 12. Add forty-nine, seven, five, three, and four. 
 
 13. How many are ninety-seven, six, seYen, four, 
 seven, and three? 
 
 14. How many are forty-five, six, three, five, six, 
 four, three, and four ? 
 
 LESSON VII. 
 
 1. Frank is seven years old, his sister is eight years 
 older ; how old is his sister ? 
 
 2. Thomas killed six squirrels on Monday, and nine 
 on Tuesday ; how many squirrels did he kill on both 
 days? 
 
 3. A gardener gathered five bushels of apples from 
 one tree, and ten bushels from another ; how many 
 bushels from the two trees ? 
 
 4. 1 and 8 are how many ? 1 and 9 ? 1 and 10 ? 
 
 5. 2 and 8 ? 2 and 9 ? "2 and 10 ? 
 
 6. 3 and 8? 3 and 9 ? 3 and 10? 
 
 7. 4 and 8 ? 4 and 9 ? 4 and 10? 
 
 8. 5 and 8 ? 5 and 9 ? 5 and 10 ? 
 
 9. 6 and 8 ? 6 and 9 ? 6 and 10 ? 
 
 10. 7 and 8? 7 and 9? 7 and 10? 
 
 11. 8 and 8 ? 8 and 9 ? 8 and 10 ? 
 
 12. 9 and 8? 9 and 9 ? 9 and 10 ? 
 
 13. 10 and 8 are how many ? 20 and 8 ? 30 and 8 ? 
 &c., to 98. 
 
 14. 11 and 9 are how many ? 21 and 9 ? 81 and 9 ? 
 &c., to 10. 
 
 15. 12 and 10 are how many ? 22 and 10 ? 32 and 
 10 ? &c., to 102. 
 
 16. 13 and 8 are how many ? 23 and 8 ? 33 and 8 ? 
 &c., to 101. 
 
 17. 14 and 9 are how many ? 24 and 9 ? 34 and 9 ? 
 &c., to 103. 
 
ADDITION. IT 
 
 18. 15 and 10 are how many? 25 and 10 ? 35 and 
 10 ? &c., to 105. 
 
 19. 1 6 and 8 are how many ? 26 and 8 ? 36 and 8? 
 &c., to 104. 
 
 20. 17 and 9 are how many ? 27 and 9 ? 37 and 9 ? 
 &c., to 106. 
 
 21. 18 and 10 are how many ? 28 and 10 ? 38 and 
 10 I &c., to 108. 
 
 22. 19 and 8 are how many ? 29 and 8 ? 39 and 8 ? 
 &c., to 107. 
 
 23. 10 and 10 are how many ? 20 and 10 ? 30 and 
 10? &c., to 100. 
 
 24. How much is forty-five and eight ? , Two and 
 seven and five and eight and nine are how many ? 
 
 25. Fifty-seven and eight are how many ? 
 
 26. Ninety-three and nine ? one hundred and sixteen 
 and ten ? 5, 9, 6, 9, 3, 8, 7, 2, and 1 are how many? 
 Ninety-seven, eight, nine, and ten are how many ? 
 
 LESSON VIII. 
 
 1. Count by 8's, beginning with 1, 2, 3, 4, 5^6, 7, and 
 8 respectively. Thus, 1, 9, 17, &c. 2, 10. 18, &c., 
 &c. 
 
 2. Count by 9's, beginning with 1, 2, 3, 4, 5, 6, 7, 
 8, and 9 respectively. Thus, 1, 10, 19, &c. 2, 11, 
 20, &c., &c. 
 
 3. Count by 10's, beginning with 1, 2, 3, 4, 5, 6, 7, 
 8, 9, and 10 respectively. Thus, 1, 11, 21, &c. 2, 
 12, 22, &c., &c. 
 
 4. A farmer sends sheep to market in two droves, 
 one drove has twenty sheep in it and the other eighty ; 
 how many sheep in both droves ? 
 
 5. A man makes five hundred bushels of wheat on 
 one farm, and nine hundred on another ; how many 
 hundred bushels does he make on both farms? 
 
 6. A grazier has one hundred and seventeen head of 
 
18 ADDITION. 
 
 cattle in one field, and ten head in another ; how many 
 head in the two fields ? 
 
 7. 3 tens and 8 tens are how many tens ? 3 tens 
 and 9 tens ? 3 tens and 10 tens ? 
 
 8. 8 tens and 8 tens are how many tens ? 8 tens 
 and 9 tens? 8 tens and 10 tens ? 
 
 9. 2 hundred and 8 hundred are how many hun- 
 dred ? 2 hundred and 9 hundred ? 2 hundred and 10 
 hundred ? 
 
 10. 7 hundred and 8 hundred ? 7 hundred and 9 
 hundred ? 7 hundred and 10 hundred \ 
 
 LESSON IX. 
 
 1. A father has three books, and wishes to give 
 them to his three sons ; how many would he give to 
 each son? 
 
 2. If he wished to give them to two sons, how many 
 would he give to one and how many to the other? 
 
 3. Three can be broken up how ? 
 
 ANS. Into three ones, or into two and one. 
 
 4. 4: can be separated into what numbers ? 
 ANS. 3 and 1, 2 and 2, 1 and 1 and 1 and 1. 
 
 5. What different numbers added together make 
 5? 
 
 6. A boy has six marbles ; into what different heaps 
 can he distribute them ? 
 
 7. 6 can be separated into what numbers ? 
 
 8. 7 can be separated into what numbers ? 
 
 9. What different numbers added together will 
 makeS? 9? 10? 
 
 When a number is greater than ten, and less than one hundred, the 
 inost usual and useful separation in our system of numbers is into 
 tens and units ; or, if greater than one hundred, and less than a thou- 
 sand, into hundreds, tens, and units. The notation or mode of writing 
 numbers, which has been explained in a previous lesson, makes this 
 separation very easv. 
 
 18 
 
ADDITION. 1 9 
 
 10. Separate 11, 12, 13, 14, 15, 27, 49, 58, and 97 
 into tens and units. 
 
 11. Separate 137, 142, 156, 207, 320, 197 into hun- 
 dreds, tens, and units, and into tens and units. 
 
 12. 25 and 19 are how many? 
 
 PROCESS. Separate 19, and we have 1 ten and 9 
 units ; 25 and 10 are 35 ; 35 and 9 are 44. Hence, 25 
 and 19 are 44. 
 
 13. 34 and 14 are how many ? 31 and 18 ? 32 and 
 17 ? 32 and 16 ? 
 
 14. 14 and 19 ? 95 more 13 ? 46 and 12 ? 
 
 15. 11 and 11 are how many ? 22 and 11 ? 33 and 
 11? 
 
 16. Count by ll's to 99, beginning with 11. 
 
 17. 12 and 12 are how many ? 24 and 12 ? 36 and 
 12? 
 
 18. Count by 12's to 96, beginning with 12. 
 
 19. 40 and 20 are how many ? 50 and 30 ? 60 and 
 40 ? 30 and 60 ? 70 and 20 ? 90 and 50 ? 
 
 20. 40 and 27 are how many ? 
 
 PROCESS. 27 ^ 20 andl; 40 and 20 are 60, 60 and 
 7 make 67. 
 
 21. How much is 60 and 32? 70 and 29? 90 
 and 67 ? 50 and 48 ? 80 and 22 ? 
 
 22. 47 and 20 are how many? 58 and 30? 65 
 and 40 ? 33 and 30 ? 72 and 20 ? 
 
 23. 48 arid 27 are how many ? 
 
 PROCESS. 40 and 20 are 60; 8 and 7 are 15; 60 
 and 15 are 75. Or thus, 27 is 20 and 7, 48 and 20 
 make 68, 68 and 7 make 75. 
 
 24. 72 and 25 are how many ! 64 and 56 ? 31 
 and 22 ? 48 and 63 ? 84 and 26? 
 
20 SUBTRACTION. 
 
 CHAPTEE II. 
 
 SUB TRACTION. 
 LESSON I. 
 
 1. A boy has one apple, and gives it away ; how 
 many has he left ? Another has twenty apples and 
 gives them away ; how many has he left ? 
 
 2. 1 from 1 leaves how many ? 20 from 20 leaves 
 how many ? 
 
 3. Any number taken from itself leaves what ? 
 
 4. A boy has five peaches, and eats one of them ; 
 how many has he left ? 
 
 5. Mary had five books, but she has lost two of 
 them ; how many has she now? 
 
 6. "William had ten canary birds in a cage, but al- 
 lowed two of them to escape ; how many has he left ? 
 
 7. What do you call the result obtained by adding 
 one number to another ? 
 
 ANS. Their sum. 
 
 8. What do you call the result obtained by taking 
 one number from another ? 
 
 ANS . Their differ en ce. 
 
 9. 1 from 1 leaves how much ? 
 
 2 less 2 is ? 
 
 10. 1 from 2 leaves ? 
 
 3 less 2 ? 
 
 11. 1 from 3 leaves? 
 
 4 less 2 ? 
 
 12. 1 from 4 leaves? 
 
 5 less 2 ? 
 
 13. 1 from 5 leaves ? 
 
 6 less 2 ? 
 
 14. 1 from 6 leaves ? 
 
 7 less 2 ? 
 
 15. 1 from 7 leaves ? 
 
 8 less 2 ? 
 
 16. 1 from 8 leaves ? 
 
 .9 less 2 ? 
 
 17. 1 from 9 leaves? 
 
 10 less 2 ? 
 
 18. 1 from 10 leaves ? 
 
 
 19. 11 less 1 is ? 21 less 1 ? 31 
 
 less 1 ? 41 less 1 ? 
 
 51 less 1? 61 less 1? 71 less 1? 81 
 
 lessl? 91 less 1? 
 
SUBTRACTION. 21 
 
 20. 11 less 2 ? 21 less 2 ? &c., to 101. 
 
 21. 22 less 1 ? 23 less 1 ? 24 less 1 ? 35 less 1 ? 
 46 less 1 ? 
 
 22. 57 less one? 68 less 1? 79 less 1 ? 110 less 1 ? 
 
 23. 12 less 2 is how much? 22 less 2? &c., to 
 102. 
 
 24. 13 less 2 ? 23 less 2 ? &c., to 103. 
 
 25. 14 less 2 ? 24 less 2 ? &c., to 94. 
 
 26. 15 less 2 ? 25 less 2 ? &c., to 95. 
 
 27. 16 less 2 ? 26 less 2 ? &c., to 96. 
 
 28. 18 less 2? 28 less 2? &c., to 98. 
 
 29. 19 less 2? 29 less 2? &c., to 99. 
 
 30. Begin at 100, and count backward down to 1. 
 
 31. Begin with 100, and count backwards by 2's 
 down to 2. 
 
 32. Begin with 99, and count backwards by 2's 
 down to 1. 
 
 33. 1 ten from 2 tens leaves how many? 1 ten 
 from 3 tens? 
 
 34. 1 ten from 6 tens ? 2 tens from 8 tens ? 
 
 35. 1 hundred from 4 hundred leaves how many ? 
 1 hundred from 8 hundred ? 2 hundred from. 7 hun- 
 dred I 2 hundred from 9 hundred 3 
 
 LESSON II. 
 
 1. George had ten chickens, but y a weasel killed 
 three of them ; how many remained ? 
 
 2. One farmer has seven horses and another has 
 four ; how many more has the first than the second ? 
 
 3. A man planted twelve trees, but five of them 
 died ; how many lived ? 
 
 4. A man owing thirteen dollars, paid three of 
 them ; how much did he still owe ? 
 
 5. A farmer had fifteen cows in one pasture, and five 
 in a second ; how many more were in the first pas- 
 ture than in the second 'I 
 
SUBTRACTION. 
 
 6. 
 7. 
 8. 
 9. 
 10. 
 11. 
 12. 
 13. 
 14. 
 15. 
 
 3 less 
 4 less 
 5 less 
 6 less 
 7 less 
 8 less 
 9 less 
 10 less 
 11 less 
 12 less 
 
 3 
 3? 
 3? 
 3? 
 3? 
 3? 
 3? 
 3? 
 3? 
 3? 
 
 
 
 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 
 less 
 less 
 less 
 less 
 less 
 less 
 less 
 less 
 less 
 less 
 
 4? 
 
 4? 
 4? 
 4? 
 4? 
 4? 
 4? 
 4? 
 4? 
 4? 
 
 5 less 5 ? 
 6 less 5 ? 
 7 less 5 ? 
 8 less 5 ? 
 9 less 5 ? 
 10 less 5 ? 
 11 less 5 ? 
 12 less 5 ? 
 13 less 5 ? 
 14 less 5 ? 
 
 16. 
 
 13 
 
 less 
 
 3? 
 
 
 
 
 14 
 
 less 
 
 4? 
 
 15 less 5 ? 
 
 17. 
 
 10 
 
 less 
 
 3? 
 
 20 
 
 less 
 
 3? 
 
 30 
 
 less 
 
 3? 
 
 &c., 
 
 to 90. 
 
 18. 
 
 11 
 
 less 
 
 3? 
 
 21 
 
 less 
 
 3? 
 
 31 
 
 less 
 
 3? 
 
 &c., 
 
 to 91. 
 
 19. 
 
 12 
 
 less 
 
 3? 
 
 22 
 
 less 
 
 3? 
 
 32 
 
 less 
 
 3? 
 
 &c., 
 
 to 92. 
 
 20. 
 
 13 
 
 less 
 
 3? 
 
 23 
 
 less 
 
 3? 
 
 33 
 
 less 
 
 3? 
 
 &c., 
 
 to 93. 
 
 21. 
 
 14 
 
 less 
 
 3? 
 
 24 
 
 less 
 
 3? 
 
 34 
 
 less 
 
 3? 
 
 &c., 
 
 to 94. 
 
 22. 
 
 15 
 
 less 
 
 3? 
 
 25 
 
 less 
 
 3? 
 
 35 
 
 less 
 
 3? 
 
 &c., 
 
 to 95. 
 
 23. 
 
 16 
 
 less 
 
 3? 
 
 26 
 
 less 
 
 3? 
 
 36 
 
 less 
 
 3? 
 
 &c., 
 
 to 96. 
 
 24. 
 
 17 
 
 less 
 
 3? 
 
 27 
 
 less 
 
 3? 
 
 37 
 
 less 
 
 3? 
 
 &c., 
 
 to 97. 
 
 25. 
 
 18 
 
 less 
 
 3? 
 
 28 
 
 less 
 
 3? 
 
 38 
 
 less 
 
 3? 
 
 &c., 
 
 to 98. 
 
 26. 
 
 19 
 
 less 
 
 3? 
 
 29 
 
 less 
 
 3? 
 
 39 
 
 less 
 
 3? 
 
 &c., 
 
 to 99. 
 
 NOTE. The teacher should cause the pupil to go through several of 
 the above subtractions with 4 and 5 respectively. 
 
 27. Count by 3's backwards from 100 to 1. 
 
 28. Count by 4's backwards from 100 to 0. 
 
 29. Count by 5 's backwards from 100 to 0. 
 
 30. 3 tens from 8 tens leaves how many ? 4 tens 
 from 7 tens ? 5 tens from 9 tens ? 
 
 31. One veteran regiment has 400 men in it, and 
 another 500 men ; how many more in the last than in 
 the first ? 
 
 32. 3 hundred from 6 hundred leaves how many ? 
 7 hundred less 5 hundred is ? 
 
 33. 4 hundred from 8 hundred leaves how many? 
 6 hundred from 9 hundred ? 2 hundred from 7 
 hundred ? 
 
SUBTRACTION. 23 
 
 LESSON III. 
 
 1. A horse travelled six miles in one hour, and nine 
 miles the next ; how many more miles did he travel 
 in the second hour than in the first ? 
 
 2. A farmer paid twelve dollars to one laborer, and i 
 eight dollars to another; how much less did he pay 
 the second laborer than the first ? 
 
 3. Julian found thirteen eggs, and gave John seven ; 
 how many had he left? 
 
 4. A hunter killed ten bears in the canebrake one 
 day, and six on the next day; how many more did he 
 kill on the first day than the second ? 
 
 5. 6 less 6 7 less 7 ? 8 less 8 ? 
 
 6. 7 less 6 ? 8 less 7 3 9 less 8 ? 
 
 7. 8 less 6 \ 9 less 7 \ 10 less 8 ? 
 
 8. 9 less 6 \ 10 less 7 \ 11 less 8 ? 
 
 9. 10 less 6 ? 11 less 7 ? 12 less 8 1 
 
 10. 11 less 6 ? 12 less 7 ? 13 less 8 ? 
 
 11. 12 less 6 \ 13 less 7 ? 14 less 8 1 
 
 12. 13 less 6 ? .14 less 7 ? 15 less 8 ? 
 
 13. 14 less 6 ? 15 less 7 ? 16 less 8 ? 
 
 14. 15 less 6 ? 16 less 7 2 17 less 8 2 
 
 15. 16 less 6 ? 17 less 7 ? 18 less 8 ? 
 
 16. 20 less 7 is how much ? 30 less 7 ? &c., to 90. 
 
 17. 21 less 7 is how much ? 31 less 7? &c., to 91. 
 
 18. 22 less 7 ? 32 less 7 ? &c., to 92. 
 
 19. 23 less 7 ? 33 less 7 ? &c., to 93. 
 
 20. 24 less 7 ? 34 less 7 ? &c., to 94. 
 
 21. 25 less 7 2 35 less 7 ? &c., to 95. 
 
 22. 26 less 7 ? 36 less 7 ? &c., to 96. 
 
 23. 27 less' 7 ? 37 less 7 ? &c., to 97. 
 
 24. 28 less 7 ? 38 less 7 ? &c., to 98. 
 
 25. 29 less 7 ? 39 less 7 ? &c., to 99. 
 
 NOTE. The pupil should be required to go through several of the 
 above with 6 and 8 respectively. 
 
 26. Count by 6's backward from 100 to 4. 
 
24 SUBTRACTION. 
 
 27. Count by 7's backwards from 100 to 2. 
 
 28. Count by 8's backwards from 100 to 4. 
 
 29. A man has 60 sheep in one fold, and 90 in a 
 second fold ; how many more in the second than in 
 the first ? 
 
 30. 6 tens from 9 tens leaves how many ? 7 tens 
 from 10 tens leaves ? 8 tens from 12 tens ? 
 
 31. There were 700 bison in one herd on the 
 prairie, and 900 in another herd ; how many more in 
 the last herd than the first ? 
 
 ' 32. 600 from 900 leaves how many ? 8 hundred 
 from 11 hundred ? 
 
 LESSON IV. 
 
 1. A man agreed to labor for a farmer 15 days, 
 and left at the end of nine days ; how many days' 
 labor did he still owe the farmer ? 
 
 2. A hunter on the Texas prairies killed 10 deer 
 on one day, and 14 the next ; how many more did he 
 kill on the second day than on the first ? 
 
 3. 9 less 9 are how many ? 10 less 10 are how 
 many ? 
 
 4. 10 less 9? 11 less 10? 
 
 5. 11 less 9 ? 12 less 10 ? 
 
 6. 12 less 9 ? 13 less 10 ? 
 
 7. 13 less 9? 14 less 10? 
 
 8. 14 less 9? 15 less 10? 
 
 9. 15 less 9 ? 16 less 10 ? 
 
 10. 16 less 9 ? 17 less 10 ? 
 
 11. 17 less 9? 18 less 10? 
 
 12. 18 less 9 ? 19 less 10 ? 
 
 13. 19 less 9? 20 less 10? 
 
 14. 20 less 9 are how many ? 30 less 9 ? &c., to 90. 
 
 15. 21 less 9 ? 31 less 9 ? &c., to 91. 
 
 16. 22 less 9 ? 32 less 9 ? &c., to 92. 
 
 17. 23 less 9 ? 33 less 9 ? &c., to 93. 
 
SUBTRACTION. 25 
 
 18. 24 less 9 ? 34 less 9 ? &c., to 94. 
 
 19. 25 less 9 ? 35 less 9 ? &c., to 95. 
 
 20. 26 less 9 ? 36 less 9 ? &c., to 96. 
 
 21. 27 less 9 ? 37 less 9 ? &c., to 97. 
 
 22. 28 less 9 ? 38 less 9 ? &c., to 98. 
 
 23. 29 less 9 ? 39 less 9 ? &c., to 99. 
 
 NOTE. The pupil should go through in the same manner with the 
 subtraction of 10. 
 
 24. Count backwards by 9's from 100 to 1. 
 
 25. Count backwards by 10's from 100 to 0. 
 
 26. One squadron of cavalry contains 90 mounted 
 men, and a second contains 110; how many more 
 mounted men does the second contain than the first ? 
 9 tens from 11 tens leaves how many? 10 tens from 
 15 tens leaves how many ? 
 
 27. In one of the butcheries in Paris, 9 hundred 
 sheep were slaughtered in one day, and in a second 12 
 hundred; how many more were slaughtered in the 
 second than in the first ? 
 
 28. 10 hundred or a thousand from 14 hundred 
 leaves how many ? 
 
 LESSON Y. 
 
 1. How much is 20 less 11 ? 
 
 PROCESS. 11 is 10 and I. 10 from 20 leaves 10, 
 and \from 10 leaves 9. 
 
 2. How much is 45 less 11 ? 67 less 11 ? 99 less 
 
 11 ? 88 less 11 ? 66 less 11 ? 
 
 3. How much is 25 less 12 ? 
 
 PROCESS. 12 is 10 and 2. 10 from 25 leaves 15, 
 \ind 2 from 15 leaves 13. 
 
 4. How much is 43 less 12 ? 54 less 12 ? 60 less 
 
 12 ? 72 less 12 ? 96 less 12 ? 84 less 12? . 
 
 5. 26 less 13? 52 less 13? 39 less 13? 65 less 
 13? 
 
26 SUBTRACTION. 
 
 6. 28 less 14 ? 42 less 14 ? 70 less 14 ? 
 
 7. 36 less 18 ? 54 less 18 ? 90 less 18 ? 
 
 8. Count backwards by ll's, beginning with 99. 
 
 9. Count backwards by 12's, beginning with 96. 
 
 10. How much is 30 less 20 ? 40 less 20 ? 50 less 
 30? 60 less 20? 70 less 50 ? 80 less 60 ? 90 less 
 30? 
 
 11. How much is 40 less 25 ? 
 
 PROCESS. 25 is 20 and 5. 40 less 20 is 20. 20 
 less 5 is 15. 
 
 12. How much is 30 less 22 ? 50 less 36 ? 60 less 
 37 ? 70 less 24 ? 80 less 48 ? 90 less 55 ? 
 
 13. How much is 55 less 20 ? 
 
 PROCESS. 50 less 20 is 30. Hence 55 less 20 is 
 35. 
 
 14. How much is 36 less 20 ? 48 less 20 ? 59 less 
 30 2 67 less 40 ? 77 less 50 ? 87 less 30 ? 
 
 15. How much is 33 less 23 ? 66 less 46 ? 56 less 
 36? 87 less 67? 97 less 57? 79 less 49? 88 less 
 28? 
 
 16. How much is 55 less 22 ? 
 
 PROCESS. 55 is 52 and 3. 52 less 22 is 30. 30 
 and 3 are 33. Or thus, 22 is 20 <md 2. 55 &* 20 
 i* 35. 35 less 2 w 33. 
 
 17. How much is 67 less 39 ? 78 less 23 ? 82 less 
 47 2 96 less 51 ? 47 less 28 ? 
 
 LESSON VI. 
 
 1. A boy had 25 marbles, and gave 5 of them to 
 one comrade, 6 to another, and 4 to another; how 
 many did he have left ? 
 
 2. A farmer raised 50 bushels of oats in one field, 
 20 in a second, %iid 30 bushels in a third, and sold 
 40 bushels ; how many bushels did he raise on the 
 three fields, and how many bushels had he after sell- 
 ing 40 ? 
 
SUBTRACTION. 27 
 
 3. Charles' father gave him 50 cents to make some 
 purchases. He bought an arithmetic for 15 cents, a 
 grammar for 10 cents, and a writing-book for 8 cents ; 
 how much of the money had he left? 
 
 4. A traveller had 100 miles to go. On the first 
 day he travelled 22 miles, on the second 31 miles, on 
 the third 17 guiles; how far did he go in the three 
 days ? How many miles had he yet to travel ? 
 
 5. In a conservatory there were 20 oranges on one 
 tree, 15 on another, 7 on a third, and 5 on a fourth. 
 Of these, 10 on the first tree were ripe, 7 on the 
 second, 2 on the third,, and 3 on the fourth ; how 
 many oranges were on all the trees, how many were 
 ripe, and how many unripe ? 
 
 6. 36 bees were sucking honey from the flowers of 
 a wild jessamine ; 15 of them flew to the blossoms of 
 a magnolia tree, 7 to the roses on a sweetbriar near 
 by, while 6, loaded with honey, flew back to the 
 hive; how many bees still lingered on the jessamine 
 vine? 
 
 7. A squadron of cavalry went into battle one hun- 
 dred strong, 20 of them were captured, 10 killed, and 
 22 wounded ; how many men were left in the squad- 
 ron ? 
 
 8. 43 and 8 and 5, less 11, are how many ? 
 
 9. Twenty-eight and four and nine, less seven, are 
 how many ? 
 
 10. Fifty-two and six and three and nine, less two 
 and three and five, are how many ? 
 
 11. Seventy-seven and eleven and three and four, 
 less seven, less eight, are how many ? 
 
 12. Ninety-eight and two and eight and three, 
 less twenty and twelve, are how many ? 
 
 13. Eighty-six less twenty, less seven, less nine, less 
 twenty-two, is how much ? 
 
 
28 . SUBTRACTION. 
 
 LESSON VII. 
 
 TABLES FOR PRACTICE IN ADDITION AND SUBTRACTION. 
 
 For the sake of brevity in these tables, we will use the sign of ad- 
 dition + , called plus, and the sign of subtraction , called minus. 
 Plus means more, minus means less. Thus, 7 + 6 is read 7 plus 
 6, which means 7 more 6, or 7 and 6. 7 6 is read 7 minus 6, 
 and means 7 less 6, or 6 taken from 7. 
 
 i.? 
 
 10 + 10 
 
 40 + 20 
 
 70 + 60 
 
 80 + 80 
 
 20 + 20 
 
 40 + 40 
 
 70 + 40 
 
 90 + 30 
 
 20 + 10 
 
 50 + 40 
 
 80 + 50 
 
 70 + 80 
 
 30 + 20 
 
 50 + 20 
 
 90 + 20 
 
 90 + 90 
 
 30 + 10 
 
 50 + 10 
 
 90 + 80 
 
 60 + 50 
 
 30 + 30 
 
 50 + 30 
 
 60 + 40 
 
 60 + 20 
 
 40 + 10 
 
 50 + 50 
 
 90 + 50 
 
 30 + 80 
 
 40 + 30 
 
 80 + 10 
 
 
 
 II.? 
 
 40 + 17 40 + 34 40 + 39 40 + 41 
 
 29 23 24 25 
 
 38 32 31 36 
 
 22 43 19 42 
 
 18 21 26 28 
 
 27 33 35 37 
 
 III.? 
 
 47 + 30 67 + 30 51 + 30 54 + 30 
 
 55 49 62 64 
 69 66 ' 44 58 
 59 57 52 63 
 
 48 61 68 53 
 
 56 46 65 45 
 
 IV.? 
 
 17 + 27 17 + 56 47 +19 49 + 28 
 29 29 55 66 
 
SUBTRACTION. 29 
 
 38 
 
 + 27 
 
 38 
 
 + 56 
 
 69 
 
 + 19 
 
 57 
 
 + 28 
 
 22 
 
 
 22 
 
 
 59 
 
 
 61 
 
 
 18 
 
 
 19 
 
 
 48 
 
 
 46 
 
 
 27 
 
 
 32 
 
 
 56 
 
 
 51 
 
 
 
 
 
 
 V.? 
 
 
 
 
 50 
 
 - 30 
 
 100 
 
 - 60 
 
 95 
 
 - 15 
 
 83 
 
 - 23 
 
 40 
 
 - 10 
 
 100 
 
 - 50 
 
 
 25 
 
 
 13 
 
 70 
 
 - 50 
 
 110 
 
 - 10 
 
 
 55 
 
 
 53 
 
 80 
 
 - 40 
 
 90 
 
 - 30 
 
 
 75 
 
 
 73 
 
 70 
 
 - 30 
 
 80 
 
 - 70 
 
 
 65 
 
 
 63 
 
 90 
 
 - 20 
 
 60 
 
 - 20 
 
 
 45 
 
 
 43 
 
 
 
 
 
 VI.? 
 
 
 
 
 58 
 
 - 30 
 
 82 
 
 - 40 
 
 100 
 
 - 58 
 
 90 
 
 - 83 
 
 67 
 
 
 77 
 
 
 
 47 
 
 
 79 
 
 42 
 
 
 45 
 
 
 
 76 
 
 
 64 
 
 75 
 
 
 66 
 
 
 
 67 
 
 
 61 
 
 84 
 
 
 94 
 
 
 
 38 
 
 
 42 
 
 97 
 
 
 58 
 
 
 
 24 
 
 
 34 
 
 YIL? 
 
 93 
 
 - 58 
 
 93 
 
 - 85 
 
 36 
 
 - 16 
 
 89 
 
 - 24 
 
 
 47 
 
 
 73 
 
 73 
 
 
 74 
 
 
 
 76 
 
 
 69 
 
 85 
 
 
 47 
 
 
 
 67 
 
 
 56 
 
 97 
 
 
 38 
 
 
 
 38 
 
 
 45 
 
 58 
 
 
 56 
 
 
 
 24 
 
 
 32 
 
 64 
 
 
 95 
 
 
 8. 44 less 27 Is how much ? 47 less 27 ? 44 less 
 30? 
 
 9. If we add 3 to the number from which we are 
 to subtract another, what change does it make in the 
 remainder ? 
 
 10. If we add it to the number which we subtract 
 from another number, what change do we make in 
 the remainder? 
 
30 
 
 MULTIPLICATION. 
 
 11. 24 16 ? 24 23 ? 31 16 ? 31 - - 23 ? 
 
 12. If we add seven to the number subtracted 
 and also to the number from which it is taken, is 
 there any change in the remainder ? 
 
 13. 5 - 3 is how much? 8-5? 12-9? 
 
 14. 40 20 is how much ? 41 21 ? 49 29 ? 
 
 15. One farmer had 32 sheep, and another 28 
 sheep ; each lost 10 sheep ; how many more sheep 
 did the first have than the second before the loss ? 
 After the loss ? 
 
 CHAPTEK III. 
 
 t 
 
 MULTIPLICATION. 
 LESSON I. 
 
 1. Five boys gave each a dime to a blind man ; 
 how many dimes did they all give ? 1 and 1, and 1 
 and 1, and 1, are how many? 5 times 1 are how 
 many ? one times 5 are how many ones ? 
 
 2. Six boys are flying kites, and each boy has one 
 kite ; how many kites have they all ? 1 and 1, and 
 1 and 1, and 1 and 1, are how many? 6 times 1 ? 
 Once 6 ? 
 
 3. Seven girls are coming from the garden, each 
 with a bunch of roses ; how many bunches of roses 
 have they all together ? 1 and 1, and 1 and 1, and 1 
 and 1, and 1, are how many ? 7 times 1 ? One time 
 7 is how many ones ? 
 
 4. Eight boys are going out to gather black- 
 berries, and each boy has one basket ; how many 
 baskets have they all \ 8 times 1 are how many ? 
 
MULTIPLICATION. 3 1 
 
 5. 1 time 1 ? 
 
 6. 2 times 1 ? 1 time 2 ? 
 
 7. 3 times 1 ? 1 time 3 ? 
 
 8. 4 times 1 ? 1 time 4 ? 
 
 9. 5 times 1 ? 1 time 5 ? 
 
 10. 6 times 1.? 1 time 6? 
 
 11. 7 times 1? 1 time 7? 
 
 12. 8 times 1? 1 time 8? 
 
 13. 9 times 1 ? 1 time 9 ? 
 
 14. 10 times 1 ? 1 time 10 ? 
 
 15. 27 times 1 are how many? 36 times 1 ? 48 
 times 1 ? 87 times 1 ? 108 times 1 ? 
 
 16. Twice 1 ten is how many tens ? 3 times 1 ten ? 
 4 times 1 ten ? 5 times 1 ten ? 
 
 17. Twice 1 hundred is how much ? 6 times 1 
 hundred ? 8 times 1 hundred ? 9 times 1 hundred ? 
 
 18. Two row boats have three oars each ? how 
 many oars have both boats? 3 and 3 are how 
 many ? Twice 3 ? 
 
 19. Hayes saw two Esquimaux sledges drawn by 8 
 dogs each ; how many do^s drew both sledges ? 8 
 and 8 are how many ? Twice 8 ? 
 
 20. There are two orange trees in a gentleman's 
 garden, each of which has 20 oranges on it ; how 
 many oranges on both trees ? 20 and 20 are how 
 many ? Twice 20 ? 
 
 21. A man bought two cloaks, and paid 9 dollars 
 apiece for them ; how much did he pay for both ? 
 
 22. How much is twice 1 ? 
 
 twice 2 ? 
 
 twice 3 ? 
 
 twice 4 ? 
 
 twice 5 ? 
 
 twice 6 ? 
 
 twice 7 ? 
 
 twice 8 ? 
 
 twice 9? 
 
o* MULTIPLICATION. 
 
 23. How much is twice 10? 
 
 twice 11 ? 
 
 PROCESS. 11 is 10 and 1. Twice 11 is twice 10 
 and twice 1, or 20 and 2 (i. e.) 22. 
 
 24. How much is twice 12? Twice 16? Twice 
 17 ? Twice 18 ? Twice 19 ? 
 
 25. How much is twice 2 tens or 20 ? Twice 
 30? Twice 40? Twice 50? Twice 60? Twice 
 70 ? Twice 80 ? Twice 90 ? 
 
 26 How much is twice 1 hundred? Twice 2 
 hundred ? Twice 3 hundred ? Twice 4 hun- 
 dred ? Twice 5 hundred ? Twice 8 hundred ? Twice 
 9 hundred? 
 
 The sign of multiplication is x , which we will read times. Thus, 
 2 x 12 or 3 x 6 means 2 times 12 and 3 tunes 6. The result ob- 
 tained by the multiplication of numbers is called their product. 
 
 LESSON II. 
 
 1. "What do 3 oranges at 2 cents apiece cost ? 2 
 + 2 + 2? 
 
 2. What do 3 oranges at 3 cents apiece cost ? 3 
 + 3 + 3 ? 
 
 3. Each wagon has four wheels ; how many wheels 
 have three wagons ? How many wheels have four 
 wagons ? How many have five wagons ? 4 + 4 + 
 4j 4 + 4 + 4 + 43 4. + 4 + 4+4 + 4J 
 
 4. There are three rows of trees, arid ten trees in 
 each row; how many trees? How many trees in 4 
 rows of ten trees each? In 5 rows of ten trees 
 each? 
 
 5. 3 times 10 are how many? 3 times 10 and 
 once 10 ? 4 times 10 and once 10 ? 
 
 6. 3 times 1 ? 4 times 1 ? 5 times 1 ? 
 
 7. 3 times 2 ? 4 times 2 ? 5 times 2 ? 
 
 8. 3 times 4 ? 4 times 3 ? 5 times 3 ? 
 
 9. 3 times 5 ? 4 times 4 ? 5 times 4 ? 
 
MULTIPLICATION. 33 
 
 10. 3 times 6 ? 4 times 5 ? 5 times 5 ? 
 
 11. 3 times 7 ? 4 times 6 ? 5 times 6 1 
 
 12. 3 times 8 ? 4 times 7 ? 5 times 7 ? 
 
 13. 3 times 9 ? 4 times 8 ? 5 times 8 ? 
 14 3 times 10 ? 4 times 9 ? 5 times 9 ? 
 
 15. 3 times 2 are how many times 3 ? 4 times 2 
 are how many times 4 ? 
 
 16. 5 times 2 are how many times 5 ? 3 times 4 
 are how many times 3 ? 
 
 17. 3 times 5 are how many times 3 ? 4 times 5 
 are how many times 4 ? 
 
 18. How much is 3 times 11 ? 
 
 PROCESS. 11 is 10 and 1. 3 times 11 is 3 foVi&j 
 10 mZ 3 times 1 ; i. e., 30 <m<# 3, or 33. 
 
 19. 3 times 12? 3 times 13? 3 times 14? 3 
 times 15 ? 3 times 16 ? 3 times 17 ? 3 times 18 ? 3 
 times 19 ? 
 
 20. How much is 4 times 11 ? 4 times 12 ? 4 
 times 13 ? 4 times 14 ? 4 times 15 ? 4 times 16 ? 4 
 times 17 ? 4 times 18 ? 4 times 19 ? 
 
 21. How much is 5 times 11? 5 times 12? 5 
 times 13 ? 5 times 14 ? 5 times 15 ? 5 times 16 ? 5 
 times 17? 5 times 18 ? 5 times 19 ? 
 
 22. How much is four times 2 tens ? 4 times 3 
 tens ? 4 tens ? 8 tens ? 9 tens ? 
 
 23. How much is 3 times 2 tens ? 3 times 3 tens ? 
 3 times 4 tens ? 3 times 5 tens ? 3 times 6 tens ? 
 8 tens? 
 
 24. How much is 5 times 2 tens? 3 tens? 4 
 'tens ? 5 tens ? 6 tens ? 7 tens ? 8 tens ? 
 
 25. How much is 4 times 2 hundred ? 3 hundred? 
 6 hundred ? 5 times 3 hundred ? 5 hundred ? 7 
 hundred ? 3 times 2 hundred ? 9 hundred ? 4 hun- 
 dred? 
 
34: 
 
 MULTIPLICATION. 
 
 LESSON III. 
 
 1. I planted 6 rows of trees with 6 trees in each 
 row ; how many trees did I plant ? 5 times 6 and 
 once 6 are how many ? 7 rows of trees, 6 trees in a 
 row, are how many trees ? 8 rows of trees, 6 trees in 
 a row, are how many trees ? 6x6 and once 6 ? 7 
 
 x 6 and once 6 ? 
 
 2. What do 6 pounds of sugar cost at 7 cents a 
 pound ? at 8 cents ? at 10 cents ? 
 
 3. There are 7 days in one week ; how many days 
 in 7 weeks ? in 8 weeks ? 7 and 7 and 7 and 7 and 
 7 and 7 and 7 are how many? 7 times 7 and once 7 
 are how many ? 
 
 4. If flour is 8 dollars a barrel, how much will 5 
 barrels cost ? 8 barrels ? 9 barrels ? 10 barrels ? 
 
 '5. 6 times 2? 7 times 2? 8 times 2? 
 
 6. 6 times 3? 7 times 3? 8 times 3? 
 
 7. 6 times 4 ? 7 times 
 
 8. 6 times 5 ? 7 times 
 
 9. 6 times 6 ? 7 times 
 6 times 7 ? 7 times 
 6 times 8 ? 7 times 
 6 times 9 ? 7 times 
 
 5? 
 6? 
 7? 
 8? 
 9 ? 
 
 many 
 
 4? 
 
 5? 
 
 6? 
 
 7? 
 
 8? 
 
 9? 
 
 8 times 10 ? 
 times 6 ? 7 times 
 
 8 times 
 8 times 
 8 times 
 8 times 
 8 times 
 8 times 
 
 10. 
 11. 
 12. 
 
 13. 6 times 10 ? 7 times 10 
 
 14. 6 times 5 are how 
 are how many times 7 ? 
 times 8 ? 
 
 15. 6 times 7 are how many times 6 ? 8 times 7 
 are how many times 8 ? 6 times 8 are how many 
 times 6 ? 
 
 6 times 11? 
 19? 
 
 7 times 11? 
 19? 
 
 } times 11? 
 19? 
 
 16. How much is 
 15? 16? 17? 18? 
 
 17. How much is 
 15? 16? 17? 18? 
 
 18. How much is 
 15? 16? 17? 18? 
 
 8 times 5 are how many 
 
 12 ? 13 ? 14 ? 
 12? 13? 14? 
 12? 13? 14? 
 
 
MULTIPLICATION. 
 
 35 
 
 19. How much is 6 times 2 tens ? 3 tens ? 5 tens ? 
 
 7 tens ? 8 tens ? 9 tens ? 
 
 20. .How much is 7 times 2 tens ? 3 tens ? 4 tens ? 
 
 6 tens ? 8 tens ? 9 tens ? 
 
 21. How much is 8 times 3 tens ? 5 tens ? 7 tens ? 
 
 8 tens? 9 tens? 11 tens? 
 
 22. How much is 6 times 2 hundred ? 5 hundred ? 
 
 8 hundred ? 
 
 23. How much is 7 times 3 hundred ? 4 hundred? 
 
 9 hundred? 
 
 24. How much is 8 times 2 hundred ? 6 hundred ? 
 
 7 hundred ? 
 
 LESSON IV. 
 
 1. An elephant has two tusks ; how many tusks 
 have 9 elephants? How many tusks have 10 ele- 
 phants ? 
 
 2. In a certain village there are 9 cottages in a 
 row, and each cottage has four windows in front ; how 
 many front windows have all the cottages ? 
 
 3. Ten laborers received each 5 dollars for a week's 
 labor ; how much did they all receive together ? 
 
 4. At 8 dollars a barrel, how much will 9 barrels 
 of flour cost ? Ten barrels ? 
 
 2 ? 10 times 
 10 times 
 10 times 
 10 times 
 10 times 
 10 times 
 10 times 
 10 times 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 
 13. 9 times 10 ? 10 times 10 ? 
 
 14. 9 times 8 are how many times 9 ? 10 times 8 
 are how many times 10 ? 
 
 15. 9 times 10 are how many times 9 ? 10 times 2 
 are how many times 10 ? 
 
 9 times 
 9 times 
 9 times 
 9 times 
 9 times 
 9 times 
 9 times 
 9 times 
 
 3? 
 
 4? 
 5? 
 6? 
 7? 
 8? 
 9? 
 
 2? 
 3? 
 
 4? 
 5? 
 6? 
 7? 
 8? 
 9? 
 
36 aiTJLTIPLICATIOIT. 
 
 16. 9 times 2 tens are how many ? 3 tens ? 5 
 tens i 7 tens ? 8 tens ? 10 tens ? 
 
 17. 10 times 3 tens ? 4 tens ? 5 tens ? 6 tens ? 8 
 tens ? 9 tens ? 
 
 18. How much is 9 times 4 hundred ? 7 hundred ? 
 
 8 hundred ? 
 
 19. How much is 10 times 3 hundred ? 6 hundred ? 
 
 9 hundred ? 
 
 20. How much is 9 times 11? 12? 13? 14? 
 15? 16? 17? 18? 19? 
 
 21. How much is 10 times 11? 12? 13? 14? 
 15? 16? 17? 18? 19? 
 
 22. How much is 11 times 11 ? 
 
 PROCESS. 10 times 11 and once 11 gives 121. 
 
 23. How much is 12 times 11 ? 12 times 12 ? 
 
 24. What will 11 dozen eggs cost, at 10 cents a 
 dozen ? At 11 cents ? At 12" cents ? 
 
 DEFINITION. The number by which we multiply is called the mul- 
 tiplier, and the number multiplied is called the multiplicand. The 
 two are called factors. 
 
 LESSON V. 
 
 1. How much is 5 times 26 ? 
 
 PROCESS. 5 x 26 is 5 x 2 tens, + 5x6, which 
 is 10 tens, or 100, + 30. 
 ANS. 130. 
 
 TABLES FOR PRACTICE. 
 
 IJ 
 
 2 x 40 
 
 5 x 90 
 
 2 x 13 
 
 3 x 13 
 
 9 x 30 
 
 .6 x 80 
 
 2 x 23 
 
 3 x 21 
 
 4 x 70 
 
 3 x 50 
 
 2 x 14 
 
 3 x 25 
 
 9 x 60 
 
 4 x 60 
 
 2 x 42 
 
 3 x 31 
 
 8 x 80 
 
 7 x 30 
 
 2 x 35 
 
 3 x 28 
 
 7 x 50 
 
 2 x 90 
 
 2 x 49 
 
 3 x 32 
 

 ^ MULTIPLICATION. 37 
 
 II.? 
 
 4 x 13 5 x 35 6 x 45 8 x 58 10 x 22 
 
 6 x 14 6 x 58 6 x 36 8 x 47 10 x 34 
 
 4 x 16 7 x 26 6 x 83 8 x 76 10 x 45 
 
 5 x 22 4 x 23 7 x 42 8 x 67 10 x 56 
 
 6 x 23 7 x 5*4 7 x 34 9 x 45 10 x 65 
 4 x 42 5 x 62 7 x 53 9 x 36 10 x 72 
 4 x 35 6 x 56 7 x 82^ 9 x 64 10 x 91 
 
 4. A man sold 57 barrels of apples at 5 dollars a 
 barrel ; how much did he get for all ? 
 
 5. If he had sold them at 6 dollars a barrel, how 
 much would he have received? at 7 dollars, how 
 much ? 
 
 6. What distance does a man travel in 9 days who 
 goes 42 miles each day? what distance in ten days? 
 
 7. Out of a flock of sheep, 34 ewes had two lambs 
 each ; how many lambs did they all have ? 
 
 8. In an orchard there are 8 rows of apple trees, 
 and each row contains 22 trees ; how many apple trees, 
 in all ? 
 
 LESSON VI. 
 
 1. How much is 2x2x2? 2x2x3? 2x 
 2x4? 2x2x5? 2x2x6? 2x3x4? 
 2x5x6? 
 
 2. What are these products called ? 
 ANS. Continued products. 
 
 3. What is the continued product of 3, 4, and 5 ? 
 5, 4, and 3 ? 3, 5, and 6 ? 5, 6, and 3 ? 
 
 4. In a continued product, can we begin the multi- 
 plication with any two of the numbers, without chang- 
 ing the product ? 
 
 5. What single number multiplied by 5 will give 
 the same product as 2 x 3 multiplied by 5 ? as 2 x 2 
 
 x5? as 2x4x5? as 3x4x5? as 2x2 
 x 2 x 5? 
 
38 MULTIPLICATION. 
 
 6. How much is 4 x5x6? 4x6x7? 3x6 
 x7? 2x2x2x3x5? 2x2x2x4x4? 
 
 7. The square of any number is what we obtain by 
 multiplying that number by itself. The square of 1 
 is 1 x 1 ; of 2, is 2 x 2, &c. 
 
 8. What is the square ofl? 21 3? 4? 5? 6? 
 7? 8? 9? 10? 11? 12? 
 
 9. A man cuts two oranges into two parts each, and 
 then cuts each of the parts into two parts ; into how 
 many parts were both oranges cut ? 
 
 10. A miller wished to divide 3 bushels of meal 
 among some beggars. He divided each bushel in 
 four heaps, and then each heap in two parts. In. 
 how many parts were the three bushels divided ? 
 
 11. If I cut a melon in 6 parts, and cut each of 
 these parts in 5 pieces, and each of these pieces in 4 
 parts ; in how many parts will the melon be divided ? 
 
 12. A man had 10 orchards ; each orchard con- 
 tained 10 rows of trees, and each row of trees con- 
 tained 10 trees ; how many trees in all ? 
 
 LESSON VII. 
 
 MISCELLANEOUS. 
 
 1. I subtracted 15 from a certain number, and ob- 
 tained 5 for a remainder ; what is that number ? 
 
 2. 22 is 13 less than what number ? 40 is 15 less 
 than what number ? 
 
 3. What number is composed of the two parts 34 
 and 26? 
 
 4. A farmer sold 5 barrels of flour at 5 dollars a 
 barrel, and 10 bushels of wheat at 2 dollars a bushel ; 
 how much did he receive for the flour and wheat ? 
 
 5. The difference between 46 and a greater number 
 is 15 ; what is that number? 
 
 6. John has 6 marbles, and William has 3 times 
 as many, less 10 ; what number has William ? 
 
MULTIPLICATION. 39 
 
 7. A farmer sold 3 lambs at 2 dollars each ; 5 pigs 
 at 2 dollars each ; 8 bushels of wheat at 3 dollars 
 a bushel, and 5 barrels of corn at 8 dollars a bar- 
 rel ; what sum of money did he receive for all ? 
 
 8. July, August, and October have each 31 days ; 
 September, March, and June have each 30 days ; how 
 many more 5ays in the first three than the last three ? 
 
 PROCESS. In the first three, there are 3 x 31, or 3 
 x 30 -f- 3 x 1 ; and in the last three there are 3 x 
 30 ; hence, there are 3 x 1, or 3 more days in the first 
 three than in the last. 
 
 9. How much greater is 3 x 32 than 3 x 30 ? 
 ANS. 3 x 2. 
 
 10. How much less is 5 x 10 than 5 x 15? 
 ANS. 5 x 5. 
 
 11. How much less is 5 x 35 than 5 x 40 ? 
 ANS. 5 x 5. 
 
 12. How much greater is 9 x 10 than 9x3? 
 Than 9x4? Than 9x5? 
 
 13. How much more is 5 x 50 than 5 x 40? 
 Than 5 x 41 ? Than 5 x 45 ? Than 5 x 49 ? 
 
 14. How much more is 8 x 8 than 7x7? 
 
 15. If a man buys 5 barrels of flour at dollars 
 a barrel, and sells them at 7 dollars, how much will 
 he gain ? How much if he sells at 8 dollars a barrel ? 
 9 dollars a barrel ? 10 dollars a barrel ? 
 
 16. Frank has 30 chinquepins. William has 3 
 times as many, less 40, and Charles has twice as many, 
 less 30 ; how many have they all together ? 
 
 17. Multiply the sum of 8 and 9 by 9 ? Find the 
 sum of the two products 8 x 9and9 x 9? 
 
 18. Two travellers set out from the same point at 
 the same time, and travel in opposite directions. One 
 travels 5 miles an hour, and the other 4 miles an 
 hour ; how far are they apart at the end of 5 hours ? 
 At the end of 6 hours'? 7 hours ? 8 hours ? 
 
 19. 5x5 + 5x4: is how much ? 
 
4:0 DIVISION. 
 
 20. Two travellers set out from the same point at 
 the same time, and travel in the same direction. One 
 travels 6 miles an hour, and the other four miles an 
 hour; how far will the slowest one be behind the 
 other in 5 hours ? 6 hours ? 9 hours ? 
 
 21. 5x6, less 5 x 4, is how much? 
 
 22. Two travellers set out at the same moment, one 
 from Richmond and one from Washington, and travel 
 toward each other. One travels 4 miles an hour, 
 and the other 5 miles an hour. Putting the distance 
 between Richmond and "Washington at " 100 miles ; 
 how far will the travellers be apart 3 hours' after 
 setting out ? 4 hours ? 5 hours ? 
 
 23. 100 less 3x4 + 3x5 is how much ? 
 
 24. Ten times eight, plus eighteen, less twenty, less 
 six, are how many ? 
 
 25. Five times twenty, plus fifty, plus eight, are 
 how many ? 
 
 26. Six times twenty, less five times ten, are how 
 many? 
 
 27. If a man earns 60 cents a day, and pays out 20 
 cents a day, how much will he have made at the end 
 of 6 days'? 
 
 CHAPTER IY. 
 
 DIVISION. 
 LESSON I. 
 
 We have seen how to separate large numbers 
 into different sets of smaller ones. Thus, 5 is sepa- 
 rated into 2 + 3, 2 +2 + 1, 2 + 1 + 1 + 1, 1 + 
 1 + 1 + 1 + 1. 5 is said to contain these smaller 
 numbers. A number can be separated into smaller 
 
DIVISION. 41 
 
 numbers, all of which are equal to each other. We 
 say that the first number contains the other a certain 
 number of times. 
 
 1. Into what equal numbers may every number be 
 seperated ? 
 
 ANS. Into ones. 
 
 2. How many ones in 6 ? In 7 ? 8 ? 10 ? 105 ? 
 
 3. 1 is contained in 6 how many times ? In 7 ? 
 8? 9? 110? 
 
 4. Distribute six cannon balls into heaps of two 
 balls each ; how many heaps will you have ? 
 
 5. Distribute them into heaps of three balls each ; 
 how many heaps ? 
 
 6. 2 is contained in 6 how many times ? 
 
 7. 3 is contained in 6 how many times ? 2 -f 2 -f 
 2 are how many? 
 
 8. How many times can you take 2 from 6 and 2 
 from the remainder until nothing remains ? 3 + 3 is 
 how much ? 
 
 To divide one number by another is to find how 
 often the latter is contained in the former. 
 
 9. How do we divide 12 by 2 ? 
 
 ANS. We distribute 12 into 2's, and then count 
 the number of 2's. 
 
 10. How is this done ? 
 
 ANS. Either, 1", By adding 2'* thus : 2 + 2 -f 2 
 -f2 + 2 + 2, until we get 12 ; or, 2^ ? by subtracting 
 'Zfrom 12 and 2 from the remainder, and so forth, 
 until nothing remains or, 3 n *, we may remember, 
 from multiplication, how many times 2 make 12, or 
 what number taken 2 times 'makes 12. 
 
 11. Which of these is the shortest process? 
 
 The number which is divided by the other is 
 called the dividend. The number by which we divide 
 is called the divisor. The result obtained is called 
 the quotient. 
 
 12. How many i>air of spurs will make eight spurs ? 
 
42 DIVISION. 
 
 13. How many pair of spectacles require ten glasses ? 
 
 14. An ivory hunter gets two tusks from each ele- 
 phant which he kills ; how many elephants must he 
 kill to get 12 tusks ? 
 
 15. If Frank studies two hours *a day, in how many 
 days will he study 14 hours ? 
 
 16. How many 2's in 4? 6? 8? 10? 12? 
 14? 16? 18? 20? 
 
 17. In what number is 2 contained exactly 2 times ? 
 3 times ? 4 times ? 5 times ? 6 times ? 7 times ? 8 
 
 'times ? 9 times ? 10 times ? 
 
 18. How much is 2 divided by 2 ? 12 divided by 2 ? 
 
 4 divided by 2 ? 14 divided by 2 ? 
 
 6 divided by 2 ? 16 divided by 2 ? 
 
 8 divided by 2 ? 18 divided by 2 ? 
 10 divided by 2 ? 20 divided by 2 ? 
 
 The numbers which contain 2 an exact number of 
 times we say are divisible by 2. 
 
 19. How many pair of spurs will five spurs make ? 
 
 20. If we endeavor to distribute 5 into 2's what do 
 we find ? 7 into 2's ? 9 into 2's ? 
 
 21. "What is the 1 which is left in each case called ? 
 ANS. The remainder. 
 
 22. How much is 3 divided by 2 ? 
 ANS. 1 and 1 remainder. 
 
 23. How much is 5 divided by 2 ? 13 divided by 2 ? 
 
 7 divided by 2 ? 15 divided by 2 ? 
 
 9 divided by 2 ? 17 divided by 2 ? 
 11 divided by 2 ? 19 divided by 2 ? 
 
 24. "What are the numbers called which are divisible 
 by 2? 
 
 ANS. Even numbers. 
 
 25. What are the numbers called which are not 
 divisible by 2 ? 
 
 ANS. Odd numbers. 
 
 26. What remainder results always when we divide 
 the odd numbers by two ? 
 
DIVISION. 43 
 
 27. Give the even numbers from 2 to 100. 
 
 28. Give the odd numbers from 1 to 99. 
 
 29. How much is 4 tens or 40 divided by 2 ? 
 ANS. 2 tens or 20. 
 
 30. 4 hundred divided by 2 ? 
 
 31. 6 tens or 60 divided by 2 ? 
 
 32. 6 hundred divided by 2 ? 
 
 33. 8 tens or 80 divided by two? 8 hundred 
 divided by 2 ? 
 
 34. 10 tens or 100 divided by 2 ? 10 hundred 
 divided by 2 ? 
 
 35. 12 tens or 120 divided by 2 ? 14 tens divided 
 by 2 ? 16 tens divided by 2 ? 18 tens divided by 2 ? 
 
 36. How much is 30 divided by 2 ? 
 PROCESS. 30 is 20 -f 10. 20 divided ly 2 is 10, 
 
 and 10 divided l)y 2 is 5. 
 ANS. 10 + 5 or 15. 
 
 37. How much is 50 divided by 2 ? 70 divided by 
 2? 90 divided by 2? 
 
 38. How much is 21 divided by 2 ? 
 
 PROCESS. 21 is 20 + 1. Hence, 21 divided ~by 2 
 t's 10 and 1 remainder. 
 
 39. How much is 23 divided by 2 ? 24 ? 25 ? 36 ? 
 45? 
 
 40. If we multiply 5 by 2, and divide the product 
 by 2, what will be the quotient ? 
 
 41. If we divide 8 by 2, and multiply the quotient 
 by 2, what will be the product ? 
 
 42. If we divide any number by 2, and multiply 
 the quotient by 2, what will be the result ? 
 
 LESION II. 
 
 *1. If one flour barrel holds 3 bushels, how many 
 barrels will hold 9 Bushels ? 12 bushels ? 15 
 bushels ? 30 bushels ? 
 
 2. Into how many ranks of four men eaeh can you 
 
44 
 
 DIVISION. 
 
 divide a squad of 8 soldiers ? a squad of 16 soldiers ? 
 a company of 30 soldiers ? of 40 soldiers ? 
 
 3. Into how many heaps of five balls each can you 
 
 15? 18? 21? 
 
 distribute 25 cannon balls ? 30 cannon balls ? " 45 
 cannon balls ? 50 cannon balls ? 
 
 4. How many 3's in 6 ? 9 ? 12 ? 
 24? 27? 30? 
 
 many 4's in 8 ? 12? 16? 20? 24? 
 
 36 ? 40 ? 
 
 many 5's in 10? 15? 25? 20? 30? 
 
 40?' 50? 
 
 5. 
 28? 
 
 6. 
 35? 
 
 7. In 
 Twice ? 
 times ? 
 
 8. In 
 Once ? 
 
 5 times? 
 
 How 
 
 32? 
 How 
 45? 
 
 what number is 3 contained exactly once? 
 6 times? 4 times ? 7 times ? 5 times ? 9 
 8 times ? 10 times ? 3 times ? 
 what number is 4 contained exactly twice ? 
 4 times ? 8 times ? 10 times ? 6 times ? 
 7 times ? 3 times ? 9 times ? 
 
 9. In what number is 5 contained exactly 4 times? 
 Twice ? 6 times ? 5 times ? 3 times ? 7 times ? 
 9 times ? 8 times ? 10 times ? 
 
 3 divided by 3 ? 
 
 6 divided by 3 ? 
 
 9 divided by 3 ? 
 12 divided by 3 ? 
 15 divided by 3 ? 
 18 divided by 3 ? 
 21 divided by 3? 
 24 divided by 3 ? 
 27 divided by 3 ? 
 30 divided by 3 ? 
 
 11. How much is 3 x 
 many times ? 3 in 36 ? 
 
 12. How much is 4 x 
 many times ? 4 in 48 ? 
 
 13. How much is 5 x 
 many times ? 5 in 60 ? 
 
 10. How much is 
 4 divided by 4 ? 
 8 divided by 4 ? 
 
 12 divided by 4 ? 
 
 16 divided by 4? 
 
 20 divided by 4 ? 
 
 24 divided by 4 ? 
 
 28 divided by 4? 
 
 32 divided by 4 ? 
 
 36 divided by 4? 
 
 40 divided by 4 ? 
 11? 3 x 
 
 5 divided by 5 ? 
 10 divided by 5 ? 
 15 divided by 5? 
 20 divided by 5 ? 
 25 divided by 5 ? 
 30 divided by 5? 
 35 divided by 5 ? 
 40 divided by 5 ? 
 45 divided by 5 ! 
 50 divided by 5 ? 
 12? 3 in 33 how 
 
 11? m 4 x 12? 4 in 44 how 
 11 ? 5x12? 5 in 55 how 
 
DIVISION. 45 
 
 14. 3 is contained in 4 how many times, and what 
 remainder ? in 5 how many times ? 
 
 15. How many three bushel bags can you fill with 
 7 bushels of wheat, and what will remain? How 
 many with 8- bushels? 
 
 16. What are the remainders of numbers divided by 
 3? 
 
 17. Give the numbers from 1 to 100 which are 
 divisible by 3. 
 
 18. 4 in 5 how many times ? 4 in 6? 4 in 7? 4 
 in 9 ? 4 in 10 ? 4 in 11 ? 
 
 What are the remainder of numbers divided by 4 ? 
 
 19. Give the numbers from 1 to 100 which are 
 divisible by 4. 
 
 20. 5 in 6 how many times ? 5 in 7 ? 5 in 8 ? 5 
 in 9? 5 in 11? 5 in 12? 5 in 13? 5 in 14? 
 
 What are the remainders of numbers divided by 5 ? 
 
 21. Give the numbers from 1 to 100 which are 
 divisible by 5. 
 
 22. 3 in 3 tens how many times ? in 6 tens ? in 9 
 tens? in 12 tens? &c. 
 
 23. 3 in 3 hundred how many times ? in 6 hun- 
 dred? in 9 hundred? in 12 hundred ? 
 
 24. 4 in 4 tens? in 8 tens? in 16 tens? in 24 
 tens? 
 
 25. 4 in 4 hundred? in 8 hundred? in 16 hun- 
 dred? 
 
 26. 5 in 5 tens? 10 tens? 15 tens? in 5 hun- 
 dred? 15 hundred? 
 
 27. Multiply 9 by 3, 4 and 5, and divide the pro- 
 ducts by 3, 4 and 5 respectively. 
 
 28. Divide 60 by 3, 4 and 5, and multiply the quo- 
 tients bv 3, 4 and 5 respectively. 
 
 29. A number multiplied by 3 and the product 
 divided by 3 gives what result? 
 
 
DIVISION. 
 
 LESSON III. 
 
 1. If it requires 6 horses to draw each cannon on 
 the field of battle, how many cannon will require 12 
 horses? 24 horses? 42 horses? 36 ' horses ? 60 
 horses ? 
 
 2. How many weeks do 14 days make ? 28 days ? 
 21 days? 35* days? 49 days? 56 days? 70 
 days? 
 
 3. If 8 leaves only are left to grow and ripen on 
 each tobacco plant, how many plants will yield 
 16 leaves? 32 leaves? 48 leaves? 40 leaves? 80 
 leaves ? 
 
 4. How 
 42? 54? 
 
 5. 
 42? 
 
 many 6's in 12 ? 24 ? 18 ? 36 ? 48 ? 
 60? 30? 
 
 many 7's in 14? 28? 21? 35? 49? 
 63 ? 70 ? 
 
 many 8's in 16 ? 24 ? 40 ? 32 ? 48 ? 
 72 ? 80 ? 
 
 How 
 56? 
 
 6. How 
 64? 56? 
 
 7. In what number is 6 contained exactly once ? 
 times? Twice? 5 times? 7 times ? 8 times? 
 times? 10 times? 
 
 8. In what number is 7 contained exactly once? 
 times I 6 times? 3 times? 5 times? 7 times? 
 times ? 10 times ? 8 times ? 
 
 9. In what number is 8 contained exactly once? 
 times ? 3 times ? 4 times ? 7 times ? 9 times ? 
 
 10 times ? 
 10. How much is 
 
 7 divided by 7 ? 
 14 divided by 7? 
 21 divided by 7? 
 28 divided by 7? 
 35 divided by 7? 
 42 divided by 7 ? 
 49 divided by 7? 
 56 divided by 7? 
 
 times ? 8 times ? 
 
 6 divided by 6? 
 24 divided by 6 ? 
 36 divided by 6 ? 
 12 divided by 6? 
 18 divided by 6? 
 48 divided by 6? 
 42 divided by 6? 
 54 divided by 6 ? 
 
 3 
 
 9 
 
 4 
 9 
 
 5 
 6 
 
 8 divided by 8 ? 
 24 divided by 8 ? 
 40 divided by 8 ? 
 32 divided by 8? 
 16 divided by 8? 
 48 divided by 8? 
 56 divided by 8 ? 
 72 divided by 8 ? 
 
DIVISION. 47 
 
 60 divided by 6 ? 63 divided by 7 ? 64 divided by 8 ? 
 30 divided by 6 ? 70 divided by 7 ? 80 divided by 8 ? 
 
 11. How much is 6 x 11 ? 6 x 12 ? How much 
 is 66 divided by 6? How much is 72 divided by 6 ? 
 
 12. How much is 11 x 7 ? 12 x 7 ? How much 
 is 77 divided by 7 ? 84 divided by 7? 
 
 13. How much is 11 x 8? 12 x 81 How much 
 is 88 divided by 8? 96 divided by 8 ? 
 
 14. 6 in 7 how many times, and what remainder? 
 In 8? In 9 ? In 10 ? In 11 ? In 13 ? In 14 ? In 
 15? 16? 17? 
 
 What are the remainders of numbers divided by 6 ? 
 
 15. 7 in 15 how many times, and what remainder ? 
 In 16? 17? 18? 19? 20? 22? 23? 24? 25? 
 26? 27? 
 
 What are the remainders of numbers divided by 7? 
 
 16. 8 in 9 how many times, and what remainder ? 
 In 10? 11? 12? 13? 14? 19? 25? 26? 27? 
 28? 29? 30? 31? 
 
 What are the remainders of numbers divided by 
 8? 
 
 17. Give the numbers from 1 to 100 which are 
 divisible by 6. 
 
 18. Give the numbers from 1 to 100 which are 
 divisible by 7. 
 
 19. Give the numbers from 1 to 100 which are 
 divisible by 8. 
 
 20. 6 in 6 tens how many times ? In 12 tens ? In 
 18 tens ? In 24 tens ? In 36 tens ? In 60 tens ? 
 
 21. 6 in 6 hundred how many times? In 12 hun- 
 dred ? In 18 hundred ? 
 
 22. 7 in seven* tens how many times ? In 14 tens? 
 In 21 tens ? In 49 tens ? In 56 tens ? In 70 tens ? 
 
 23. 7 in 7 hundred? In 14 hundred? In 21 
 hundred ? 
 
 24. 8 in 8 tens? 16 tens? 24 tens? 32 tens? 
 56 tens ? 80 tens ? 
 
4:8 DIVISION. 
 
 25. 8 in 8 hundred ? in 16 hundred ? in 24 hun- 
 dred? 
 
 26. How much is 6 x 30 divided by 6? How 
 much is 56 divided by 8 and the quotient multiplied 
 by 8? 
 
 27. 6 in 40 how many times ? 
 
 PROCESS. 40 is 36, and 4, 6 in 36 6 times. Hence 
 6 in 40 6 times and remainder 4. 
 
 28. 6 in 50 how many times ? 6 in 100 ? 8 in 50 ? 
 8 in 60 ? 8 in 90 ? 7 in 60 ? 7 in 80 ? 7 in 100 ? 
 8 in 100 ? 
 
 LESSON IV. 
 
 1. If I wish to plant 36 trees in parallel rows of 
 nine trees each, how many rows must I plant ? 27 
 trees will give how many rows of 9 trees each ? 45 
 trees how many rows ? 54 trees ? 72 trees ? 63 
 trees ? 81 trees ? 90 trees ? 
 
 2. A vessel was wrecked on a rocky coast, and there 
 were more than a hundred men clinging to the wreck. 
 The life-boat from the shore could only save ten men 
 on each trip to the wreck. How many trips did its 
 bold rowers have to make in order to save 30 men ? 
 40 men ? 60 men ? 100 men ? 80 men ? 
 
 3. How many 9's in 18 ? 27 ? 36 ? 45 ? 63 ? 
 54? 72? 81? 90? 
 
 4. How many 10's in 20? 80? 90? 70? 60? 
 50? 40? 30? 100? 
 
 5. What number contains 9 exactly once ? 3 times? 
 Twice ? 5 times ? 6 times ? 8 times ? 7 times ? 
 4 times ? 9 times ? 10 times ? 
 
 6. What number contains 10 exactly 10 times? 9 
 times ? 8 times ? 7 times ? 6 times ? 5 times ? 4 
 times ? 3 times ? Twice ? 
 
 7. How much is 
 9 divided by 9 ? 10 divided by 10 ? 
 
DIVISION. 49 
 
 18 divided by 9 ? 20 divided by 10 ? 
 27 divided by 9 ? 30 divided by 10 ? 
 36 divided by 9 ? 40 divided by 10 ? 
 45 divided by 9 ? 50 divided by 10 ? 
 54 divided by 9 ? 60 divided by 10 ? 
 63 divided by 9 ? TO divided by 10 ? 
 72 divided by 9 ? 80 divided by 10 ? 
 81 divided by 9 ? 90 divided by 10 ? 
 90 divided by 9 ? 100 divided by 10 ? 
 
 8. How much is 11 x 9? 12 x 9 ? How many 
 times does 99 contain 9 ? How many times does 108 
 contain 9? 
 
 9. How much is 11 x 10 ? 12 x 10 ? 10 in 110 
 how many times ? 10 in 120 how many times ? 
 
 10. 9 in 10 how many times, and what remainder ? 
 In 11? 12? 13? 14? 15? 16? 17? In 19? 
 20? 21? 22? 23? 24? 25? 26? 
 
 What are the remainders of numbers divided by 
 9? 
 
 11. 10 in 21 how many times, and what remainder? 
 In 22? 23? 24? 25? 26? 27? 28? 29? In 
 41? 42? 43? 44? 45? 46? 47? 48? 49? 
 
 What are the remainders of numbers divided by 
 10? 
 
 12. Give the numbers, from 1 to 100, which are 
 divisible by 9. 
 
 13. Give the numbers, from 1 to 200, which are 
 divisible by 10. 
 
 14. What figure do all numbers divisible by 10 end 
 in? 
 
 15. 9 in 9 tens how many times ? In 18 tens ? 27 
 tens ? 36 tens ? 45 tens ? 90 tens ? 
 
 16. 9 in 9 hundred how many times ? 18 hundred ? 
 27 hundred? 36 hundred? 45 hundred? 63 hun- 
 dred? 
 
 17. 10 in 10 tens how many times? In 20 tens? 
 30 tens ? 40 tens ? 50 tens ? 90 tens ? 
 
50 DIVISION. 
 
 18. 10 in 20 hundred ? In 3000 ? 4000 ? 60 hun- 
 dred ? In 7000 ? In 9000 ? 
 
 19. 9 in 86 how many times? 
 
 PROCESS. 86 is 81 and 5. 9 in 81, 9 times. Hence, 
 9 in 86, 9 times, and 5 remainder. 
 
 20. 9 in 97 how many times? In 105? In 111? 
 113? In 960? 963? 999? In 456? 
 
 21. 10 in 770 how many times ? 220 ? 440 ? 1010? 
 77? 78? 101? 105? 362? 456? 
 
 22. When a number is not divisible by 10, which 
 figure of the number is always the remainder ? 
 
 23. How much is 7 x 8 x 10, divided by 10? 
 How much is 60 x 9, divided by 9 ? 
 
 24. How much is 9x8 + 3? 9x8 + 2? 9x 
 7 + 5? 9x6 + 7? 
 
 25. How much is 10 x 3 + 4 ? 10 x 4 + 5 ? 
 10x7 + 6? 10x8 + 9? 
 
 LESSON V. 
 
 1. A workman, desiring to finish a piece of work 
 speedily, labors 11 hours a day. In how many days 
 will he have worked 33 hours ? 44 hours ? 55 hours ? 
 77 hours? 22 hours? 88 hours? 99 hours? 110 
 hours ? 
 
 2. Twelve eggs make a dozen. How many dozen 
 eggs are 24 eggs ? 36 eggs ? 72 eggs ? 48 eggs ? 
 60 eggs? 84 eggs? 96eg<?s? 108 eggs ? 120 eggs? 
 
 3. How many ll's in 22? 33? 44? 55? 66? 
 77? 88? 99? 110? 
 
 4. How many 12's in 24 ? 36 ? 48 ? 60 ? 72 ? 
 84? 96? 108? 120? 
 
 5. What number contains 11 exactly once? Twice ? 
 4 times ? 7 times ? 3 times ? 5 times ? 9 times ? 
 
 6 times ? 8 times ? 10 times ? 
 
 6. What number contains 12 exactly twice ? Once ? 
 3 times? 5 times? 6 times? 4 times? 8 times? 
 
 7 times ? 9 times J 10 times ? 
 
DIVISION. 51 
 
 
 
 7. How much is 
 
 11 divided by 11 ? 12 divided by 12 ? 
 22 divided by 11 ? 24 divided by 12 ? 
 33 divided by 11 ? 36 divided by 12 2 
 44 divided by 11 ? 48 divided by 12 ? 
 55 divided by 11 ? 60 divided by 12 ? 
 66 divided by 11 ? 72 divided by 12 ? 
 77 divided by 11 ? 84 divided by 12 ? 
 88 divided by 11 ? 96 divided by 12 ? 
 99 divided by 11 ? 108 divided by 12 ? 
 110 divided by 11 ? 120 divided by 12 ? 
 
 8. How much is 11 x 11 ? 12 x 11 ? 12 x 12? 
 
 11 in 121 how many times ? 11 in 132 ? . 12 in 132 ? 
 
 12 in 144? 
 
 9. 11 in 12 how many times, and what remainder ? 
 11 in 13? in 14? in 15 2 in 16? in 17 ? in 18 ? 
 in 19 ? in 20 ? in 21 ? in 23 ? &c. 
 
 What are the remainders of numbers divided by 
 11? 
 
 10. How much is 7 x 11 + 3? 5 x 11 + 4? 5 
 x 11 + 5? 6 x 11 + 7? 6 x 11 + 10? 
 
 11. 12 in 25 how many times, and what remainder ? 
 in26? 27? 28? 29? 30? -31? 32? 33? 34? 
 35 ? 37 ? 
 
 12. How much is 2 x 12 + 1 ? 2 x 12 + 2? 2 
 x 1.2 + 3? 2 x 12 + 4? 2 x 12 + 5? 2 x 12 
 + 10? 2 x 12 + 11? 
 
 What are the remainders of numbers divided by 
 12? 
 
 13. Give the numbers, from 1 to 200, which are 
 divisible by 11. 
 
 14. Give the numbers, from 1 to 200, which are 
 divisible by 12. 
 
 15. 11 in 11 tens how many times? in 23 tens? 
 33 tens ? 66 tens ? 99 tens ? 
 
 16. 11 in 11 hundred how many times? in 22 
 hundred ? 44 hundred ? in 88 hundred ? 
 
52 
 
 DIVISION. 
 
 17. 12 in 12 tens how many times ? in 24 tens? 
 in 36 tens ? in 60 tens ? in 84: tens ? in 96 tens ? 
 
 18. 12 in 12 hundred how many times? in 24 
 hundred ? in 48 hundred ? in 72 hundred ? in 96 
 hundred ? 
 
 19. 11 in 79 how many times? in 89? in 101? 
 772? 330? 334? 67? 667? 446? 
 
 20. 12 in 38 how many times? in 360? in 362? 
 in 484 ? in 75 ? in 721 ? 964 ? 
 
 21. How much is 55 divided by 11 and the cpo- 
 tient multiplied by 11 ? 
 
 22. How much is 60 divided by 12 and the quotient 
 multiplied by 12 ? How much is 7 x 12, divided by 
 12? 
 
 LESSON VI. 
 
 1. Pythagoras, a Greek philosopher, united very 
 
DIVISION. 
 
 53 
 
 ingeniously in a square table all the products of the 
 numbers from 1 to 12. This table can be used as a 
 table of quotients. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 10 
 
 G 
 12 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 2 
 
 4 
 
 6 
 
 8 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 30 
 
 33 
 
 36 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 40 
 
 44 
 
 48 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 50 
 
 55 
 
 60 
 
 6 
 
 12 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42 
 
 48 
 
 54 
 
 GO 
 
 66 
 
 72 
 
 7 
 
 14 
 
 21 
 
 28 
 
 35 
 
 42 
 
 49 
 
 56 
 
 63 
 
 70 
 
 77 
 
 83 
 
 8 
 9 
 
 16 
 
 24 
 
 32 
 
 40 
 
 48 
 
 56 
 
 64 
 
 72 
 
 80 
 
 88 
 
 96 
 
 18 
 
 27 
 
 36 
 
 45 
 
 54 
 
 G3 
 
 72 
 
 81 
 
 90 
 
 99 
 
 108 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 GO 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 
 
 11 
 
 22 
 
 33 
 
 44 
 
 55 
 
 66 
 
 77 
 
 88 
 
 99 
 
 110 
 
 121 
 
 132 
 
 12 
 
 24 
 
 36 
 
 48 
 
 60 
 
 72 
 
 84 
 
 96 
 
 108 
 
 120 
 
 132 
 
 144 
 
 2. The first row at the top, moving from left to 
 right, contains the numbers 1, 2, 3, &c., to 12. The 
 first row on the left, moving from the top to the bot- 
 tom of the table, contains also the numbers 1, 2, 3, 
 &c., to 12. 
 
 The numbers in the second horizontal row, count- 
 ing from the top, and those in the second vertical 
 row, counting from the left, are formed by adding the 
 
 NOTE. Horizontal means here across tho page from left to right or 
 right to left Vertical means up and down the page. The horizontal 
 we will call rows ; the vertical, columns. 
 
54: DIVISIOX. 
 
 numbers in tlie first row to themselves. Those in the 
 third rows are formed by adding those in the second 
 rows to those in the first respectively. The numbers 
 in the fourth rows by adding those in the third rows 
 to those in the first respectively, &c. 
 
 3. The first row and the first column contain 1, 2, 
 3, &c. 
 
 The second row and second column contain 1, 2, 3, 
 &c., multiplied by 2. 
 
 The third row and third column contain 1, 2, 3, 
 &c., multiplied by 3, &c., &c. 
 
 4. If we wish to find the product of two numbers, 
 as 5 multiplied by 6, we find 5 in the top row, and 6 
 in the first column on the left. We come down the 
 fifth column until we get to the 6th row, and there 
 find 30, the product. 
 
 5. If we wish to find the quotient of 108 divided 
 by 9, we find 9 in the top row, come down the column 
 to 108, and find 12, the first number on the left, in 
 the same horizontal row with 108. 
 
 6. How is the table of Pythagoras formed ? 
 
 7. What do the first row and first column contain ? 
 
 8. The second row and the second column ? 
 
 9. The fourth row and the fourth column ? 
 
 10. Write the table on your slate. 
 
 11. Use the table to count by 2's from 2 to 24. 
 Count by 3's from 3 to 36. Count by 5's from 5 to 
 60. Count by 9's from 9 to 108. Count by ll's from 
 11 to 132. Count by 12's from 12 to 144. 
 
 12. Begin with the numbers 24, 36, 48, &c., in the 
 column on the right, and count backward by 2's, 3's, 
 4's, &c., respectively. 
 
 LESSON VII. 
 
 1. Seven are how many times 2 ? How many times 
 3? 4? 
 
DIVISION. 55 
 
 2. Ten are how many times 2 ? 3 ? 4 ? 5 ? 
 
 3. Nine are how many times 2? 3 ? 4 ? 5 ? 
 
 4. Eleven are how many times 3? 4? 6 ? 
 
 5. Twelve are how many times 4 ? 6 ? 3 ? 2 ? 
 5? 8? 
 
 6. Thirteen are how many times 2 ? 3 ? 4 ? 5 ? 
 6? 7? 8? 9? 
 
 7. Fourteen are how many times 2 ? 7 ? 5 ? 3 ? 
 4? 
 
 8. Fifteen are how many times 3? 5? 6? 4? 
 
 9. Sixteen are how many times 4 ? 5 ? 6 ? 3 ? 
 
 10. Seventeen are how many times 3 ? 4 ? 5 ? 2 ? 
 6? 9 ? 
 
 11. Eighteen are how many times 2? 3? 4? 6? 9? 
 
 12. IN in eteen are how many times 6? 3? 10? 2? 5? 
 
 13. Twenty are how many times 2 ? 3 ? 4 ? 5 ? 
 6? 7? 8? 9? 10? 
 
 14. Twenty-one are how many times 7 ? 3 ? 8 ? 
 2 ? 4 ? " 6 ? 10 2 
 
 15. Twenty-two are how many times 2 ? 3 ? 4 ? 
 5? 6? 7? 8? 9? 10? 112 
 
 1G. Twenty-three are how many times 2 ? 3 ? 4 ? 
 6? 8? 10? 11? 
 
 17. Twenty-four are how many times 2 ? 3 ? 4 ? 
 5? 6? 6? 8? 9? 
 
 18. Twenty-five are how many times 2? 3? 4? 
 5? 6? 7? 8? 10? 
 
 19. Twenty-six are how many times 3 ? 4 ? 5 ? 
 6? 7? 8? 9? 10? 
 
 20. Twenty-seven are how many times 3 ? 4 ? 5 ? 
 6? 7? 8? 10? 11? 
 
 21. Twenty-eight are how many times 3 ? 4 ? 5 ? 
 6 ? 8 ? 10 ? 
 
 22. Twenty-nine are how many times 3 ? 4 ? 5 ? 
 6? 7? 8? 9? 10? 
 
 23. Thirty are how many times 3 ? 4 ? 5 ? C ? 
 7? 8? 9? 10? 
 
56 DIVISION. 
 
 24. Thirty-one are how many times 3 ? 4 ? 5 ? 
 6? 7? 8? 9? 11? 
 
 25. Thirty-two are how many times 9 ? 10 ? 8 ? 
 7? 6? 5? 4? 3? 
 
 26. Thirty- three are how many times 6 ? 8 ? 7 ? 
 4? 9? 5* 10? 3? 
 
 27. Thirty-four are how many times 7 ? 3 ? 9 ? 
 10? 6? 8? 4? 5? 
 
 28. Thirty-five are how many times 3? 4? 5? 
 6? 7? 83 9? 10? 
 
 29. Thirty-six are how many times 3 ? 4 ? 5 ? 6 ? 
 7? 8? 9? 11? 
 
 30. Thirty-seven are how many times 3 ? 4 ? 5 ? 
 7? 8? 9? 10? 
 
 31. Thirty-eight are how many times 2? 4? 3? 
 5? 6? 7? 8? 9? 
 
 32. Thirty-nine are how many times 3 ? 4 ? 5 ? 
 6? 7? 8? 9? 10? 11? 
 
 33. Forty are how many times 8 ? 10 ? 6 ? 4 ? 
 3? 9? 5? 7? 
 
 34. Forty-one are how many times 4 ? 5 ? 6 ? 7 ? 
 8? 9? 10? 
 
 35. Forty-two are how many times 4 ? 5 ? 6 ? 7 ? 
 8? 9? 10? 11? 
 
 36. Forty-three are how many times 4 ? 5 ? 6 ? 
 
 7 ? 8 ? 9 ? 
 
 37. Forty-four are how many times 4 ? 5 ? 6 ? 
 7? 8? 9? 11? 
 
 38. Forty-five are how many times 4 ? 5 ? 6 ? 7 ? 
 
 8 ? 9 ? 10 ? 
 
 39. Forty-six are how many times 4 ? 5 ? 6 ? 7 ? 
 8? 9? 10? 
 
 40. Forty-seven are how many times 4 ? 5 ? 6 ? 
 7? 8? 9? 10? 
 
 41. Forty-eight are how many times 4? 5? 6? 
 7? 8? 9? 10? 12? 
 
 42. Forty-nine are how many times 4 ? 5 ? 6 ? &c. 
 
DIVISION. 57 
 
 43. Fifty are how many times 4 ? 5? 6? 7? 8? 
 9? 10? 11? 
 
 LESSON VIII. 
 
 1. Fifty-one are how manv times 10? 5 ? 3 ? G? 
 9? 7? 8? 4? 
 
 2. Fifty-two are how many times 4 ? 8 ? 7 ? 6 ? 
 12? 10? 13? 
 
 3. Fifty-three are how many times 9? 10? 6? 
 7? 8? 12? 11? 
 
 4. Fifty-four 'are how manv times 9? 10? 6? 7? 
 8? 12? 11? 
 
 5. Fifty-five are how many times 4 ? 8? 10? 12? 
 
 11 ? 2 ? 
 
 6. Fifty-ei^ht are how many times 2 ? 3 ? 4? 5 2 
 
 6 ? 7 ? 8 ? 9 ? 10 ? 
 
 7. Sixty are how many times 3? 4? 12? 15? 
 30? 
 
 .8. Sixty-four are how many times 4? 8? 16? 12? 
 30? 
 
 9. Sixty-five are how many times 5 ? 6 ? 7 ? 8 ? 
 10? 12? 13? 
 
 10. Sixty-nine are how many times 6 ? 7 ? 10 ? 
 20? 
 
 11. Sixty -eight are how many times 11? 12? 8? 
 
 7 ? 10 ? 6 ? 
 
 12. Sixty-seven are how many times 8 ? 7 ? 11 ? 
 
 12 ? 5 ? 9 ? 
 
 13. Seventy are how many times 2 ? 6 ? 7 ? 8 ? 
 ^ ? 11 ? 10 ? 
 
 14. Seventy -two are how many times 3 ? 5 ? 7 ? 
 6? 8? 9? 12? 
 
 15. Seventy-three are how many times 6 ? 7 ? 8 ? 
 9? 10? 
 
 16. Seven tv-six are how many times 3 ? . 4 ? 5 ? 
 6? 7? 8? "9? 10? 11? 
 
 3* 
 
58 DIVISION. 
 
 17. Seventy-seven are how many times 6? 7 ? 8 ? 
 9? 10? 11? 
 
 18. Eighty are how many times 7 ? 8 ? 9 ? 10 ? 
 11? 20? 40? 
 
 19. Eighty-one are -how many times 7? 8? 9? 
 10? 11? 123 6? 
 
 20. Eighty-four are how many times 6 ? 7 ? 8 ? 
 9? 10? 11? 12? 
 
 21. Eighty-five are how many times 6 ? 7 ? 9*? 
 9? tOf 111 12? 
 
 22. Eighty-eight are how many times 7 ? 8 ? 9 
 10? 11? 12? 
 
 23. Eighty-nine are how many times 7 ? 11 ? 12? 
 8? 9? 101 
 
 24. Ninety are how many times 10 ? 12 ? 7? 9 ? 
 8 ? 6 ? 30 ? 
 
 25. Ninety-one are how many times 7? 8? 9? 
 10? 11? 12? 30? 
 
 26. Ninety-four are how many times 12 ? 7 ? 10 ? 
 9? 8? 11? 
 
 27. Ninety-live are how many times 9 ? 10 ? 7 ? 
 8 ? 11 ? 12 ? 
 
 28. Ninety-six are how many times 9 ? 7 ? 8 ? 
 10? 11? 12? 
 
 29. Ninety-seven are how many times S? 9? 10? 
 11? 12? 
 
 30. Ninety-eight are how many times 8 ? 9 ? 10 ? 
 11 ? 12 ? 30 ? 
 
 31. Ninety-nine are how many times 8 ? 9? 10 ? 
 
 32. One hundred are how many times 4 ? 5 ? 20 ? 
 50? 9? 10? 2? 8? 25?. 
 
 33. One hundred and eight are how many times 9 ? 
 12? 3? 6? 25? 20? 
 
 34. One hundred and ten are how many times 10 ? 
 11? 5? 2? 20? 
 
 35. One hundred and twenty are how many times 
 10? 12?' 6? 20? 5? 8? 
 
MISCELLANEOUS. v 59 
 
 36. One hundred and forty-four are how many 
 times 12? 9? 6? 20? 50? 
 
 LESSON IX. 
 
 1. Divide 126, 180, 162, and 138 each by 2. 
 
 PROCESS. 126 is 12 tens and 6 units. 12 tens con- 
 tain 2 6 tens, or 60, times, and 6 contains 2 3 times. 
 Hence, 126 contains 2 63 times. 
 
 2. Divide 114, 150, 168, 243, and 108 each by 3. 
 PROCESS. 114 is 11 tens and 4 units, or 9 tens and 
 
 2 tens and 4 units. 3 in 9 tens, 3 tens, or 30 times. 
 
 3 m 24, 8 times. Hence, 3 in 114, 38 times. 
 
 3. Divide 140, 180, 408, 252, and 156 each by 4. 
 
 4. Divide 120, 240, 315, 225, and 550 each by 5. 
 
 5. Divide 96, 126, 162, 186, 198, and 216 each by 
 6. 
 
 6. Divide 91, 105, 154, 224. and 308 each by 7. 
 
 7. Divide 104, 176, 200, 216, and 404 each by 8. 
 
 8. Divide 126, 207, 333, 306, and 405 each by 9. 
 
 9. Divide 500, 690, 780, 320, and 450 each by 10. 
 
 10. Divide 154, 198, 253, 374, and 495 each by 11. 
 
 11. Divide 192, 300, 480, 492, and 708 each by 12. 
 
 , 
 
 CHAPTER Y. 
 
 MISCELLANEOUS. 
 LESSON I. 
 
 'or the sake of abbreviation, the sign _- is used in division to in- 
 dicate " divided by." 30 JL. 6 is read, 30 divided by (j. This sign, as 
 well as the x for " times," or "multiplied by," will be used hereafter, 
 when it is more convenient to do so. 
 
 1. Thomas divided 12 apples equally among his 
 
60 * MISCELLANEOUS. 
 
 three brothers ; how many apples did he give each 
 of them ? 
 
 2. If 4 yards of cloth cost 20 dollars, how much 
 does one yard cost ? 2 yards ? 
 
 3. Mary divided 10 roses equally among her 5 play- 
 mates, how many did she give to each one ? 
 
 4. If I pay 35 cents for 5 pounds of sugar, how 
 much does one pound cost ? 
 
 5. 42 bushels of corn meal were divided equally 
 among Y persons; how many bushels did each receive ? 
 
 6. 60 dollars are paid for 12 cords of wood ; how 
 much does the wood cost per cord '? 
 
 Y. If 10 oranges are worth as much as 40 apples, 
 how many apples are worth as much as one orange ? 
 
 8. If 63 men reap a field in one day, how many 
 men, laboring at the same rate, will reap it in 9 days ? 
 
 9. How many barrels of flour can be bought for 84 
 dollars, at 12 dollars a barrel ? at Y dollars a barrel? 
 at 14 dollars a barrel ? 
 
 10. Two boats on the Mississippi are 100 miles 
 apart. The hindmost boat gains on the other 5 miles 
 an hour. In how many hours will it overtake the 
 other ? 
 
 11. A man, having 86 dollars, bought 12 cords of 
 wood, and had 2 dollars left ; how much did he give 
 for each cord ? 
 
 12. What is the price of a pound of beef, if 12 
 pounds can be bought for 60 cents ? for 84 cents ? 
 96 cents ? 
 
 13. One man can do a piece of work in 25 
 days ; how long will it take 5 men to do the same 
 work ? 
 
 14. 5 men can do a piece of work in 10 days ; ho^jv 
 long will it take one man to do the same work 1 
 
 KOTE. Each day's labor of each man is considered the same in 
 these examples. 
 
MISCELLANEOUS. 61 
 
 15. If one pipe discharges a cistern of water in 84 
 hours, how long will it take 7 pipes of the same size 
 to discharge the cistern ? 
 
 16. If 12 pipes discharge a cistern in 96 hours, how 
 long will it take one pipe of the same dimensions to 
 discharge it ? 
 
 17. Two boats are 50 miles apart, approaching each 
 other. One is moving at the rate of 6 miles an hour, 
 and the other 4 miles an hour ; how long before they 
 will come together ? 
 
 18. For 65 dollars, how many sheep can you buy, 
 at 6 dollars apiece, and how much money would you 
 have left ? 
 
 19. If 51 dollars be divided among 3 men, how 
 many dollars will each receive ? 
 
 20. If 12 tons of coal cost 96 dollars, how much 
 does one ton cost ? 2 tons ? 
 
 21. In how many days will 5 men reap a field of 
 wheat, if 40 men reap it in one day ? 
 
 22. A laborer engaged to work 4 months for 96 
 dollars; how much c did he receive for each month? 
 How much for each week, allowing 4 weeks to each 
 month ? How much each day, allowing 6 working 
 days to each week ? 
 
 23. Six whaling vessels brought from their voyages 
 1800 barrels of sperm oil, and each vessel brought the 
 same number of barrels ; how many barrels did each 
 bring ? 
 
 24. How much butter at 12 cents a pound can be 
 bought with 6 dozen eggs at 10 cents a dozen? 6 
 
 les 10 are how many times 12 ? 
 . 4 times 9 are how many times 3 ? 
 
 26. 4 times 10 are how many times 5 ? 8 ? 
 
 27. 5 times 12 are how many times 15 ? 6 ? 
 6 times 7 are how many times 14 ? 
 How many times 12 are 9 times 4? 
 
 30. A man travels 8 miles an hour for 9 hours, 
 
62 MISCELLANEOUS. 
 
 how many hours must a man travel 6 miles an hour 
 in order to go the same distance ? 
 
 31. How many times 7 are 3 times 14 ? 
 
 32. How many times 12 are 9 times 4 + 24: ? 
 
 33. 5 times 9 + 9 are how many times 6 ? 
 
 34. 7 times 12 14 are how many times 5 ? 
 
 35. How many yards of cloth at 12 dollars a yard 
 can be bought for 360 dollars ? 
 
 36. How many tons of coal at 11 dollars a ton can 
 be bought for 136 dollars, and how many dollars will 
 remain ? 
 
 37. 8 times 8 are how many times 4 ? 32 ? 16 ? 
 
 38. 7 times 10 are how many times 5 ? 7 ? 14 ? 
 
 39. How long will it take a man to ride 72 miles, 
 if he rides 8 miles an hour ? 9 miles an hour ? 3 miles 
 an hour ? 6 miles an hour ? 4 miles an hour ? 
 
 40. 10 times 6 are how many times 12 ? 20 ? 5 ? 
 4 ? 30 ? 
 
 41. How much is 67 less 9 times 3 ? less 4 times 
 9? 
 
 42. A boat sailed up stream 8 miles an hour for 
 12 hours, and came back over the same distance in 6 
 hours ; how many miles an hour did she sail down 
 stream ? 
 
 43. How much is 25 less 45 -=- 5 ? 
 
 44. How much is 27 ~ 3 and the quotient multi- 
 plied by 4 't 
 
 LESSON II. 
 
 1. If 3 oranges cost 9 cents, how much will 5 
 oranges cost ? 
 
 PROCESS. 1. If 3 oranges cost 9 'cents, 1 orange 
 costs 3 cents. 2. If 1 orange costs 3 cents, 5 oranges 
 cost 15 cents. 
 
MISCELLANEOUS. 63 
 
 2. If 4 lemons cost 24 cents, how much will 7 lem- 
 ons cost ? 
 
 3. If 12 barrels of apples cost 108. dollars, how 
 much will one barrel cost? If 1 barrel of apples 
 costs 9, dollars, how much will 3 barrels cost '{ 4 
 barrels ? 
 
 4. A man travels 16 miles in 4 hours ; how far does 
 he travel in 1 hour ? How far in 6 hours ? 7 hours ? 
 9 hours ? 
 
 5. How many workmen can do in 1 day what 4 
 workmen can do in 12 days ? 
 
 ANS. 12 times as many, or 48 workmen. 
 
 6. How many workmen will require 12 days to do 
 what 48 workmen can do in 1 day ? 
 
 ANS. 48 is 12 times as many as the number of 
 workmen required. 
 
 Hence 48 divided by 12, or 4 workmen. 
 
 7. 3 workmen in 15 days will do as much as 15 
 workmen in how many days? 
 
 PROCESS. 1. The work done by 3 workman in 15 days 
 'can be done in one day by 15 x 3, or 45 workmen. 
 2. What 45 workmen can do in one day, 15 workmen 
 can do in as many days as 45 -j- 15, or 3 days. 
 
 8. How many workmen will do in 4 days what 2 
 workmen require 28 days to accomplish ? 
 
 PROCESS. 1. The work which 2 workmen can do in 
 28 days, can be done l>y one workman in 2 x 28, or 
 56 days. 2. What one workman can do in 56 days, 
 can be done in 4 days by as many workmen as 56 ~ 
 4, 6>^ 14 workmen. 
 
 9. 15 laborers in 4 days do as much as 5 laborers 
 in days ? 
 
 10. 24 laborers in 2 days do as much as 16 laborers 
 in days ? 
 
 11. 12 laborers in 6 days do as much as 9 laborers 
 in days? 
 
 12. 9 laborers in 10 days do as much as 30 laborers 
 in days 3 
 
64 MISCELLANEOUS. 
 
 13. 10 laborers in 18 days do as much as 60 laborers 
 in days ? 
 
 14. 10 laborers in 12 days do as much as 20 labor- 
 ers in days ? 
 
 15. 12 laborers in 12 days do as much as 18 laborers 
 in days ? 
 
 16. 40 laborers in 3 days do as much as 6 laborers 
 in days ? 
 
 17. 15 days' work of 3 men is equal to 9 days' 
 work of men ? 
 
 18. 15 days' work of 4 men is equal to 5 days' 
 work of men ? 
 
 19. 3 days' work of 10 men is equal to 16 days' 
 work of men ? 
 
 20. 6 days' work of 6 men is equal to 12 days' 
 work of men ? 
 
 21. 5 days' work of 15 men is equal to 15 days' 
 work of men ? 
 
 22. 24 days' work of 5 men is equal to 12 days' 
 w r ork of men ? 
 
 23. 24 days' work of 4 men is equal to 48 days' 
 w^ork of men ? 
 
 24. How much will 30 pounds of sugar cost if 6 
 pounds cost 42 cents ? 
 
 25. If 10 pencils cost 30 cents, what will be the 
 cost of 7 pencils ? of 9 ? of 15 ? of 12 ? 
 
 26. If 7 days' board cost 14 dollars, what will be 
 the cost of 30 days' board ? 
 
 27. If 8 quarts of milk cost 48 cents, how much 
 will 30 quarts cost? 
 
 28. A merchant bought 20 yards of cloth for 100 
 dollars ; he gave 10 yards of it for some flour which 
 cost 10 dollars a barrel ; how many barrels did he 
 get? 
 
 29. A greyhound, chasing a rabbit, is 30 yards be- 
 hind the rabbit, and gains on him 6 yards every 3 
 minutes ; in how many minutes will he overtake him ? 
 
MISCELLANEOUS. 65 
 
 30. A man hired a laborer and agreed to give him 
 10 dollars for every 3 days' work ; how much did he 
 have to pay him each week of 6 working days, and 
 how much for each month of 4 weeks ? 
 
 31. If 6 oranges are worth 1 pine-apple, and 2 pine- 
 apples are worth 1 melon, how many oranges are worth 
 4 melons ? 
 
 32. If 9 bushels of apples cost 18 dollars, how much 
 do 100 bushels cost? 
 
 33. If 60 pounds of pork cost 300 cents, how much 
 will 9 pounds cost ? 10 pounds ? 3 pounds ? 
 
 34. If a man travels 56 miles in 8 hours, how far 
 does he travel in 3 hours ? 4 hours ? 5 hours? 
 
 35. On the 12th of November, 25 meteors appear 
 and shoot across the sky every 2 hours ; how many ap- 
 pear in 3 hours ? In 5 hours ? 
 
 36. A certain number of rations last 20 soldiers 5 
 days ; how long would they last 2 soldiers ? 50 sol- 
 diers? 25 soldiers? 
 
 37. 40 days' rations for 5 men are 8 days' rations 
 for men ? 
 
 38. 16 days' rations for 5 men are 10 days' rations 
 for men ? 
 
 39. 18 days' rations for 6 men are 12 days' rations 
 for men ? 
 
 40. 30 days' rations for 60 men are 18 days' rations 
 for men ? 
 
 41. 3 days' rations for 90 men are 10 days' rations 
 for men ? 
 
 42. 24 days' rations for 10 men are 3 days' rations 
 for men ? 
 
 43. 16 days' rations for 6 men are 12 days' rations 
 for men ? 
 
 44. 8 days' rations for 8 men are 2 days' rations 
 
 men ? 
 
 5. 10 yards of cloth at 8 dollars a yard will pay 
 how many barrels of flour at 5 dollars a barrel ? 
 
 s 
 
66 MISCELLANEOUS. 
 
 46. 9 times 8 are how many times 24 -=- 2 ? 
 
 47. 9 times 11 less 18 are how many times 36 
 
 48. 10 times 6 and 12 are how many times 40 8 ? 
 
 49. 12 times 7 and 15 are how many times 44 4? 
 
 50. 4 times 12 and 8 are how many times 49 7 ? 
 
 51. Bought 20 barrels of flour, at 6 dollars a barrel, 
 and 6 yards of cloth, at 5 dollars a yard. Paid for 
 the whole with 15 loads of hay. How many dollars 
 did I get for each load of hay ? 
 
 52. If 4 cords of wood are worth 24 dollars, and 
 10 cords of wood are given for 40 thousand feet of 
 plank, how many dollars will each thousand feet of 
 plank cost ? 
 
 53. 5 men bought a horse for 63 dollars, and paid 
 two dollars a week for keeping him ; at the end of 8 
 weeks, they sold him for 54 dollars. How much did 
 each man lose by the bargain ? 
 
 54. A man bought 12 bales of cotton at 5 hundred 
 dollars for every two bales. He sold all the cotton at 
 6 hundred dollars for every three bales. How many 
 hundred dollars did he lose ? 
 
 55. What is the rent of 12 acres of land, that of 
 16 acres being 20 dollars ? 
 
 56. How long will 10 months' rations for 300 sol- 
 diers, issued at the same rate, last a garrison of 500 
 soldiers \ 
 
 
MULTIPLES AND DIVISORS. 67 
 
 CHAPTER VI. 
 
 MULTIPLES AND DIVISORS. 
 LESSON I. 
 
 1. 12 contains 6 an exact number of times. By 
 adding 6 to itself once, or multiplying it by 2, we 
 obtain 12. Hence 12 is called a multiple of 6. Thus, 
 24 is a multiple of 6, and 50 is a multiple of 5. 
 
 2. When is one number said to be a multiple of 
 another number ? 
 
 ANS. When one number contains another number 
 an exact nuinber of times, it is said to be a multiple 
 of that number. 
 
 3. 6 divides 12 exactly, and hence is called a divisor 
 of 12, or one of its factors. In the same manner, 4 
 is & factor or divisor of 12. 5 is one of the factors or 
 divisors of 60. 
 
 4. What is meant by a factor or divisor of a given 
 number ? 
 
 ANS. Any number which is contained an exact 
 nujnber of times in the given number. 
 
 5. Every number contains itself once. 5 contains 5 
 once, 100 contains 100 once. Hence, every number 
 is a multiple and divisor of itself. 
 
 6. What number divides exactly all numbers? 
 How often is 1 contained in 5 ? In 6 ? In 500 ? 
 
 7. What two divisors has every number, at least ? 
 ANS. The number itself, and one. 
 
 8. Give 4 numbers which are multiples of 9. 
 
 9. Give 5 numbers which are multiples of 10. Of 
 11. Of 12. 
 
68 MULTIPLES AND DIVISORS. 
 
 10. Give 3 multiples of each of the numbers, 20, 30, 
 and 40. 
 
 11. Find some numbers which are factors or divisors 
 of 15. Of 20. Of 40. Of 50. 
 
 Including one and the numbers themselves, respec- 
 tively 
 
 12. Find the 3 divisors of 4. Of 9. Of 25. Of 49. 
 
 13. What are the four divisors of 8 ? of 10 ? of 
 14? 15? 21? 22? 26? 33? 34? 35? 38? 
 39? 46? 51? 55? 57? 58? 62? 65? 69? 
 74? 77? 82? 85? 86? 87? 91? 93? 94? 
 95? 
 
 14. What are the five divisors of 16 ? of 81 ? 
 
 15. What are the six divisors of 12 ? 18 ? 20 ? 
 28? 32? 44? 45? 50? 52? 63? 68? 75? 
 76? 92? 98? 99? 
 
 16. What are the seven divisors of 64? 
 
 17. What are the eight divisors of 24? of 30? 
 40 ? 42 ? 54 ? 56 ? 66 2 70 ? 78 ? 88 ? 
 
 18. What are the nine divisors of 36 ? of 100 ? 
 
 19. What are the ten divisors of 48 ? of 80 ? 
 
 20. What are the 12 divisors of 60 ? of 72 ? of 
 84 ? of 90 ? of 96 ? 
 
 LESSON II. 
 
 A number which has several divisors is called a 
 common multiple of these divisors. Thus, 10 is a 
 common multiple of 2 and 5, because 2 and 5 are 
 both contained in 10 an exact number of times. 4 is 
 a common multiple of 2 and 4. 12 is a common 
 multiple of 1, 2, 3, 4, 6, and 12. 
 
 1. What is a common multiple of two or more 
 numbers ? 
 
 ANS. It is any number which contains each of the 
 given numbers an exact number of times. 
 
 2. Find some common multiples of 5 and 4. Of 6, 
 3, and 2. Of 3 and 4. 
 
MULTIPLES AND DIVISORS. 
 
 69 
 
 3. What is the least common multiple of two or 
 more numbers ? 
 
 ANS. It is the smallest number which these num- 
 bers will exactly divide. 
 
 4. What is the least common multiple of 2 and 3 ? 
 of 3 and 4 ? of 4 and 5 ? of 5 and 10 ? 
 
 5. of 2 and 4 ? of 3 and 5 ? of 5 and 6 ? of 5 and 
 100? 
 
 6. of 2 and 5'? of 3 and 6 ? of 4 and 7 ? of 6 and 
 0? 
 
 7. of 2 and 6 ? of 3 and 7 ? of 4 and 8 ? of 7 and 
 1? 
 
 8. of 2 and 7? of 3 and 8 \ of 4 and 9 ? of 8 and 
 
 9. of 2 and 8 ? of 3 and 9 ? of 4 and 10 ? of 9 
 id 30 ? 
 
 10. of 2 and 9 ? of 3 and 10 ? of 4 and 11 ? of 
 and 8 ? 
 
 11. What is the least common multiple of 2, 3, and 
 
 PROCESS. The least common multiple of 2 and 3 
 1 6, of 6 and 4 is 12. Hence , 12 is the least common 
 \dtiple of 2, 3, and 4. 
 
 12. What is the least common multiple 
 
 of 2, 3, and 5 ? of 3, 4, and 5 ? 
 of 2, 3, and 6 ? of 3, 4, and 6 ? 
 of 2, 3, and 7 ? of 3, 4, and 7 ? 
 
 13. What is the least common multiple of 2, 3, 4, 
 id 6? 
 
 PROCESS. The least common multiple of 2 and 3 
 6, of 6 and 4 is 24, ^ 24 #;i6? 6 is 24. .SWtf, ^ 
 ast common multiple of 2, 3, 4, 6 is 24. 
 
 14. What is the least common multiple 
 
 of 2, 3, 4, 5 ? of 3, 4, 5, 6 ? of 4, 5, 6, 8 ? 
 of 2, 3, 4, 7 ? of 3, 4, 5, 10 ? of 4, 5, 6, 9 ? 
 of 2, 3, 4, 10 ? of 3, 4, 5, 8 ? of 4, 5, 6, 10 ? 
 
 15. What is the least common multiple of 2, 3, 4, 
 
70 MULTIPLES AND DIVISORS. 
 
 LESSON III. 
 
 1. Give two or more numbers of which 3 is a 
 divisor. 3 is called a common divisor or common 
 factor of these numbers. 
 
 2. What numbers are common divisors of 8 and 
 12? 
 
 ANS. 2 and 4 because they divide 8 and 12 with- 
 out a remainder. 
 
 3. What numbers are common divisors of 6, 12, 
 and 24 ? 
 
 ANS. 2, 3, and 6. 
 
 4. When is a number said to be a common divisor 
 or common factor of two or more numbers ? 
 
 ANS. When it is contained an exact number of 
 times in each of these numbers. 
 
 5. What number is a common divisor of all num- 
 bers ? 
 
 6. Which is the greatest common divisor of 8 and 
 12? 
 
 ANS. 4. Because it is the greatest number which 
 divides 8 and 12 without a remainder. 
 
 7. What is the greatest common divisor of 6, 12, 
 and 24 ? 
 
 ANS. 6. 
 
 8. What is the greatest common divisor of two or 
 more numbers ? 
 
 ANS. The greatest number which is contained ex- 
 actly in these numbers. 
 
 9. What is the greatest common divisor of 15 and 
 20? of 6 and 24 ? of 12 and 30 ? of 25 and 45 ? 
 of 22 and 33 ? 
 
 10. What is the greatest common divisor of 4, 8, 
 and 16? 
 
 PROCESS. The greatest common divisor of 8 and 
 16 is 8 ; of 8 and 4 -is 4. Hence , 4 is the greatest 
 common divisor of 4, 8, and 16. 
 
MULTIPLES \XD DIVISORS. 71 
 
 11. What is the greatest common divisor of 5, 10, 
 and 15 ( of 
 
 12. What is the srreatest common divisor of IS and 
 of '24: and 3' and 54: i of 24: and 100* 
 
 LESSOR IV. 
 
 A nnmber which has no other divisors except the 
 number 1 and the number itself is called a prime 
 number. Thus, 2 has no divisor except itself and 1 ; 
 3 has no divisor except 3 and 1 ; 5 has no divisor ex- 
 cept 5 and 1. tire. Hence, 2, 3. and 5 are prime num- 
 bers, 1 is also a prime number. 
 
 1. Are any of the even numbers which are greater 
 than 2, prime numbers Why not \ 
 
 2. Find the prime numbers from the number 1 to 
 the number 
 
 3. Find the prime numbers between 25 and 50. 
 4:. Find the prime numbers between 50 and 
 
 5. Find the prime numbers between 75 and 
 The numbers which are not prime are the prod:: 
 
 of two or more numbers: I > _ _ > _ x 3. 
 
 _ 4:, or 2 x 2 x 2. i-i >- x 3, or 2 x 
 3, or 2 x 2 x 2 x 3. 
 
 The divisors or fac: .\re called the//r/ffi 
 
 fact 2. 2, 2, 3. the /////!<:- factors of L'4r. 
 
 6. 2i is the continued product of what 4: numK 
 
 9 is the product of what 2 prime numbe il - is the 
 continued product of what three prime fact-; - 
 
 7. What are the prime factors of any number ? 
 Axs. The smallest numbers whose continued pro- 
 duct id II given number. 
 
 ^ Find the 2 prime factors of each of the number- 
 , U, 15, 21, 
 
 I":. "". --. B&, ^ 87, 
 (Thirty-four numbt 
 9. Find 3 prime f :' each of the nuniK 
 
72 FRACTIONS. 
 
 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 
 75, 76, 78, 92, 98, 99. (Twenty-two numbers.) 
 
 10. What are the 4 prime factors of 16, 24, 36, 40, 
 54, 56, 60, 81, 84, 88, 90, 100 ? (Twelve numbers.) 
 
 11. What are the 5 prime factors of 32, 48, 72, 80 ? 
 (Four numbers.) 
 
 12. Find the 6 prime factors of each of the num- 
 bers, 64 and 96. (Two numbers.) 
 
 13. There are 32 + 22 + 12 + 4+2 numbers, 
 from 1 to 100, which have factors ; how many are 
 there ? 
 
 14. How many prime numbers from 1 to 100 ? 
 
 15. What number multiplied by itself gives 4? 
 25? 9? 49? 16? 36? 64? 81-? 100? 121? 
 144 ? 169 ? 196 ? 
 
 CHAPTER VII. 
 
 FRACTIONS. 
 
 LESSON I. 
 Halves, Thirds, and Fourths, or Quarters. 
 
 1. IF a line be divided into two equal parts, what is 
 each one of these parts called ? 
 
 ANS. One-half of the line. 
 
 2. How do you get the half of an apple or orange, 
 or any thing ? 
 
 ANS. Divide it into two equal parts and take one 
 of these parts. 
 
 3. How many half cents in 1 cent ? How many 
 half cents in two cents ? How many half apples in 
 
FRACTIONS. 73 
 
 three apples? How many half cents in 50 cents? 
 How many half dollars in 8 dollars ? 
 
 4. How many halves in 1 ? 2 ? 3 ? 4 ? 5 ? 6 ? 
 7 ? 8 ? 9 ? 10 ? 61 ? 40 ? 500 ? 
 
 5. How do you find the number of halves in any 
 number ? 
 
 ANS. Multiply the number by 2. 
 
 6. How many half apples can you make of 2 
 half apples? How many cents are equal to 10 
 half cents? How many dollars do 12 half dollars 
 make? 
 
 7. How many ones in 2 halves ? 4 halves ? 8 
 halves? 6 halves 10 halves? 12 halves? 24 
 halves ? 48 halves ? 100 halves ? 
 
 18. How do you get the number of ones or units in 
 y number of halves ? 
 ANS. Divide the number of halves by 2. 
 
 9. How many half apples in 3 apples and one half 
 apple ? How many dollars in 5 dollars and a half 
 dollar ? How many half yards of cloth in 7 yards 
 and a half yard ? 
 
 10. How many halves in 10 and one-half? 11 and 
 one-half? 15 and one-half? 50 and one-half? 60 
 and one-half? 4 and one-half? 19 and one-half? 
 
 11. How many apples in 3 half apples? How 
 many dollars in 7 half dollars ? What remains over 
 in each case? 
 
 12. How many ones in 7 halves? 9 halves? 11 
 halves? 13 halves? 27 halves? 99 halves? 57 
 halves ? 45 halves ? 23 halves ? 31 halves ? 
 
 t!3. 7 divided by 2 is how much? 
 ANS. 3 and one-half. 
 14. 2 in 27 how many times ? 2 in 99 ? 2 in 31 ? 
 n 23 ? 2 in 81 ? 2 in 11 ? 
 15. 1 half of 2 apples is - - ? one-half of 3 dol- 
 's is how much ? 
 16. How much is one-half of 2 ? of 3 ? of 5 ? of 
 
74 FRACTIONS. 
 
 6? of 7? of 8? of 9? of 10? of 11? of 12? 
 of 25? of 100? 50? 40? 32? 22? 400? 61? 
 
 17. How do we get the half of a number ? 
 ANS. Divide it by 2. 
 
 18. 3 is the half of what number? 6 is the half 
 of what number ? the half of what number is 7 ? 
 9? 12? 20? ^30? 25? 60? 15? 19? 
 
 PROCESS. 3 is the half of two 3's, or 2 x 3. 7 is 
 the half of 2 x 7, &c., &c. 
 
 19. How much is three halves of 12 ? 
 
 PROCESS. One-half of 12 is 6 ; hence, three halves 
 of 12 is three times 6. 
 
 20. How much is three halves of 24 ? of 50 ? of 
 30? of 18? of 20? 
 
 21. How much is 7 halves of 20 ? 18 ? 30 ? 12 ? 
 
 22. How much is 9 halves of 18 ? 24 ? 10 ? 90 ? 
 
 23. How much is 11 halves of 12 ? 14 ? 30 ? 22 3 
 
 24. If a line be divided into three equal parts, what 
 is each part called ? 
 
 ANS. One-third. 
 
 25. How do you get the third of an apple, orange, 
 or of any thing ? 
 
 26. How do you get two-thirds of any thing ? 
 
 27. How many thirds of an apple in one apple ? 
 in 2 apples ? How many thirds of a cent in 3 cents ? 
 How many thirds of a yard of cloth in 6 yards of 
 cloth ? 
 
 28. How many thirds inl? 2? 3? 4? 5? 8? 
 7? 6? 9? 10? 30? 40? 12? 11? 300? 
 
 29. How do you find the number of thirds in any 
 number ? 
 
 30. How many apples do 3 thirds of an apple make ? 
 How many dollars are equal to 6 thirds of a dollar? 
 How many oranges do 12 thirds of an orange make ? 
 
FRACTIONS. 75 
 
 31. How many ones in 3 thirds ? 6 thirds ? 9 
 thirds? 15 thirds? 12 thirds? 18 thirds? 24 
 thirds ? 21 thirds ? 27 thirds ? 99 thirds ? 60 thirds ? 
 
 32. How do you find the number of ones or units 
 in any number of thirds ? 
 
 33. How many thirds of an orange in 3 oranges 
 and 1 third of an orange ? in 3 oranges and 2 thirds ? 
 How many thirds of a bushel of meal in 6 bushels 
 and one-third of a bushel ? How many thirds of a 
 dollar in 8 dollars and 2 thirds of a dollar ? 
 
 34. How many thirds in 3 and one-third ? in 3 and 
 2 thirds ? in 5 and 2 thirds ? in 30 and a third ? 
 21 and 2 thirds ? in 40 and one-third ? in 11 and 2 
 thirds ? in 100 and one-third ? 
 
 35. How many oranges can you make of 4 thirds of 
 an orange ? How many dollars do 8 thirds of a dollar 
 make? How many thirds remain in each case? How 
 many bushels of meal do 10 thirds of a bushel make ? 
 What remains ? 
 
 36. How many ones in 4 thirds ? 5 thirds ? 7 
 thirds? 8 thirds? 11 thirds? 13 thirds? 16 thirds? 
 20 thirds ? 61 thirds ? 100 thirds ? 
 
 37. 7 divided by 3 is how much ? 
 ANS. 2 and one-third. 
 
 38. 3 in 13 how many times? in 14? 80? 62? 
 44? 22? 8? 
 
 39. How much is one-third of 2 ? Show by the 
 picture above that one-third of two is 2 thirds of one. 
 How much is one-third of 3 ? 4 ? 5 ? 6? 7 ? 8 ? 
 9? 10? 11? 12? 24? 27? 15? 18? 30? 
 60? 90? 600? 900? 99? 22? 17? 31? 
 
 40. How do you find one-third of any number ? 
 How much is 2 thirds of 6 ? 
 
 PROCESS. One-third of is 2. Hence, 2 thirds of 
 6 is 2 x 2, or 4. 
 
 41. How much is two-thirds of 21 ? 2 thirds of 9 ? 
 2 thirds of 5 ? of 7 ? of 100 ? of 60 ? 
 
76 FRACTIONS. 
 
 42. How much is 4 thirds of 6 ? 
 
 PROCESS. One-third of 6 is 2. Hence, 4 thirds of 
 6 is 4: x 2, or 8. 
 
 43. Which is the greater, 4 thirds of a number or 
 the number itself? How much is 3 thirds of 6 ? 
 
 44. How much is 4 thirds of 21 ? 24 ? 27 ? 60 ? 
 15? 10? 12? 
 
 45. How much is 5 thirds of 5 ? 12 ? 21 ? 24 ? 
 27? 60? 33? 
 
 46. How much is 6 thirds of 27 ? 9 ? 12 ? 18 ? 
 24? 30? 42? 36? 
 
 47. How much is 10 thirds of 15 ? 12 ? 21 ? 27 ? 
 20? 60? 90? 99? 
 
 48. 2 is one-third of what number ? 6 is one-third 
 of what number ? 
 
 49. One-third of what number is 7 ? 9 ? 12 ? 8 ? 
 20? 30? 60? 19? 15? 100? 300? 600? 
 
 PROCESS. 2 is one-third of three 2's, or 3 times 2. 
 6 is one-third of 3 times 6, dec. 
 
 50. If a line be divided into 4 equal parts, what is 
 each one of these parts called ? 
 
 ANS. One-fourth of the line, or sometimes one- 
 quarter of the line. 
 
 51. How do you get one-fourth or a quarter of an 
 apple or orange, or of any thing ? How do you get 2 
 fourths or quarters ? 3 fourths or quarters ? 
 
 ANS. Divide it into four equal parts, and take one 
 of the parts, two of the parts, and three of the parts, 
 respectively. 
 
 52. How many fourths of an apple in one apple ? 
 How many fourths of a bushel of meal in 2 bushels of 
 meal ? How many fourths of a pint of wine in one pint 
 of wine ? How many quarter dollars in 5 dollars ? 
 
 53. How many fourths in 1 ? 2 ? 3 ? 4 ? 5 ? 6 ? 
 
FRACTIONS. 77 
 
 7? 8? 9? 10? 11? 12? 13? 15? 20? 30? 
 40? 100? 200? 202? 
 
 54. How do you find the number of fourths in any 
 Dumber ? 
 
 55. How many fourths in 3 and one-fourth ? 
 PROCESS. There are 12 fourths in 3, and these, 
 
 added to one-fourth give 13 fourths. 
 
 56. How many fourths in 6 and 2 fourths ? 7 and 
 3 fourths ? 20 and one-fourth ? How many quarter 
 dollars in 10 dollars and 3 quarter dollars? How 
 many fourths in 30 and 3 fourths ? in 100 and 2 
 fourths ? in 12 and 3 quarters ? 
 
 5 7. How many oranges do 4 fourths of an orange 
 make ? How many dollars do 12 quarter dollars 
 make ? How many gallons of wine do eight quarter 
 gallons make ? 
 
 58. How many ones in 4 fourths ? 8 fourths? 12 
 fourths ? 24 fourths ? 32 fourths ? 28 fourths ? 60 
 fourths ? 40 fourths ? 400 fourths ? 88 fourths ? 
 
 59. How do you find the number of ones in any 
 number of fourths ? 
 
 60. How many oranges do 5 quarter oranges make ? 
 How much over? 
 
 61. How many dollars do 6 quarter dollars make? 
 How much over ? 
 
 62. How many yards of cloth do 17 quarter yards 
 make ? How much over ? 
 
 63. How many ones in 5 fourths? 6 fourths? 7 
 fourths ? 21 fourths ? 22 fourths ? 61 fourths ? 49 
 fourths ? 37 fourths ? 39 fourths ? 
 
 64. If we divide 9 apples among four boys, how 
 many whole apples would each boy get ? In order to 
 give each boy an equal share, what must we do with 
 the remaining apple ? 9 divided by 4 is how much ? , 
 
 ANS. 2 and one fourth. 
 
 65. If we divide 10 apples equally among 4 boys, 
 what would be the share of each one ? 
 
78 FRACTIONS. 
 
 ANS. 2 apples and 2 fourths of an apple. 
 
 66. If we divide 11 apples equally among 4 boys, 
 what would be the share of each ? 
 
 67. 4 in 13 how many times? in 14 ? 15? 22? 
 21? 25? 27? 30? 62? 63? 43? 103? 101? 
 401? 
 
 68. How much is one-fourth of 2 ? of 3 ? Show 
 by the picture of lines above that one-fourth of 2 is 
 2 fourths of 1. Show that one-fourth of 3 is 3 fourths 
 of 1. 
 
 69. How much is one-fourth of 2 ? 3 ? 4 ? 5 ? 
 6? 7? 8? 9? 10? 11? 12? 13? 16? 18? 
 20? 44? 60? 64? 80? 400? 444? 
 
 70. How do you -find one- fourth of any number ? 
 
 71. How much is 2 fourths of 8 ? 3 fourths of 8 ? 
 4 fourths of 8 ? 5 fourths of 8 ? 10 fourths of 8 ? 
 
 72. 5 fourths of eight is how many times 8 ? 
 ANS. Once 8 and one-fourth of 8. 
 
 73. How much is 3 fourths of 12 ? of 9 ? of 24 ? 
 of 28? 32? 80? 100? 400? How much is 2 
 fourths of 8 ? 16 ? 12 ? 40 ? 100 ? 400 ? 
 
 74. How much is 5 fourths of 56 ? 60 ? 92 ? 80 ? 
 20? 28? 36? 40? 48? 
 
 75. How much is 7 fourths of 12 ? 96 ? 20 ? 16 ? 
 24? 52? 40? 44? 
 
 76. How much is 10 fourths of- 8 ? 12 ? 24 ? 6 ? 
 16? 80? 400? 
 
 77. 2 is one-fourth of what number ? 3 is one- 
 fourth of what number? One-fourth of what num- 
 beris4? 5? 6? 7? 8? 9? 10? 11? 12? 60? 
 40? 100? 200? 15? 400? 900? 
 
 78. 8 is -what part of 16 ? 24 ? 32 ? 6 is what 
 part of 24 ? 12 ? 18 ? 7 is what part of 21 ? 28 ? 
 14? 10 is what part of 40? 20? 30? 60 is what part 
 of 120 ? 240 ? 180 ? 9 is what part of 27 ? 36 ? 
 18 ? 12 is what part of 36 ? 15 is what part of 60 ? 
 14 is what part of 28 ? 100 is what part of 400 ? 
 
 
FRACTIONS. 79 
 
 79. 3 halves is 3 -f- ? 5 halves is 5 -4- ? 4 thirds 
 is 4 + i 5 fourths is 5 -f- ? 100 fourths is 100 H- ? 
 
 We use figures to write briefly halves, thirds, and 
 fourths, or in general, fractions. (The sign signifies 
 equal to.) 3 halves is written f ; 5 fourths is written 
 ; 20 thirds is written 2 ^-. 
 
 80. What does the number above the line si^nd 
 for? 
 
 ANS. It gives the number of halves, fourths, or 
 thirds. 
 
 81. What does the number below the line show ? 
 ANS. To write halves we put 2 below the line / to 
 
 write thirds we put 3 ; to write fourths we put 4 
 below the line. 
 
 82. Give the figures and their places which will 
 represent 7 halves, 10 thirds, 14 fourths. Read the 
 
 frflftinnq - 6 -- 4_4 2 - 7 3. 5. 
 
 & ,2 ) 35 45453' 
 
 83. 5 and one-half, and 2 thirds, 8 and 3 fourths, 
 &c., are written without any sign between the frac- 
 tion and the whole number, thus : 5|-, 6f, 8f, and 
 these are called mixed numbers. 
 
 Write 100 and one-fourth, 20 and 2 fourths, 5 and 
 2 thirds, 7 and a half. Read 25|, 42i, 8j. 
 
 NOTE. In the chapters preceding Chapter VII. the word number 
 means whole number. In this and succeeding chapters it is applied 
 to fractions as well as whole numbers. 
 
 
80 FRACTIONS. 
 
 LESSON II. 
 
 1. If a line be divided in five equal parts, what do 
 we call one part of it ? 
 
 ANS. One-fifth of the line. 
 
 2. If a line be divided into 6 equal parts ? Seven 
 equal parts ? 
 
 ANS. One part is called one- sixth and one-seventh, 
 respectively. 
 
 3. How do you get one seventh of anything? one- 
 sixth ? one-fifth ? 2 sevenths ? 3 sevenths ? 4 
 sevenths ? 5 sevenths ? 2 fifths ? 3 fifths ? 4 fifths ? 
 3 sixths? 5 sixths? 
 
 4. If we conceive the number 1 divided into five 
 equal parts, what will these parts be called ? If it is 
 divided into six equal parts, what do we call them ? 
 into 7 equal parts ? 
 
 5. How many fifths inl? 2? 3? 4? 5? 6? 
 7? 8? 9? 10? 11? 12? 20? 15? 40? 60? 
 24? 32? 100? 200? 
 
 6. How many fifths in 2? 3? 4? 5? 6? 7? 
 8? 9? 12? 10? 11? 20? 15? 40? 30? 31? 
 50 ? 100 ? 400 ? 
 
 7. How many sevenths inl? 2? 3? 4? 5? 6? 
 7? 8? 9? 10? 11? 12? 75? 16? 19? 20? 
 30? 41? 100? 120? 
 
 8. How do you find the number of fifths in any num- 
 ber ? the number of sixths ? the number of sevenths ? 
 
 9. How many apples do 5 fifths of an apple make ? 
 6 fifths of an apple ? How many dollars are equal to 
 12 sixths of a dollar? to 14 sevenths of a dollar ? 
 
 10. How many ones in 5 fifths ? 6 fifths ? 7 fifths ? 
 8 fifths ? 9 fifths ? 10 fifths ? 15 fifths ? 20 fifths ? 
 
FRACTIONS. 81 
 
 25 fifths ? 26 fifths ? 2T fifths ? 28 fifths ? 29 fifths ? 
 30 fifths ? 60 fifths ? 100 fifths ? 200 fifths ? 
 
 11. How many ones in 6 sixths ? 7 sixths ? 8 
 sixths ? 9 sixths ? 10 sixths ? 11 sixths ? 12 sixths 2 
 21 sixths? 30 sixths? 36 sixths? 48 sixths? 60 
 sixths? 72 sixths? 100 sixths? 31 sixths? 32 
 sixths ? 34 sixths ? in 600 sixths ? 
 
 12. How many ones in 7 sevenths ? 8 sevenths ? 9 
 sevenths? 10 sevenths? 11 sevenths? 12 sevenths? 
 13 sevenths ? 14 sevenths ? 15 sevenths? 21 sevenths ? 
 42 sevenths ? 35 sevenths ? 70 sevenths ? 84 sevenths ? 
 in 100 sevenths? 490 sevenths ? 210 sevenths ? 
 
 13. How do you find the number of ones in any 
 number of fifths ? in any number of sevenths ? in 
 any number of sixths ? 
 
 14. 7 in 8 how many times? in 10 ? in 26 ? in 30 ? 
 
 15. 6 in 13 how many times ? in 14 ? in 15 ? in 
 28 ? in 29 ? 
 
 16. 5 in 11 how many times ? in 12 ? in 13 ? in 
 24 ? in 39 ? 
 
 17. If I wish to divide 13 oranges among 5 boys, 
 how many oranges and what parts of an orange 
 would the equal share of each boy be ? What would 
 be the equal share of each of 6 boys ? of each of 7 boys ? 
 
 18. How many fifths in 3 and one-fifth? 20 and 
 3 fifths ? 60 and 4 fifths? 5 and one-sixth how many 
 sixths y4 and 3 sixths ? 100 and 5 sixths? 8 and 2 
 sevenths how many sevenths ? 10 and 3 sevenths ? 
 1 5 and 5 sevenths ? 40 and 6 sevenths ? 
 
 19. How much is one-fifth of 2 ? Draw lines on 
 3ur slate, and show that one-fifth of 2 is 2 fifths of 
 ne. Show that one-sixth of 5 is 5 sixths of 1. 
 liow that one-seventh of 3 is equal to 3 sevenths 
 '1. 
 
 20. How much is one-fifth of 3 ? 4? 5? 10? 
 >? 20? 24? 25? 30? 31? 35? 37? 40? 
 I? 50.? 100? 200? 500? 150? 250? 
 
82 FRACTIONS. 
 
 21. How much is one-sixth of 2 ? 3 ? 4 ? 5 ? 6 ? 
 12? 13? 14? 18? 24? 28? 29? 30? 36? 
 58? 72? 70? 100? 600? 120? 300? 
 
 22. How much is one-seventh of 2 ? 3 ? 4 ? 5 ? 
 6? 7? 11? 12? 13? 14? 28? 30? 35? 36? 
 42? 49? 70? 63? 100? 84? 490? 700? 
 
 23. How much is 2 fifths of 10 ? 3 fifths of 45 ? 4 
 fifths of 50? 6 fifths of 30? 4 fifths of 80? 5 fifths of 25? 
 
 24. How much is 3 sixths of 12 ? 4 sixths of 18 ? 
 5 sixths of 96 ? 6 sixths of 72 ? 7 sixths of 42. 
 
 25. How much is one-seventh of 84? 2 sevenths of 
 84 ? 3 sevenths of 84 ? 4 sevenths of 21 ? 5 sevenths 
 of 28 ? 6 sevenths of 35 ? 7 sevenths of 98 ? 10 
 sevenths of 70 ? 
 
 26. 3 is one-fifth of what number ? One-fifth of 
 what number is 4 ? 5? 6? 7? 8? 9? 10? 12? 
 60? 100? 25? 33? 200? 
 
 27. 3 is one-sixth of what number ? One-sixth of 
 what number is 2 ? 4 ? 5 ? 6 ? 7? 8? 9 ? 10 ? 
 11? 12? 13? 14? 15? 20? 22? 30? 40? 
 50? 100? 45? 200? 400? 
 
 28. 2 is one-seventh of what number? One- 
 seventh of what number is 3? 4? 5? 6? 7? 8? 
 9? 10? 11? 12? 13? 14? 20? 21? 30? 33? 
 40? 100? 200? 300? 
 
 29. What part of 15 is 3? What part of 30 is 
 5? 6? What part of 42 is 6? 7? WhaP^rt of 
 35 is 5 ? 7 ? What part of 72 is 12 ? What part of 
 63 is 9 ? of 80 is 16 ? 8 is what part of 16 ? 24 ? 
 32? 40? 48? 56? 1 is what part of 2? 3? 4? 
 5 ? 6 ? 7 ? 100 is what part of 200 ? 300 ? 400 ? 
 500 ? 600 ? 700 ? 
 
 30. 4 fifths of 1 is one-fifth of 4, or 4 divided by 
 what ? 6 sevenths is one-seventh of 6, or 6 divided by 
 what I 5 sixths is one-sixth of 5, or 5 divided by what \ 
 
 31. 4 fifths is written f 6 sevenths is written ^ ; 
 4 sixths is written |. What does the number under 
 
FRACTIONS. 83 
 
 the line show ? What does the number over the line ? 
 What figures, and how placed, stand for 1 sixth? 
 
 2 fifths? 3 sevenths? 4 sevenths? 3 sixths? 1 
 fifth ? 1 seventh ? 4 and 2 sevenths ? 5 and 5 
 sixths? 100 and 3 fifths? 
 
 32. Read the following fractions: J, f, J T % ^S Y-? 
 V, f, I, J, -4*, 47i, 3i, 5f, 20i. 
 
 LESSON III. 
 
 1. If a line be divided into 8 equal parts, what is 
 each one of those parts called ? ANS. One-eighth of 
 the line. If the line is divided into nine equal parts, 
 what is each part ? ANS. One-ninth. If the line be 
 divided into 10 equal parts, what is each part? ANS. 
 One-tenth of the line. 
 
 2. How do you find the eighth, ninth, and tenth of 
 any thing? How do you get 4 eighths of a thing? 
 
 3 ninths ? 7 tenths ? 
 
 3. How many eighths of a dollar in one dollar? 
 in 2 dollars ? 4 dollars ? How many ninths of a 
 gallon of wine in one gallon ? 3 gallons ? How many 
 tenths of a dollar in 5 dollars? 
 
 4. How many eighths inl? 2? 3? 4? 5? 6? 
 7? 8? 9? 10? 11? 12? 20? 30? 40? 22? 35? 
 44 ? 100 ? 200 ? 
 
 5. HW many ninths inl? 2? 3? 4? 5? 6? 7? 
 8? 9? 10? 11? 12? 13? 14? 15? 16? 20? 
 24? 30? 40? 100? 120? 300? 900? 
 
 6. How many tenths inl? 2? 3? 4? 5? 6? 7? 
 8? 9? 10? 11? 12? 13? 14? 15? 16? 20? 
 37? 49? 56? 100? 240? 500? 1000? 
 
 7. How many bushels do 16 eighths of a bushel 
 make ? How many dollars are equal to 21 dimes or 
 tenths of a dollar ? How many yards of cloth are 
 
 Iual to 27 ninths of a yard ? 
 8. How many ones in 8 eighths ? 10 eighths ? 12 
 
84: 
 
 FRACTIONS. 
 
 eighths ? 16 eighths ? 24 eighths ? 32 eighths ? 
 eighths ? 35 eighths ? 39 eighths ? 40 eighths ? 
 eighths ? 56 eighths ? 64 eighths ? 71 eighths ? 
 eighths ? 100 eighths ? 800 eighths ? 
 
 9. How many ones in 9 ninths ? 
 ninths? 18 ninths? 25 ninths? 
 ninths? 36 ninths? 38 ninths? 
 ninths? 54 ninths? 58 ninths? 
 ninths? 72 ninths? 81 ninths? 
 ninths ? 126 ninths ? 
 
 1.0 ninths? 
 
 -27 ninths? 
 
 45 ninths ? 
 
 63 ninths? 
 90 ninths? 
 
 33 
 46 
 
 80 
 
 17 
 30 
 50 
 69 
 100 
 
 10. How many ones in 10 tenths ? 
 
 14 tenths? 20 
 
 tenths? 22 tenths? 30 tenths? 36 tenths? 40 
 tenths? 45 tenths? 50 tenths? 53 tenths? 60 
 tenths? 61 tenths? 70 tenths? 72 tenths? 80 
 tenths? 88 tenths? 90 tenths? 99 tenths? 100 
 tenths? 150 tenths? 200 tenths ? 305 tenths? 1000 
 tenths ? 
 
 11. If yon wish to divide 17 pounds of candy equally 
 among 8 boys, how many pounds and what parts of a 
 pound would you give to each? If you divided 17 
 pounds of candy equally among 9 boys, what would 
 be the share of each ( If it was divided equally among 
 10 boys, what would be the share of each ? 
 
 12. How many eighths in 5 and 2 eighths ? in 20 
 and 5 eighths ? in 3 and 7 eighths ? 10 and 3 eighths? 
 How many ninths in 5 and 2 ninths ? 10 and 6 ninths ? 
 12 and 8 ninths? How many tenths in 6 and 3 
 tenths ? in 25 and 7 tenths? in 30 and 9 tenths? 
 
 13. How much is 1 eighth of 3? Show by means 
 of lines that 1. eighth of 3 is 3 eighths of 1. Show 
 that 1 ninth of 2 is 2 ninths of 1. Show that 1 
 tenth of 3 is 3 tenths of 1. 
 
 14. How much is 1 ninth of 2? 3? 4? 5? 6? 
 10? 15? 9? 20? 24? 18? 27? 36? 45? 54? 
 63? 72? 75? 81? 88? 90? 100? 900? 1800? 
 
 15. How much is 1 eighth of 2 ? 3 ? 4 ? 5 ? 10? 
 15? 16? S? 21? 24? 32? 36? 40? 44? 48? 
 50? 56? 64 2 72? 80? 96? 97? 100? 800? 1600 
 
 cu . 
 
 )0? 
 
FRACTIONS. 85 
 
 1,6. How much is 1 tenth of 2 ? 3 ? 4 ? 5 ? 6 ? 
 10? 12? 14? 17? 19? 20? 30? 35? 48? 60? 
 67? 70? 80? 89? 90? 100? 111? 500? 120? 
 600 ? 1000 ? 
 
 17. How much is 1 'eighth of 24 ? 2 eighths of 24 ? 
 3 eighths of 24 ? 4 eighths of 16 ? 6 eighths of 40 ? 
 7 eighths of 32 ? 2 eighths of 72 ? 10 eighths of 80 ? 
 
 18. How much is 1 ninth of 27? 2 ninths of 27? 
 
 5 ninths of 27 ? 10 ninths of 27? 4 ninths of 36? 
 
 6 ninths of 72 ? 8 ninths of 100 ? 
 
 19. How much is 1 tenth of 50 ? 5 tenths of 50 ? 
 6 tenths of 50 ? 11 tenths of 50 ? 3 tenths of 40? 4 
 tenths of 70? 7 tenths of 210? 12 tentfis of 120? 
 9 tenths of 110? 
 
 120. 3 is one-eighth of what number ? One-eighth of 
 hat number is 4? 2? 5? 7? 6? 8? 9? 10? 
 H 14? 25? 100? 200? 80? 
 21. One-ninth of what number is 1 ? 2 ? 3 ? 4 ? 
 ? 6? 7? 8? 10? 12? 19? 22? 31? 100? 
 )0? 90? 
 
 22. One-tenth of what number is 1 ? 2? 3? 4? 
 5? 6? 8? 9? 10? 11? 12? 17? 20? 36? 45? 
 100 ? 120 ? 200 ? 
 
 23. 2 is what part of 20? 18? 16? 14? 12? 10? 
 8? 6? 4? 2? 5 is what part of 10 ? 5? 20? 15? 
 30 ? 25 ? 40 ? 35 ? 45 ? 50 ? 1 is what part of 8 ? 
 
 ;10 ? 10 is what part of 80 ? 100 ? 90 ? 100 is 
 hat part of 100 ? 900? 800? 1000? 
 
 24. How do you express in figures one-ninth ? one- 
 snth ? one-eighth ? five-eighths ? six-ninths ? four- 
 inths ? eight and three-tenths ? nine and four-eighths ? 
 
 o hundred and seven-tenths. 
 
 25. Eead the fractions |, -f^-, f , ^ 5|, 7-f, 8^-, 
 M$. 
 
 26. The numbers above the lines in a fraction are 
 ailed what ? 
 
 ANS. The numerators. 
 
86 FRACTIONS. 
 
 27. The numbers below the line are called what ? 
 ANS. The denominators. 
 
 28. The denominator shows in how many parts the 
 unit is divided. It denominates or names the parts. 
 And the numerator shows hovP many of these parts 
 form the fraction. It numbers the parts taken. In 
 how many parts is the unit divided, and how many of 
 these parts are taken in -f ? ? f ? ^ ? I ? f i ? 
 
 29. If we divide any thing into 11 equal parts, what 
 is each one of these parts called ? Into 12 equal parts ? 
 Into 20 equal parts? Into 35 equal parts? Into 60 
 equal parts ? Into 100 equal parts ? 
 
 30. How many twelfths in 3 ? 4? 5? 12? How 
 many twentieths in 1? 2? 3? 20? How many 
 Imndredths in 5 ? 6? 7? 10? 
 
 31. How many ones in ff ^, fjft, |f ? 
 
 32. How much is -^ of 12 ? 24:? 25 '{ 60 ? How 
 much is ^5- of 60 ? of 90 ? 120 ? How much is 
 
 of 200? 300? 360? 
 
 33. We know now three sorts of fractions. 1st. 
 Proper fractions, as ^, f , f , -^, in which the numera- 
 tor is less than the denominator. 2d. Improper frac- 
 tions, as |-, f, -2^-, -{-f , -2-lp-, in which the numerator is 
 equal to or is greater than the denominator. 3d. 
 Mixed numbers, as 4f , 5^-, 7|, &c., made up of whole 
 numbers and fractions. 
 
 REMARK. An improper fraction always stands for 
 either a whole or mixed number. 
 
 34. If you increase the denominator of a fraction, do 
 you increase or diminish the fraction ? If you increase 
 the numerator of a fraction, do you increase or dimin- 
 ish the fraction ? 
 
FRACTIONS. 87 
 
 LESSON IV. 
 
 1. If an orange costs 9 cents, what is the cost of % 
 of it? f of it? 
 
 2. If a bushel of potatoes costs 60 cents, what is 
 the cost of ^ of a bushel ? of f of a bushel ? of 4 of a 
 bushel? off of a bushel? of f of a bushel ? What is 
 the cost of T V of a bushel ? of -^ of a bushel ? of -J- of 
 a bushel ? of f of a bushel ? of T ^- of a bushel ? of T 7 T 
 of a bushel ? of ^ of a bushel ? of V of a bushel ? of 
 fg- of a bushel ? 
 
 3. If an acre of land costs 72 dollars, what is the 
 cost of -| of the acre ? of f of it ? of ^ of it? of -^ 
 of it? of i of it? of \r of it? off of it? off of it? of 
 T *g-ofit? of -fa of it? off of it? 
 
 4. If a railway car travels 40 miles an hour, how 
 many miles does it go in ^ of an hour ? in -f- of an 
 hour ? in -|~J of an hour? in ^ of an hour ? 
 
 5. If a man earns 48 dollars a month, how much 
 does he earn in % of a month ? in f of a month ? in -fa 
 of a month ? in f of a mouth ? in |- of a month ? in 
 ^ of a month? in f of a month ? in -^ of a month ? 
 in -/ of a month ? 
 
 6. If a man can do a piece of work in 90 days, how 
 long will it take him to do the work ? ^ of the work ? 
 | of the work ? -j 3 ^ of the work ? How long will it 
 take him to do -fa of the work ? f of the work ? f of the 
 work ? f of the work ? -^ of it ? ^ of it ? f -| of it ? 
 
 7. 84 gallons of water flow from a spring in one hour, 
 how many gallons flow out in ^ an hour '( in f of an 
 hour? in*f of an hour ? in f of an hour ? in f of an 
 hour? in \% of an hour? in ||- of an hour? in ^\ of 
 an hour ? in |-f of an hour ? in ^f of an hour ? 
 
 8. If -^ of an acre of land costs 5 dollars, what is 
 the whole acre worth ? What is -$ of the acre worth ? 
 fa of the acre ? 5 is -fa of what number j 
 
88 FRACTIONS. 
 
 9. If ^ of a yard of cloth costs 4 dollars, what is the 
 cost of the whole yard ? the cost of | of a yard ? 
 
 10. If 20 gallons of water flow from a spring in \ 
 of an hour, how many gallons flow in one hour ( in | 
 of an hour ? in f- of an hour ? 
 
 11. If -J- of an ounce of the otto of roses costs 5 
 guineas, what is the cost of 1 ounce ? of | of an ounce ? 
 of -^- of an ounce ? 
 
 12. If one pound of sugar costs 30 cents, what part 
 of a pound can be bought for 2 cents ? for 3 cents ? 
 5 cents ? 6 cents ? 10 cents ? 15 cents ? 
 
 13. If one yard of ribbon costs 40 cents, what part 
 of a yard costs 10 cents ? 4 cents ? 8 cents \ 20 cents ? 
 
 5 cents ? 1 cent ? 2 cents ? 
 
 14. A minute contains 120 half seconds, what part 
 of a minute do 12 half seconds make ? 10 half seconds ? 
 
 6 half seconds? 15 half seconds? 8 half seconds? 20 
 half seconds? 40 half seconds? 30 half seconds? 60 
 half seconds ? 1 half second ? 2 half seconds ? 3 half 
 seconds ? 
 
 15. A whole hive of bees yield 36 pounds of honey 
 in one year, what part of the hive yields 6 pounds ? 9 
 pounds ? 12 pounds ? 4 pounds ? 2 pounds ? 18 pounds ? 
 
 16. How long will it take a vessel which sails 10 
 miles an hour, to sail 67 miles ? 45 miles? 102 miles? 
 507 miles? 422 miles? 53 miles? 
 
 17. At 9 dollars a barrel, how many barrels of flour 
 can be bought for 28 dollars ? 39 dollars \ 69 dollars ? 
 71 dollars ? 
 
 18. If corn costs 5 dollars a barrel, how much can be 
 bought for 47 dollars? 54 dollars? 61 dollars? 23 
 
FRACTIONS. 89 
 
 LESSON V. 
 
 1. If 5 bushels of wheat cost 9 dollars, what is the 
 cost of 10 bushels ? 
 
 PROCESS. 10 bushels is 2 times 5 bushels, hence cysts 
 twice as much, or 18 dollars. 
 
 2. If 12 bushels of wheat cost 20 dollars, what is 
 the cost of 3 bushels ? 4 bushels ? G bushels ? 2 
 bushels? 1 bushel? 
 
 PROCESS. 3 bushels is ^ of 12 bushels, and costs 
 erefore ^ of 20 dollars, or 5 dollars. 
 
 3. How many dozen in 10T ? 112 ? 145 ? 26 ? 
 
 4. 4 men can do a piece of work in 45 days,- in how 
 many days can 12 men do the same work ? 20 
 men ? 40 men ? 30 men ? 60 men ? 32 men ? 28 
 
 len? 
 
 5.. If each yard of a certain cloth cost 2^ dollars, 
 what is the cost of 5 yards ? Twice 5 and 1 fifth of 5 
 are how much ? 
 
 6. If a man travels 7 miles an hour, how far does 
 he travel in 1^- hour? in 4 and f hours? in 6| 
 hours ? 8 f- hours ? 
 
 7. How much is 3 times 10 and -^ of 10 ? 5 times 10 
 and -& of 10 ? 8 times 10 and -& of 10 ? 10 times 10 
 and T 8 oflO? 
 
 8. How much is 9 times 8 and -| of 8 ? 10 times 8 
 and | of 8 ? 7 times 8 and f of 8 ? 
 
 9. At 2 dollars a bushel, what will be the cost of 
 \\ bushel of wheat? 5 bushels? 100 bushels? 
 25 bushels ? 
 
 10. A boy gave away 7 apples, which was -J of the 
 number of apples which he had ; how many apples 
 had he ? 
 
 11. Frank is 10 years old and is -J- as old as his father, 
 how old is his father ? 
 
 12. If | of an acre of land costs 25 dollars, what is 
 
90 FRACTIONS. 
 
 the cost of -|- of the acre ? Of the whole acre ? Of -f- 
 of the acre ? 
 
 13. If T 9 of a gallon of water leaks from a vessel in 
 27 hours, in how many hours will -fa of a gallon leak 
 out.? In how many hours will one gallon leak out ? 
 -fa of a gallon ? -^f of a gallon ? 
 
 14. Two boats are moving in the same direction ; 
 one gains on the other % of a mile in 35 minutes ; in 
 how many minutes does it gain -J- of a mile ? 1 mile ? 
 f of a mile ? 
 
 15. A merchant gains ^ of a dollar in each sale of 
 8 dollars ; what sales must he make in order to gain -J- 
 of a dollar ? 1 dollar ? of a dollar \ 
 
 16. 8 is |^ of what number? Is -J of what number? 
 -ig 9 - of what number ? ^ of what number ? f of what 
 number ? |- of what number? 
 
 PROCESS. If 8 is - of the number, then ^ of 8, or 
 2, is ^ of the number. If \ of the number is %,%of it, 
 or the entire number, is 6 times 2, or 12. 
 
 17. 3 is f of what number? Is f of what number? 
 if- of what number ? 
 
 18. 8 is f of what number? of what number? 
 -^ of what number? T \- of what number? yf^j- of 
 what number ? -^ of what number ? -| of what num- 
 ber ? 
 
 19. 12 is f of what number? -| of what number? 
 -ff of what number? |f of what number? -j^- of 
 what number ? 
 
 20. 9 is f of what number? |- of what number? 
 -^ of what number ? T 9 g- of what number ? -| of what 
 number ? ^ of what number ? 
 
FRACTIONS. 
 
 91 
 
 LESSON VI. 
 
 lV IjJT A "A T 6 TV TIT A A 
 
 1. If a line be divided into two equal parts, and 
 of these parts be divided into two equal parts. 
 
92 FRACTIONS. 
 
 into how many equal parts will the line be divided ? 
 One half is equal to how many fourths ? 
 
 2. If a line be divided into two equal parts, and 
 each of these parts into three equal parts, what part of 
 the whole line will each of those parts be ? One half 
 is equal to how many sixths ? 
 
 3. If we divide each half of the line into 4 equal 
 parts, how many of these parts will the whole line con- 
 tain ? % is equal to how many eighths ? ^ is equal to 
 how many tenths ? 
 
 4. In what fractions can halves be expressed ? 
 ANS. In fourths, sixths, eighths, tenths, cfe., whose 
 
 denominators are multiples of 2. 
 
 5. If a line be divided into three equal parts, and 
 each of these parts successively into 2, 3, 4, 5, &c., 
 equal parts, what part of the line will each of these 
 parts be respectively ? 
 
 6. How many sixths, ninths, twelfths, fifteenths, 
 twentieths ? 
 
 7. In what equivalent fractions can thirds be ex- 
 pressed ? 
 
 ANS. In sixths, ninths, twelfths, &c., whose denomi- 
 nators are multiples of 3. 
 
 8. If a line be divided into 4 equal parts, and 
 each of these parts into 2 equal parts, in how many 
 parts will the line be divided ? % is equal to how 
 many eighths ? How many sixteenths ? How many 
 twelfths ? 
 
 9. One fifth is how many tenths ? How many 
 twentieths ? How many fifteenths. 
 
 10. One sixth is how many twelfths ? How many 
 eighteenths ? How many twenty-fourths ? 
 
 11. What are the denominators of equivalent frac- 
 tions in which fourths may be expressed ? 
 
 12. What are the denominators of equivalent frac- 
 tions in which fifths may be expressed ? 
 
 13. What are the denominators of equivalent frac- 
 tions in which sixths may be expressed ? 
 
FRACTIONS. 93 
 
 14. Express in parts of a unit 3 times as small. 
 In parts 5 times as small. 10 times as small. In 
 parts 100 times as small. 
 
 15. Express f in parts 5 times as small. Find four 
 fractions, each of which is equal to f . f is equal to 
 how many forty-fourths? 
 
 16. How many parts of a unit, 10 times as small, is 
 % equal to ? How many parts of a unit 10 times as 
 small is ^ equal to ? Find 5 fractions, each equal to 
 f . f is equal to how many sixtieths ? 
 
 17. Find 10 fractions, each of which is equal to f. 
 How many parts of a unit 15 times as small is -J- equal 
 to ? Is f "equal to ? Find 6 fractions, each of which 
 is equal to -| . 
 
 18. 1 = = ^ = ^ = \ = \= \ === T^ = -gV = 
 
 1 _ 2 __ _ JLQ.Q. 
 
 4 Q n - ft o 0' T ~ ! 
 
 (Read 1 = how many halves, &c., &c. ? 1=2 
 what? 3 what? &c.) 
 
 1 Q Q ! ? _ ! f _ 12. 16. JJ fifi. 
 
 1J. ^ -g- 3- -j-g- TO"0~~ I ? ' 
 
 90 q ? _ t -I & 12 4i 
 
 ^U. O = -3 T7 T Q- -3-^- , - 
 
 21. What multiples must the denominators of frac- 
 tions be in which sevenths can be expressed ? In 
 which eighths can be expressed ? In which ninths can 
 be expressed ? In which tenths can be expressed ? 
 
 22. Give 5 fractions, equal to \. Give 5 equal 
 to |. Give 5 fractions equal to . Give 5 equal 
 to T V 
 
 23. Express f in parts 4 times as small. Express 
 f in parts 6 times as small. T 9 F is equal to how many 
 twentieths ? how many fiftieths ? how many hun- 
 dredths ? how many two hundredths ? | is how many 
 forty-eighths ? 
 
 24. To change fifths into tenths, into how many 
 equal parts must each fifth be divided ? To change 
 fifths into fifteenths, into how many ? into twentieths, 
 into how many ? into thirtieths, into how many must 
 each be divided ? 
 
94 FRACTIONS. 
 
 25. T f- are how many forty-eighths ? how many 
 sixty-fourths ? how many ninety-sixths ? 
 
 26. Change f, y^, y 7 -^, and -J-J-, each to thirtieths. 
 
 27. Change f, f, If, and ^ each to ninetieths. 
 
 28. Change f , f , f , and f to tenths. 
 
 29. Change y^ to three-hundredths. Change T 5 
 to thirty-ninths. Change -J to ninety-sixths. 
 
 30. How many 72ds in f ? f? f ? f? 1? -U-? 
 
 fl? H *** 
 
 31. How many IGOths in ? f ? ? f ? 
 
 32. How many lOOths of a dollar make y 1 ^ of a dol- 
 lar ? How many half a dollar ? 
 
 33. How do you find a fraction of a greater denom- 
 inator equal to a given fraction ? 
 
 LESSON VII. 
 
 1. A line is divided into 10 equal parts; what part 
 of the line will T 5 ^ make ? If a line is divided into 12 
 equal parts, what part will T 6 ^- make ? 
 
 2. Reduce A; *5 A; i ; WV; W; to halves. 
 What must be done in order to bring T 6 ^- back to -J- ? 
 
 3. Change ^ to fifths. Change |-f- to fifths. Change 
 -^ to halves. Change |-|- to thirds ; also to fourths. 
 
 4. T 8 2- of a bushel is how many sixths of a bushel ? 
 How many thirds of a bushel ? 
 
 5. Express ^- in parts of a unit twice as great. 
 Express it in parts of a unit 5 times as great. Ex- 
 press it in parts of a unit 10 times as great. 
 
 6. |-f is equal to what fraction with a denominator 
 half as" great ? One-third as great ? as great ? -J- 
 as great 2 -|- as great ? T ^- as great ? -g^ as great ( 
 
 7. How do you change a fraction into an equal 
 fraction of smaller denominator ? In order to express 
 a fraction in parts 3 times as great, the numerator and 
 
FRACTIONS. 95 
 
 denominator must be divisible by what ? In parts 4 
 times as great, divisible by what ? In parts 6 times 
 as great ? 
 
 8. Can -f-, f, f 3 f, |, TV b e expressed exactly in 
 greater parts of a unit ? Can they be expressed in 
 smaller parts of a unit ? 
 
 9. ^ = f = |- = 1-g-g-, &c. |- is said to be in lower 
 terms than any of its equivalent fractions, being ex- 
 pressed in smaller figures. In the same manner -f- is 
 a fraction at its lowest terms, or simplest form. 
 
 10. When is a fraction in its simplest form ? 
 
 ANS. When its numerator and denominator have 
 no common divisor greater than 1. 
 
 11. Reduce f f to its simplest form. 
 
 12. Reduce -J-f to its simplest form. 
 
 13. Reduce f|- and y 2 ^ each to its simplest form. 
 
 14. Reduce -f|- to its simplest form. 
 
 15. Reduce |f to its simplest form. 
 
 16. Reduce -f/-^ to its simplest form. 
 
 17. How do you reduce a fraction to its simplest 
 form ? 
 
 18. If you multiply the denominator of the fraction 
 f by 4, what change do you make in the value of the 
 fraction \ If you multiply the numerator of the frac- 
 tion f by 4, what change do you make in the value? 
 
 19. If you multiply both the numerator and de- 
 nominator of f by 4, the fraction | * | = = | ; why ? 
 
 ANS. Because we take 4 times as many parts , each 
 of which is one fourth as great. 
 
 20. If you divide the denominator of the fraction T 8 j 
 by 4, what change of value do you make in the frac- 
 tion ? If you divide the numerator by 4 what change 
 do you make ? If you divide both numerator and de- 
 nominator by 4 you make no change ; why ? 
 
 ANS. Because you take one fourth as many parts, 
 each of which is 4 times as great. 
 
 NOTE. The teacluT should ilhr^trato this with other examples until 
 it is perfectly clear to each punil. 
 
96 FRACTIONS. 
 
 21. How many 9ths in -|| ? How many 7ths in 
 
 22. How many 8ths in f ? In f f ? 
 
 23. Eeduce to the lowest terms the following frac- 
 3ns : 
 
 tions : 
 
 A? 
 
 *- i. VJ 1 T V-> 
 
 lower terms, 
 
 24. Give tive fractions which cannot be reduced to 
 >wer terms. 
 
 LESSON VIII. 
 
 1. How do you express % and -J- by two other 
 fractions having tKe same denominator how do you 
 proceed ? 
 
 ANS. f $ = | = T V <fic. t = f = | A, 
 dec., a*# ^<3 6'^;? express either ^ oy f <m6? -J 5?/ f ? ^^ i 
 ^y yV? a?7y ^ i fy/ 1^5 an d g n higher and find others. 
 
 2. What is the 6 or 12 called in this case ? 
 
 ANS. 77/^ common denominator ', a;i<^ mw6'^ J^ # 
 common multiple of 2 an^ 3. 
 
 3. What is the least common denominator which 
 any two fractions equal to ^ and ^ can have ? 
 
 ANS. 6, /^ least common multiple of 2 and 3. 
 
 4. What is the least common denominator of ^ and 
 i? Of I- and |? Oft and i? Of^andf? 
 
 5. What are 6 of the common denominators of f 
 andf? 
 
 PKOCESS. 6 common multiples of 3 and 4 are 12, 
 24, 36, 48, 60, 72. Hence these are 6 of the common 
 denominators of fractions equal to f and f respectively. 
 
 6. What is the least common denominator of f 
 and f ? 
 
 7. Reduce the fractions f and to the least common 
 denominator. 
 
 PROCESS. The least common multiple of 8 and 5 is 
 
 JA 3 15 
 
 ^ u 
 
FRACTIONS. 97 
 
 8. Reduce ^, ^, and to the least common denomi- 
 nator. 
 
 9. What is the least common denominator of f and 
 
 A? 
 
 10. What is the least common denominator of -- and 
 
 A* 
 
 11. What is the least common denominator of -^ and 
 
 _9_2 * 
 
 10* 
 
 12. What is the least common denominator of T 4 T 
 and ^ ? 
 
 13. What is the least common denominator of f and 
 
 A? 
 
 14. What is the least common denominator of ^, J-, 
 and f ? 
 
 15. What is the least common denominator of ^ 
 
 ru> and i 2 
 
 16. What is the least common denominator of -|-, T ^-, 
 and -^ ? 
 
 Reduce to common denominators 
 
 17. f and f. 20. T V and -^. 
 
 18. f andf 21. 1 and ^ 
 
 19. f and f. 22. ||- and ^. 
 
 23. Reduce to the least common denominator % 
 and T V 
 
 24. Reduce to the least common denominator y 1 ^ 
 and ^-. 
 
 25. Reduce to the least common denominator ? T 3 , 
 and -J-. 
 
 26. Reduce to the least common denominator , 5 
 
 27. Reduce ^, -j^, and ^ to the least common de- 
 nominator. 
 
 28. Reduce , f , f, |, -j^- and ||- to the least coin- 
 
 >n denominator. 
 29. How do you reduce fractions to common de- 
 minators ? 
 
 30. How do you reduce fractions to the least com- 
 7 
 
 
98 FRACTIONS. 
 
 mon denominators, when the denominators have no 
 common factor? 
 
 31. How do yon find the least common denominator, 
 when, the denominators have a common factor ? 
 
 LESSON IX. 
 
 1. A boy gave his brother ^ of an apple, and his 
 sister of an apple ; how many fourths did he give to 
 both? 
 
 2. A man sold a bushel of apples for -| of a dollar, 
 a bushel of potatoes for | of a dollar, and a bushel of 
 meal for -| of a dollar. How many dollars did he re- 
 ceive for all ? 
 
 3. A farmer divided his land into several lots. One 
 lot contained | of an acre, a second lot ^ of an acre, 
 a third lot -f- of an acre, and a fourth lot if- of an acre. 
 How many acres in all the lots ? 
 
 4. One man pays me 8| dollars, and another pays 
 me 1-J- dollar ; how much do both together pay me ? 
 
 5. | and f are how much? $ + f ? J / + * ? 
 
 T 9 o 
 
 7. A workman receives from his employer at dif- 
 ferent times if of a dollar, 4^- of a dollar, 3^ of a 
 dollar, 11 T 5 T of* a dollar, and 2y 9 ^- of a dollar. How 
 many dollars did he receive altogether ? 
 
 8. A vintner has in his cellar 4 casks ; the first con- 
 tains Sy 1 ^- gallons of wine, the second 7 T 6 Q-, the third 
 f-^, and the fourth 5^ gallons ; how many gallons of 
 wine in the four casks \ 
 
 9. f-f 1 L = ? Can these be added without changing 
 them first into the same denomination ? 
 
 PROCESS. J f ; | ; + |=f.+_f = |. 
 
 10. How do you add fractions having a common 
 
FRACTIONS. 99 
 
 demominator? Plow do you add fractions of different 
 denominators? 
 
 11. i+|- f+t 
 
 13. . 
 
 PROCESS. 1|- i'<s f. Hence we add J- a^<# -J-, or 
 
 . 4i-+f=? 3f+f=? 10 + 6^4-5-1-=? 
 
 15. What is the sum of 14|, 10, and 5J ? 
 
 16. What is the sum of 6^, 8|, and 5f ? 
 
 17. What is the sum of ^ 12|, 15}, and 9| ? 
 
 18. Add f , 3J, 4, and 5f. 
 
 19. A man planted 2^- acres in corn, sowed lOf- 
 acres in oats, and 5f acres in wheat. How many 
 acres had he in corn, wheat, arid oats ? 
 
 20. I bought 2|- bushels of potatoes of one man, 6f 
 bushels of another, and 4} bushels of a third. How 
 many bushels did I buy from the three men ? 
 
 LESSON X. 
 
 1. I purchased a lot containing | of an acre, and cut 
 off from it a lot containing | of an acre. How many 
 fourths of an acre did the other part contain ? 
 
 2. Mary had-| of an orange, and gave f away. 
 How much had she left ? 
 
 3. Charles had a melon, and gave T 3 of it away. 
 How much had he left ? 
 
 4. Julian had 5 oranges, and gave away f of one of 
 them. How many had he left ? 
 
 5. From $ take -f$. From y 2 ^ take yW. From 
 take T V From 18f take 13|, From lOf take 6f . 
 
 6. How much is 1 less $ ? 2 less -J- ? 3 less f ? 15 
 lessf? 16 less 2-J- ? 5 less If? 10 less T V? 5 less 4 f? 
 
 H 
 
 
100 FRACTIONS. 
 
 T^S 
 
 7. How much is less ^ ? 
 SOLUTION. |- 1==|- f f 
 
 How much is | less ? -f less f ? -f- less | ? f less 
 1? flessj-? 
 
 8. What is the difference between ^ and | ? -| and 
 A a*d ft? A and -ft f and -^ ? 
 
 9. From 3 take 2f . 
 
 PROCESS.-^- 2f _ y- * _ ft- ff -. A- 
 From 3| take 2. How much is 3| 1| J What 
 is the difference between lOf and 9|- ? 
 How much is 
 
 10. 3J-2J? 13. 7J~ 4f? 
 
 11. 4|-2f? 14. 9-i 8f ? 
 
 12. 3f--lf ? B 15. 12-i| 5f 
 
 16. What is the difference between 19 and ^Q? Be- 
 tween 12 and f ? Between 16 and 1|? 
 
 17. ^ + ^ is how much less than 1 ? 
 
 18. -|- + f + f is how much less than 2? 
 
 19. i sold ^ of my land to one man, ^ of it to an- 
 other, and -^ of it to a third. How much of it did I 
 have left ? 
 
 20. I conceive two numbers, one is J- and the other 
 is f greater what is the number? What number is 
 f less than f ? 
 
 21. Of two numbers f is the greater and -J- is the 
 difference between them what is the smaller ? 
 
 22. How much greater is % than f ? 
 
 23. How do you subtract one fraction from an- 
 other ? 
 
 24. How do you subtract a fraction from a whole 
 number ? 
 
 25. How do you subtract one mixed number from 
 another ? 
 
 
FRACTIONS. 101 
 
 LESSON XI. 
 
 1. "What is the cost of 5 pounds of tea at of a 
 dollar a pound ? 5 times three-fourths is how many 
 fourths? |+i+|.+|.+i4i| 
 
 2. If |- of a barrel of flour last a family one month, 
 how much flour will last the family 6 months? f + f 
 
 +*+*+*+*? 6 times f ? 
 
 3. How much is 15 times -|? 5 times 1? 11 times 
 
 I? 
 
 4. How much is 5 times f ? 19 times f ? 11 times 
 | ? 26 times f ? 14 times f ? 25 times f ? 10 times 
 |? 15 times |? 20 times f?^-? ? &? A? 24 
 times |? f? I? I? |? 1? >T V? 
 
 5. What is meant by multiplying 5 by ^-? 
 ANS. To take ^ of 5. 
 
 6. If a number be multiplied by a number less than 
 unity, is the product greater or less than the first 
 number ? Multiply 5 by f-. 
 
 7. How much is 10 times f ? f xlO? f of 10? 
 How much is 32 times f? |x32? f of 32? How 
 much is 23 times f ? f x 23 ? f of 23 ? 
 
 8. How do you find a given fraction of any number ? 
 
 9. A man sold 10 yards of cloth at 2-J- dollars a 
 yard ; what did he receive for the 10 yards ? 
 
 10. At 2 dollars a yard, what will 3-J- yards of cloth 
 cost? 
 
 11. At 5 dollars a barrel, what will 4 barrels of 
 flour cost? 
 
 12. If a man walks 5 miles an hour, how far will 
 he walk in 6|- hours ? > 
 
 13. How much is 5 x 6|? 6 x 7-J? 6f x 6? 9f 
 X 62 4^x3? 1^x4? 5x4A 31^x10? 
 
 x 9? 91|x 8? 20f x Y? 
 
 14. How much is 12 times 10 and T % of 10? 
 
 15. How much is 9 times 7 and of 7? 
 
102 FRACTIONS. 
 
 16. 10X T V= ? ^Xl0=? -^oflO ? 11 X 
 
 ? Wxl2? 
 
 17. How much do you get when you multiply 
 i by 1 ? 
 
 18. How much do you get when you multiply -J- 
 by ^ ? How much is -J- x \ ? 
 
 ANS. % of \. 
 
 19. If you take a line and divide it into 3 equal 
 parts, and take ^ of one of these parts, what part of 
 the whole line do you take ? If you divide the line 
 into two equal parts, and take -J- of one of these 
 parts, what part of the whole line do you take? 
 i x |- = % of -J- = -J- of |-. Show this by a line on 
 your slate. 
 
 * TT === T7T X y-Q- = . 
 
 21. f multiplied by f is 2 times -J- x J = ? How 
 much is I- off? Off? Of-jV? Of^? Of||? 
 
 PROCESS. -J x f, o/ 1 -^ of ^ is 3 x ^ ^/"^- = ^C-. 
 
 22. How much is I- of --? |of|? fof|? 
 How much is | of ^ ? -J-off? j-off? | of *f ? 
 of |? 
 
 23. How much is | xf ? fx | = f of f fof f ? 
 How much is -f x f ? -f- x -^ ? yV x iV - I x I 
 
 4 X 3 ? | X | ? | X 3 ? ^ x , f| f>< 7 ? 
 
 24. How much is -f- x | ? ^ of f ? f of | ? ^ 
 
 25. A man sold a -bushel of wheat for 1 dollars. 
 What is the cost of -J- of the bushel ? 
 
 26. How much is 1 multiplied by J ? What does 
 it mean to multiply 1|- by -J ? 
 
 ANS. To take -i- of ^^. PROCESS. 1^ is f . % of 
 
 27. Sold a barrel of flour for 7| dollars. What was 
 the cost of -- of the barrel ? 
 
FRACTIONS. 103 
 
 28. How much is 8$ x | = | of 8 ? How much 
 is 2 x i? 9i x f? 3 x | ? 4f x f ? If x 3 T V? 
 fx3f? JxlOi? 
 
 29. What is the cost of 3 yards of cloth at 1% dol- 
 lar a yard? How much is 3 x 1J? 
 
 SOLUTION. 3 x 1 =\ 3 - x f = f of *- ^- or 4f 
 
 30. How much is If x 1? 1J x 2f ? 2f x 7J-? 
 2fxl? 6fx2 T V? Hx6f? 2|x3? 1 
 
 31. Find the sum, difference, and product of f and -J. 
 Of 1| and 2. Of 2J- and 3f . Of 1 and 2f Of 7| 
 and 3f 
 
 32. How do you multiply a fraction by a whole 
 number ? How do you multiply a fraction by a frac- 
 tion, or take a part or parts of a fraction ? How do 
 you multiply a mixed number by a mixed number ? 
 
 33. A man bought a ton of coal for 17f dollars a 
 ton. What was the cost of -J- of a ton ? 
 
 34 A man bought 10 bushels of wheat for 27| dol- 
 lars. What did he pay for each bushel ? 
 
 35. 5 yards of cloth cost 18f dollars. What is the 
 cost of one yard 2 
 
 LESSON XII. 
 
 1. If an apple is divided into two equal parts, each 
 part is ^. 1 divided by 2 = ? If 2 apples each be 
 divided into 3 equal parts and ^ part of each be 
 taken, we will .have f of an apple. 2 divided by 3 
 what ? 
 
 2. What do we mean by dividing 1 by 2 ? 2 by 3 ? 
 ANS. To divide 1 into two equal parts and take 
 
 one of them, which gives ^. In the same manner to 
 divide 2 ly 3 gives f . 
 
 3. What fraction is equal to 1 divided by 5 ? l-~6 ? 
 7? i^s? 1-^-9? 
 
 4. If you divide half an orange into two equal 
 parts, what part of the orange do you get ? divided 
 by 2 is what ? . 
 
104 FRACTIONS. 
 
 5. How much is divided by 3 ? 4 ? 5 ? 6 ? 7 ? 
 8 ? 9 ? 10 ? 
 
 ANS. J- divided l>y 3 is ^ of -J-, or -|-, cfee. 
 
 6. How much is divided by 3? 4? 5? How 
 much is f divided by 3? 4 ? 5 ? 6? 7? 8? 
 
 7. A man had |- of a barrel of flour, and divided it 
 equally among 6 persons. How much did he give to 
 each ? How much is ^^6 = -J- of ? 
 
 8. A person wished to divide f of an acre of land 
 into 10 equal lots ? How much land would each lot 
 contain ? 
 
 9. Divide by 9. By 2. By 5. By 7. By 6. Divide 
 | by 11. J by 5. fl by 4. ff by 5. 
 
 10. How many halves make 1 ? How often is % 
 contained in one ? How many halves in 2 ? How 
 often is contained in 2 ? 
 
 11. A person wishes to distribute 4 barrels of flour 
 so as to give ^ of a barrel to each of a number of 
 families. How many families could he supply ? 
 
 12. How many thirds in 10 ? -J- is contained in 10 
 how many times ? is contained in 5, 6, 7, 8, 9, how 
 many times ? 
 
 13. How much is 1 divided by -J- ? 2 divided by 
 -J- ? 5 divided by -J- ? 9 divided by | ? 11 divided by 
 -iV ? 12 divided by ^ ? 
 
 14 How much is 10 divided by ^ ? f is contained 
 in 10 how many times ? 
 
 SOLUTION. J- is contained in 10, 50 fo'm<?s / hence 
 f is contained half as many times, or 25 fa'ratfs. 
 
 15. How much is 5 divided by -| ? 
 
 SOLUTION. 5 divided l>y \ gives 45 ; hence 5 di- 
 vided by -| must give a quotient 6 times as small 
 
 = 4 / = 7f 
 
 16. How much is 6 divided by f ? 6 is f of what 
 number ? How much is 6 divided by 5 ? 6 is 5 times 
 what number ? How much is 10 divided by -f- ? 10 
 is % x by what number ? 10 is -f of what number ? 
 
FRACTIONS. 105 
 
 17. How much is 6-f-f ? 6 is f of what number ? 
 
 18. How much is 4-^-J- ? 4 is ^ of what number ? 
 
 19. How much is 16-^f ? 16 is | of what number ? 
 
 20. How much is 12-^ ? 10 -Hr ? 6 divided by 
 | ? 9-T--& ? 5 divided by f ? 11 divided by ^ ? 
 
 21. How much is 100 divided by f ? | ? f ? ? 
 
 22. 2 is f of what number ? 5 ? 3 ? 7 ? 10 ? 11 ? 
 12? 15? 
 
 23. 60 is f of what number ? f of what number ? 
 -f of what number ? T ^- of what number ? 
 
 24. How do you divide a fraction by a whole num- 
 ber ? How do you divide a whole number by a frac- 
 tion? 
 
 25. If one soldier's ration of flour is 1-J- pound, how 
 many rations will 8 pounds of flour make ? 1^- is con- 
 tained how often in 8 ? 
 
 SOLUTION. 1-J- is |-, and 8-f-f is 6. 
 
 26. How often is 6f contained in 9 ? How much is 
 60 divided by 3|-? By 1-JJ 3f ? How much is 2 
 divided by 3^ ? 4^ ? How much is 5 divided by 4 ? 
 
 ^27. How much is 2f + 2 ? 8f -f- 7 ? 14f -f- 12 ? 
 
 28. If f of a ton of coal is burned in a stove in one 
 month, how long will f of a ton last for the same 
 stove? f is contained how often in f ? 3 in 9 how 
 many times ? -jf- in how many times ? -f$ in T 9 ? 
 -A- in 6 9 ? 
 
 29. f in - 2 5 - how many times ? -^ divided by f is 
 how much? How much is -i/- divided by f ? - 1 1 ^ 
 contains ^ how many times? -i|-Q--f-^-= ? -^^-v- 
 14 = ? -^gQ-~|-g= ? J^--^.|=? . contains -| how 
 many times ? f divided by f is how much ? 6 con- 
 tains 5 how often ? 5 divided by 6 is how much ? 
 
 30. 1 divided by ^ is how much ? |- divided by -^ 
 is how much ? 
 
 ANS. Since 1 -=- ^ is 3, |- -f- -J ?m^ l)e ^ of 3, 0r f . 
 
 
106 FRACTIONS. 
 
 31. How much is divided by -J- ? 
 
 Asrs. J- = ^ <md -J- = ^.;^--f-^ 5 . =3 6 divided ly 
 
 5 I = H- 
 
 32. How much is J- divided by \ ? 
 
 PROCESS. -^-J-= |. Hence, ^~\ _ 5 ^'m0s f = - 1 /- 
 
 33. How much is % divided by |- ? 
 
 PROCESS. | divided by ^ is f , <mrf % divided ~by f 
 ^tf be | </ f ^. 
 
 34. How much is f divided by f ? 
 
 ANS. f divided by ^ is ^ and f divided by |- ^ 
 
 . 
 
 35. How much is-^-==f x|? 
 
 36. How much is -f^-f^ =& x ^ ? 
 
 37. How much is -^-J = T \ of f = T 3 3- x f ? 
 
 38. How much is divided by f ? by f ? f 
 
 f A? 
 
 39. How much is f divided by T V ? by f ? | ? 
 
 40. How much is | divided by T V ? | ? f 'i f ? 
 
 41. How do you divide one fraction by another ? 
 
 42. A tailor having 8f yards of cloth, cut it into 
 pieces, each 2|- yards long how many such pieces 
 did he get ? 2J is contained how often in 8| ? 
 
 PROCESS. 2 is f, 8f is -^ -^ divided ~by f =- a /- x 
 
 .- 
 
 43. How much is 1|- divided by 2f| 24- divided by 
 1|- ? If divided by If ? If -f- 2f ? 6| Vlf 
 
 44. How much is 2^ divided by 64 ? 5|- divided by 
 4f? 5|- divided by 2f ? divided by 3| ? 4f divided 
 
 45. If one bushel of wneat cost f of a dollar, what 
 is the cost of -f- of a bushel ? -i^- of a bushel ? 5 
 bushels ? lOf bushels ? If bushel ? 
 
 46. If one bushel of wheat cost f of a dollar, how 
 much wheat can you buy for -| of a dollar? -^ of a 
 dollar? 5 dollars? 10-} dollars? If dollar? 
 
 NOTE. Let the pupil make 5 problems like the above, and solve 
 them. 
 
FRACTIONS. 107 
 
 47. Add % and -J, and multiply the sum by f . 
 
 48. From 3 take f, and divide the remainder by f . 
 
 49. Multiply by 1^, and divide the product by f. 
 
 50. Multiply 2^ by J-, and divide the product by 4^. 
 
 51. Is the dividend always larger than the divisor? 
 Is the quotient always smaller than the dividend ? 
 
 Give examples. 
 
 52. Does the quotient always give the number of 
 times the divisor is contained in the dividend ? 
 
 LESSON XIII. 
 
 1. 40 is |- of what number ? 
 
 ANALYSIS. 40 is -| x the number. Hence the num- 
 ber is 40 divided ~by |= 40 -=- 5 and quotient x 8. 
 
 2. 27 is T ?g- of what number ? J- of what number ? 
 
 3. 32 is |- of what number? f of what number? 
 
 4. 60 is f- of what number ? -ff- of what number ? 
 
 5. 72 is |- of what number ? ^ of what number ? 
 
 6. 48 is f of what number? -^ of what number? 
 
 7. 49 is T^ of what number? J- of what number? 
 
 8. 54 is T 9 T of what number ? -f- of what number ? 
 
 9. 63 is -| of what number ? |- of what number ? 
 
 10. 20 is f of what number ? ^ of what number? 
 
 11. 10 is f of what number? 
 
 SOLUTION. T/ 7 10 is % of a number f then % of 10, or 
 31, i<? of the same number ; and 3^- is % of 2 x 3-J-, 
 or 6f . 
 
 16 is $ of what number? 
 
 12. 15 is f of what number ? f of what number ? 
 
 13. 50 is f of what number ? -y- of what number ? 
 
 14. 9 is | of what number? | of what number ? 
 
 15. 11 is f of what number ? f of what number ? 
 
 16. 5 is i of what jmmber ? f of what number ? 
 
 17. 15f is f of what number ? of what number ? 
 
 18. 7| is f of what number? T V of what number? 
 
 19. f is f of what number ? | of what number ? 
 
 I 
 
108 FRACTIONS. 
 
 20. ^5- is f of what number ? ^ of what number ? 
 
 21. TT * s f f what number? f of what number? 
 
 22. A man paid 37 dollars for f of an acre of land. 
 What was the cost of the whole acre at this rate ? 
 
 23. A man sold his horse for thirty Collars, which 
 was |- of what it cost him. What was the cost of the 
 horse ? 
 
 NOTE. The pupil will form four questions like the above, and give 
 the solutions. 
 
 LESSON XIV. 
 
 1. 20 is ^ of how many times 5 ? 
 SOLUTION. 20 is -fa of 50, and 50 is 10 times 5. 
 
 2. 72 is y 9 ^ of how many times 16 ? 
 
 3. 60 is -j^- of how many times 18 ? 
 
 4. 24 is f of how many times 2? 
 
 5. 48 is -f of how many times 19? 
 
 6. 12 is |- of how many times 6 ? 
 
 7. 11 is f of how many times 5 ? 
 
 8. 36 is -^ of how many times 11 ? 
 
 9. 20 is -^ of 5 times what number ? 
 SOLUTION. 20 is -^ of 50, and 50 is 5 times 10. 
 
 10. 16 is f of 6 times what number? 
 
 11. 40 is -fy of 16 times what number? 
 
 12. 18 is -f- of i.0 times what number? 
 
 13. 60 is 4 of 12 times what number? 
 
 14. 20 is how many tenths of 10 times 5 ? 
 
 15. 20 is 4 times what part of 10 times 5 ? 
 
 16. What number is -^ of 10 times 5 ? 
 
 17. 48 is of how many thirds of 36 ? 
 
 18. 54 is T 6 T of how many times -^ of 48 ? 
 
 19. 55 is -| of how many sixths of 24? 
 
 20.* 64 is f-f of how many nineteenths of 38 ? 
 
 21. 9 is | of how many sixths of 18 ? 
 
 22. f is f of how many times of 10 ? 
 
 23. 8 J is f of how many times J of 24 ? 
 
FRACTIONS. 109 
 
 24. 7i is | of how many times -J- of 10 ? 
 
 25. i of is f of how many times ^ of 100 ? 
 
 26. f of f is ^ of how many times -J- of ^ ? 
 
 27. A man bought a horse for $60, which was - of 
 10 times what it cost to keep the horse one month. 
 How much did it cost to keep the horse a month ? 
 
 NOTE. The pupil will frame three questions similar to question 27, 
 and give the solutions. The teacher will vary some of the questions, 
 beginning with question 17, after the manner in which question 1 is 
 varied in 9, 14, 15, and 16. 
 
 LESSON XV. 
 
 1. f of 15 is how many fifths of 100 ? 
 SOLUTION. | of 15 is 20. 20 is \ of 100. 
 
 2. f of 15 is -|- of what number ? 
 
 3. f of what number is -J- of 100? 
 
 4. f- of 48 is how many fourths of 12 ? 
 SOLUTION. f of 48 is 30. % of 12 is 3. 30 is 10 
 
 times 3, or 10 times of 12, or 10 fourths of 12. 
 
 5. ^ of 60 is how many tenths of 30 ? 
 
 6. -| of 49 is how many thirds of 42 ? 
 
 7. f of 9 is how many fourths of 12 ? 
 
 8. f of 21 is how many fourths of 60 ? 
 
 9. | of 16 is how many sixths of 36 ? 
 
 10. f of 12 is how many fifths of 20 ? 
 
 11. f of 48 is how many tenths of 40 ? 
 
 12. T 5 T of 77 is -J of what number? 
 SOLUTION. fV of 77 is 35. 35 is -J of what number f 
 
 13. -jV of 96 is ^ of what number ? 
 
 14. f of 44 is T V of what number t 
 
 15. | of 63 is % of what number ? 
 
 16. T \ of 72 is f of what number? 
 
 17. I of 11 is f of what number ? 
 
 18. $ of 36 is -& of how many times 5 ? 
 19. 4 of 56 is f of how many times 10 ? 
 
 20. | of 54 is T V of how many times 7? 
 
 21. f of 27 is f of how many times 14 ? 
 
110 FRACTIONS. 
 
 22. | of 36 is f; of how many times 10 ? 
 
 23. - of 12 is -J- of how many times 5 ? 
 
 24. -^ of ^ is ^ of how many times ^ ? How much 
 is f of f of 36 ? 
 
 25. f of f of 60 is f of what number ? 
 
 SOLUTION. | of 60 w 40. f of 40 w 15. 15 is 4 o/ 
 
 T 4 -L 9 * J 
 
 vial number? 
 
 26. ^ of -J of 45 is ||- of what number? 
 
 27. f of f of 35 is \ of what number ? 
 
 28. | of -^ of 100 is f of what number? 
 
 29. | of of 84 is -J- of f of what number? 
 
 30. f of | of 48 is f of t of what number? 
 
 31. -| of 36 is -fa of 4 times what number? 
 
 32. \ of 36 is how many tenths of 4 times 5 ? 
 
 33. f of 63 is -j^- of 5 times what number ? 
 
 34. f of 80 is | of 12 times what number ? 
 
 35. f of 81 is |f of 8 times what number ? 
 
 36. f of 11 is f of 10 times what number? 
 
 37. f of f of 24 is |- of 4 times what number ? 
 
 38. |- of -fa of 50 is |f of 5 times what number? 
 
 39. |- of -fa of 50 is how many tenths of 5 times 6 ? 
 
 40. |- of fa of 50 is 12 times what part of 5 tynes 6 ? 
 
 NOTE. The teacher will vary the above questions after the manner 
 indicated in 32, 39, 40. The pupil will form four questions similar to 
 the above, and give the solutions. 
 
 LESSON XVI. 
 
 1. |- of 15 is -fa of how many thirds' of 21 ? 
 SOLUTION. f of 15 is 12 ; 12 is -faof2(}; of 21 is 
 7; 20-^-7 gives 2f. Hence, iqflb is -fa of 2f thirds 
 
 2. f of 40 is 4 of how many ninths of 63 ? 
 
 3. -jSj- of 77 is | of how many tenths of 100 ? 
 
 4. f of 24 is T 9 Q- of how many twentieths of 80 ? 
 
 5. - of 30 is ||- of how many thirds of 36 ? 
 
 6. | of 18 is -I of how many twelfths of 72 ? 
 
FRACTIONS. Ill 
 
 7. of 9 is ^ of how many thirds of 
 
 8. of 80 is | of how many fifths of 60 ? 
 
 9. T 5 T of 39 is -^ of how many ninths of 90 ? 
 
 10. | of -5%. of 40 is 8 times what part of 36 ? 
 
 11. | of f of 48 is 10 times what part of 24? 
 
 12. \ of | of 72 is 6 times what part of 50 ? 
 
 13. T V of T V of 90 is T V time what part of 20 ? 
 
 14. 4. of f of 70 is 3 times what part of 40 ? 
 
 15. f of 48 is -| of how many times -f of 45 ? 
 
 16. f of 56 is of how many times f of 27 ? 54 is 
 how many times f of 27. 
 
 17. f of 64 is -^Q of how many times f of 35 ? 
 
 18. f of 63 is f of how many times f of 40 ? 
 
 19. f of 24 is -f- of how many times i of 20 ? 
 
 20. | of 60 is T ^ of how many times f of 24 ? 
 
 21. f of % of 96 is T V of how many times f of 12 ? 
 
 22. | of 81 is | of how many times % of f of 40? 
 How much is ^ of f of 40 ? -| of 81. is f of what 
 number ? 
 
 23. f of of 60 is ^ of how many times f of f of 
 24? 
 
 24. ^ of T 6 g- of 100 is f of f of how many times -| of 
 | of 30 ? I- of T <V of 100 is how much ? 12 is | of f of 
 what number ? How much is f of |- of 30 ? 
 
 25. f of f of 72 is f of f of how many times ^ of ^ 
 of 54? 
 
 NOTE. The pupil will form three questions similar to the above, and 
 give the answers. 
 
 LESSON XVII. 
 
 1. A farmer having 70 sheep sold ^ of them at one 
 time, and ^ of them at another. How many of them 
 had he left ? 
 
 2. A woman is 50 years old, and her age is | of the 
 age of her husband. How old is he ? 
 
 3. I bought a horse for $100 ; I paid f as much for 
 
112 FRACTIONS. 
 
 
 
 an ox, and I- as mucli for a colt. How much did I 
 pay for all three ? 
 
 4. A man bought 40 dozen eggs for 400 cents. He 
 sold f of them for 15 cents a dozen, and the remainder 
 at 10 cents a dozen. How much did he gain ? 
 
 5. Iff of a barrel of flour cost 12 dollars, what will 
 be the cost of a barrel? 
 
 6. A farm contains 60 acres of wood land, and -f| 
 of the wood land is 4 of the cleared land. How much 
 cleared land in the larm ? 
 
 7. A person owned in a town 17 lots, each lot con- 
 taining -J- of an acre. How many acres did he own ? 
 
 8. From a lot of 5 acres of land, I sold % an acre to 
 one man, f of an acre to another, and -^ to a third. 
 How many acres did I have left ? 
 
 9. I paid out of a sum of $100, at different times, 
 20^- dollars, 30f dollars, 40|- dollars. How much did 
 I have left ? 
 
 10. Frank gave f of an orange, J of an orange, and 
 f of an orange to one boy ; J of an orange, f of an 
 orange, and -% of an orange to another boy. How 
 much more did he give to the first boy than the 
 second ? 
 
 11. What is the cost of 4 yards of tape at -J of a 
 dollar a yard ? 
 
 12. What is the cost of 10 yards at -^ of a dollar a 
 yard? 
 
 13. What is the cost of 9 barrels of flour at 6f dol- 
 lars a barrel ? 
 
 14. A man paid 12 dollars for 3 yards of cloth, and 
 sold it at 5f dollars a yard. How much did he gain ? 
 
 15. What is the difference between 5| times 10 and 
 3 times 8f ? 
 
 16. 5 times 7 and -f- of 7 is how many times 10 ? 
 
 17. If a bushel of potatoes is worth ^ a dollar, how 
 touch is -J- of a bushel worth ? f of a bushel ? f of a 
 bushel ? -j^- of a bushel ? 
 
FRACTIONS. 113 
 
 18. If a yard of cloth is worth ^ of a dollar, what 
 is J of a yard worth ? f of a yard ? ^ of a yard ? 
 
 19. What is the cost of 2J- yards of cloth at f of a 
 dollar a yard ? At 1|- dollar a yard ? At 5-f dollars 
 a yard ? At 6-| dollars a yard ? At 3 dollars a yard? 
 
 20. I bought a horse for 82^ dollars, and sold him for 
 % of that sum. What did I lose on him ? 
 
 21. If a man can mow l^- acre of grass in one day, 
 how much can he mow in 5|- days ? 
 
 22. What will -^ of % of 70 gallons of wine cost at 
 If dollar a gallon I 
 
 23. At -| of a dollar a bushel, how many bushels of 
 meal can be bought for five dollars ? for 6 dollars ? 10 
 dollars ? 20 dollars ? 
 
 24. If a horse eats J- of a bushel of oats a day, how 
 long will 15 bushels of oats last him? How long 5 
 
 bushels ? 7 bushels ? 
 
 25. If 12 pounds of sugar cost -f of a dollar, what 
 is the cost of 1 pound ? 
 
 26. At f of a dollar a yard, how much cloth can be 
 bought for f of a dollar ? for f of a dollar ? 
 
 27. If 7 bushels of wheat cost 8^- dollars, what i the 
 cost of one bushel ? 
 
 28. In y 9 ^ of an acre of land, how many building 
 lots of 4- of an acre each, and how much over ? 
 
 29. If a horse eats f of a bushel of oats a day, how 
 long will |~J- of a bushel last him ? 
 
 30. If f of a pound of tea costs 1 dollar, what is the 
 cost of f of a pound ? 
 
 31. If f of a bushel of barley be given for f of a 
 bushel of corn, and the corn costs f of a dollar a 
 bushel, what is the cost of a bushel of barley? 
 
 32. In an orchard |- of the trees are apple-trees, 
 ^ peach-trees, J pear-trees, and the remaining 11 are 
 plum-trees. How many trees in the orchard? 
 
 33. What number added to 4 times f of 21 will 
 make 52 ? 
 
PROPORTION. 
 
 34. 24 is f of some number. What is ^ of the same 
 number ? 
 
 35. How long will it take two wood-choppers to 
 cut 14: cords of wood, if one of them cuts f of a cord a 
 day, and the other cuts 1 cord ? 
 
 36. Several workmen received together 6f dollars, 
 and the wages of each one was -f- of a dollar. How 
 many workmen were there ? 
 
 37. A man plants a bed of 36 square feet in cab- 
 bage, and gives to each cabbage f of a square foot. 
 How many cabbages can he plant in the bed ? 
 
 38. How many steps in a staircase 39 feet high, 
 when each step is l^ foot high ? 
 
 39. A yard of a certain cloth, 4 quarters wide, is 
 worth 3 dollars. What will 1 yard of the same cloth, 
 5 qn arters wide, be worth ? 
 
 NOTE. The pupil will frame four questions similar to the above, 
 and give the answers. 
 
 LESSON XVIII. 
 
 PROPORTION. 
 
 1. What is the difference between 6 and 2? Give 
 two other numbers which have the same difference 
 between them. 
 
 2. If we take 8 and 4 as the two numbers, we will 
 have 6 less 2 equal to 8 less 4. Now if we add the 
 greater of the first pair of numbers to the less of the 
 second pair, and the less of the first to the greater of 
 the first, what are the results? Try the same with 
 any pair whose difference is 4. 
 
 3. Find 5 pairs of numbers, the difference between 
 the two forming each pair being 5. 
 
 4. What number is as much greater than 10 as 12 
 is greater than 7 ? How is this number found ? 
 
 ANS. By adding the difference between 12 and 7 to 
 10. 
 
PROPORTION. 115 
 
 5. What is the quotient of 6-^-2? Give two other 
 numbers which have the same quotient. If we take 
 the numbers 12-^4, we will have 6-f-2 equal to 12-^-4. 
 If, now, we multiply the greater of the first pair of 
 numbers by the less of the second pair, and the less 
 of the first pair by the greater of the second, what are 
 the results ? Try the same with another pair of num- 
 bers whose quotient is 3. 
 
 6. Find 5 pairs of numbers, the quotient of the 
 numbers in each pair being 5. 
 
 7. What number is contained in 15 as often as 4 is 
 contained in 20? or 20-^4 is equal to 15-7- what num- 
 ber ? How is this number found ? 
 
 ANS. We divide 15 by the quotient of 20 divided 
 1y 4. 
 
 '8. 12 -r- 3 is equal to 16 ~- what number? 12 4 
 = 15-7- what number? 10 -f- 2 = 20 -=- what num- 
 ber? 
 
 9. 20 -r- 4 is called the ratio of 20 to 4. The line 
 between the two dots is left out for convenience ; thus, 
 20 : 4, which is read 20 to 4. 
 
 10. The ratio of 20 to 4 is - 2 /, or 5. 
 
 So, 12 --- 3 =12 :3=-V 2 - 4; 3: 12 &{; 
 5 : 7 - f; 1 : 3 - I + J- - 1 x f - f ; 3J : 1= 
 jni . l _ ^ 
 
 What is the ratio of 10 : 4? of 5 : 10 ? of 3 : 9 ? of 
 9:3? of 5 : 11 ? of 11 : 5 ? of 8-J- : 1J ? of f : f ? of 2 
 : 5? of 5 : 2|? 
 
 11. Find 6 pairs of numbers that have the same 
 ratio as 10 : 5. 
 
 12. 4 : 12 =- 13 : what number? 
 
 SOLUTION. 4 : 12 = ^. The required number is, 
 en, 13 -=- 4 = 13 x 3, or 39. Or, since 12 is 3 times 
 the required number must be 3 x 13, or 39. 
 
 13. 1:5 = 7:? 9:7 = 14 :? 6:9=8:? 2:5 
 7:? 3:7 = 6:? 3:7 = 6: 14 is called a PRO- 
 
 ORTION. 
 
116 PROPORTION. 
 
 14. What is the ratio of one number to another? 
 (See Q. 9.) 
 
 When are .four numbers said to be in proportion ? 
 Ans. When the ratio of the 1st and*%d = the ratio of 
 the 3d and 4tk. 
 
 How do you find the fourth term of a proportion 
 when three terms are known ? (See Q. 12.) 
 
 15. 4 : 10 11 : ? 5 : J == 40 : ? 6 : 4 = 7J : ? 
 1 : 2f == 4 : ? (1 : 2| f . The required number is, 
 then, 4|--t = fxf = ^ = 12). * : f = | : ? 
 
 16. 12 is f of what number ? f : 1 = 12 : what 
 number ? 18 is f of what number ? 6:7 = 18: what 
 number? f : 1 = 18 : what number? 
 
 17. 6 workmen in 2^- days will do as much as 10 
 workmen in how many days ? 
 
 SOLUTION. 6 : 10 = f . The number of days re- 
 quired is, then, 2i~ = J--f-=fxf = -V 5 -, 
 or 4-J-. 
 
 6 men in 12 \ days do as much as 15 men in days ? 
 
 12 men in 6f days do as much as 8 men in days ? 
 
 6 men in 7|- days do as much as 4 men in days ? 
 
 18. 20 days' work for six men is 10 days' work for 
 how many ? 
 
 3f days' work for 14 men is 7|- days' work for 
 men? 
 
 19. 7 days' rations for 6 men is how many days' 
 rations for 4 men ? , 
 
 6f days' rations for 5 men is how many days' rations 
 for 7 men ? 
 
 3f days' rations for 14 men is how many days' ra- 
 tions for 7 men ? 
 
 2^ days' rations for 8 men is how many days' rations 
 for men ? 
 
 20. 3 men do a piece of work in 5 days ; how long 
 will it take 7 men to do the same work ? 
 
 5 men in 6 days do as much as 11 men. in 
 days? 
 
PROPORTION. 117 
 
 21. 3 days' rations for 11 men are how many days' 
 rations for 13 men*? 
 
 22. 3 men can 'do a piece of work in 5 days ; how 
 long will it take 7 men to do twice the work ? 3 times 
 the work ? 4 times the work ? |- the work ? 
 
 23. If 5 men build a wall 50 feet long in 6 days, in 
 what time will 9 men build a similar wall 150 feet long? 
 
 24. 3 days' rations for 12 men are 16 days' half-rations 
 for how many men ? 
 
 25. If 3 horses eat 8 bushels of oats in 2 weeks, how 
 long will 40 bushels last them? How long will 40 
 bushels last 5 horses at the same rate ? 
 
 26. How many yards of cloth, 3 quarters wide, are 
 equal to 7 yards of the same cloth 5 quarters wide ? 
 
 27. How many yards of carpeting, 1 yard wide, are 
 equal to 20 yards of a yard wide ? 
 
 28. A workman laboring 11 hours a day does a 
 certain piece of work in 5 days ; in how many days 
 will he do the same work, laboring 5 hours a day ? 
 
 29. A traveller has enough money to last him 32 
 days, at J of a guinea a day ; how many days will it 
 last him if he limits his expenses to f of a guinea a 
 day? 
 
 30. A man bought 6 barrels of flour at 7 dollars a 
 barrel; at what price per barrel must he sell it in 
 order to gain 11 dollars ? 
 
 31. A man being asked how much he had given for 
 each of two horses, stated that he gave $60 for the 
 first, and f of the price of the first was J- of of the 
 price of the second. What was the price of the 
 second ? 
 
 32. A man can do a piece of work in 4f days, if he 
 labor 10 hours a day ; how many days will it take 
 him to do the work, laboring 7 hours a day ? 
 
 33. If 5 bushels of oats cost 2f dollars, what will 7 
 bushels cost? 
 
 34. If 10 yards of cloth cost 45 dollars, what will 
 
118 PROPORTION. 
 
 be the cost of 6 yards of cloth worth twice as much 
 per yard ? 
 
 35. If 6 yards of cloth cost 2|- dollars, what cost 11 
 yards ? 
 
 36. If 5 yards of cloth cost 7 dollars, what cost 13 
 yards ? 
 
 37. If 8 bushels of meal cost 10 dollars, how much 
 can be bought for 12 dollars ? 
 
 38. If 9 barrels of flour cost 58 dollars, how many 
 can be bought for 75 dollars ? 
 
 39. If 10 acres of land cost 300 dollars, what is the 
 cost of 50 acres ? 
 
 40. At the rate of 10 acres for 500 dollars, how 
 much land can be bought for 3000 dollars ? 
 
 41. A railway train travels at the rate of 90 miles 
 in 3 hours ; how many miles will it go in 10 hours ? 
 
 42. If a railway train travels at the rate of 100 miles 
 in 5 hours, how many hours will it require to go 320 
 miles ? 
 
 NOTE. In this and all the succeeding lessons the pupils should be 
 required to form questions for themselves and each other. 
 
DENOMINATE NUMBEBS. 119 
 
 CHAPTER YIII. 
 
 DENOMINATE NUMBERS. 
 
 LESSON I. 
 Table of Long Measure. 
 
 This table is used in measuring distances. 
 
 The smallest units are the inch and its subdivisions. 
 
 1 Inch. 
 
 12 inches (in.) make 1 foot marked ft, 
 
 3 feet make one yard marked yd. 
 
 5|- yards make 1 rod or pole marked rd. 
 
 40 rods make 1 furlong marked fur. 
 
 8 furlongs make 1 mile marked m, 
 
 3 miles make 1 league marked lea. 
 
 69J miles (nearly) } make 1 degree marked deg. 
 
 60 nautical miles \ of the circumference of the earth. 
 
 360 degrees make 1 circumference marked circ, 
 
 1. How many inches in 2 feet ? 3 feet ? 5 feet ? in 
 10 feet? in \ of a foot? of a foot? of a foot? \ of 
 a foot ? f of a foot ? in f of a foot ? 
 
 2. How many feet in 24 inches? 36 inches? 30 
 inches? in 15 inches? in 17 inches? in 44 inches? 
 
 3. How many feet in 2 yards ? 11 yards ? 3 yards I 
 4| yards ? in ^ of a yard ? in f of a yard ? What part 
 of a 'yard is 1 foot? What part of a yard is 1 inch? 
 
 4. How many yards in 6 feet? 8 feet? 9 feet? 12 
 feet? 13 feet? 25 feet? 
 
 5. How many yards in 3 rods ? in 5 rods ? in 10 
 rods ? in 3 rods ? in 40 rods ? 
 
 6. How many rods in 2 furlongs ? 3 furlongs ? in 6 
 
120 DENOMINATE NUMBERS. 
 
 furlongs ? in one mile ? What part of a mile is a fur- 
 long? What part of a mile is one rod? How many 
 furlongs in 40 rods? in 50 rods? in 120 rods? in 150 
 rods? 
 
 7. How many furlongs in 2 miles? Smiles? 4^ miles? 
 f of a mile how many furlongs ? -^ of a furlong is what 
 part of a mile? How many miles in 16 furlongs? in 
 24 furlongs ? in 25 furlongs ? in 40 furlongs ? in 45 
 furlongs ? How many yards in -J of a mile ? in J of a 
 mile ? in ^ of a mile ? 
 
 8. How many miles in 3 leagues ? in 4 leagues ? in 
 10 leagues ? A mile is what part of a league ? f of a 
 league are how many miles ? 
 
 9. How many leagues in 9 miles ? in 11 miles ? in 
 24 miles ? 
 
 10. If a man travels a mile in ten minutes, in how 
 many minutes will he travel 1 league? 3 leagues ? 4 
 leagues ? 
 
 11. If a ship sails 20 leagues a day, how long will 
 it take her to sail 600 miles? 300 miles? 3000 miles ? 
 
 12. How many inches in 3 yds. 2 ft. 4 in. ? in 2 yds. 
 1 ft. 5 in. ? 
 
 Table of Cloth Measure. 
 
 This measure is used in measuring cloth and other 
 articles sold by the yard. The yard and inch are the 
 same in this as in Long Measure. 
 
 
 
 2J inches (in.) make 1 nail marked na. 
 
 4 nails make 1 quarter of a yard marked qr. 
 
 4 quarters make 1 yard marked yd. 
 
 5 quarters make 1 ell English marked E. E. 
 
 1. How many inches in J of a yard ? How many 
 in |- a yard ? in f of a yard ? How many inches in an 
 
 idly 
 
DENOMINATE NUMBERS. 
 
 121 
 
 2. How many quarters in 12 nails ? 13 nails ? in 24 
 nails ? 
 
 3. How many yards in 6 quarters ? in 20 quarters ? 
 How many nails in 3 yds. 3 qrs. 3 nails ? How many 
 inches in 3 qrs. 2 na. 4 in. ? 
 
 4. If 4: nails of cloth cost 20 cents, what is the cost 
 of 5 yards ? What part of a yard is one nail ? What 
 part of an ell is a quarter ? 
 
 5. What is the cost of 12 yards of cloth at 5 dollars 
 an ell ? f of a yard is what part of an ell .? what part 
 of a rod? f of a rod is what part of a furlong? -^ of 
 a yard is what part of a foot ? T V of a foot is how 
 many inches ? 
 
 LESSON II. 
 Square Measure. 
 
 Square Measure is used for measuring surfaces. 
 The square inch is the smallest unit in the table. 
 
 Square Inch. 
 
 TABLE. 
 
 square inches (sq. in.) make one square foot 
 marked sq. ft. 
 
 9 square feet make 1 square yard marked sq. yd. 
 
 30J square yards make 1 square rod or perch 
 marked P. 
 
 40 square rods make 1 rood marked E. 
 
 4 roods make 1 acre marked A. 
 
 640 acres make 1 square mile marked sq. m. 
 
 1. How many square feet in 2 square yards? in 2 
 
122 
 
 DENOMINATE NUMBERS. 
 
 of a square foot is 
 of a square yard is 
 
 or 
 
 square yards ? in 5 square yards ; in 3 square yards ? 
 How many square yards in 36 square feet? in 81 
 square feet ? in 44 square feet ? 
 what part of a square yard ? 
 how man}' square feet ? 
 
 2. How many square yards in 3 square rods 
 perches? in 4 perches? in 10 perches? How many 
 square rods in 3 roods ? in \ of a rood ? in f of a rood? 
 in 3^- roods ? What part of a rood is a square rod ? 
 
 3. How many acres in 10 roods? 16 roods? 40 
 roods ? What part of an acre is 1 rood ? ^ of a rood ? 
 -jig- of a rood ? How many roods in f of an acre ? 20 
 square rods is what part of an acre ? 
 
 4. At 5 dollars a square rod, what is the cost of 2 
 roods and 5 sq. rods of land ? 1 A. 2 R. 15 P. are 
 how many perches ? 
 
 5. At 2f dollars a rood, what is the cost pf 2 acres 
 of land ? 
 
 Solid or Cubic Measure. 
 
 This measure is used in measuring timber and stone T 
 and gives the unit for Liquid and Dry Measure, and 
 measures weights and coins. 
 
 Cubic Inch. 
 
 TABLE. 
 
 1728 cubic inches (cu. in.) make 1 cubic foot 
 marked cu. ft. 
 
 27 cubic feet make 1 cubic yard marked cu. yd. 
 16 cubic feet make 1 cord foot marked co. ft. 
 
DENOMINATE NUMBERS. 123 
 
 8 cord feet, or 128 cubic feet, make 1 cord of wood 
 marked 0. 
 
 1. How many cubic feet in 2 cubic yards? in 3 
 cubic yards? in -J- of a cubic yard? in ^ of a cubic 
 yard ? in -^ of a cubic yard ? 
 
 2. How many cubic yards in 27 cubic feet? in 36 
 cubic feet? in 54 cubic feet? in 57 cubic feet? What 
 part of a cubic yard is 1 cubic foot ? 3 cubic feet ? 9 
 cubic feet? How many cubic feet in f of a cubic 
 yard? 
 
 3. How many cord feet in 2 cords of wood ? in 2^- 
 cords of wood ? in If cord of wood ? |- of a cord is - 
 how many cord feet ? 
 
 4. How many feet of plank, 1 inch thick, in 50 feet 
 of plank 2 inches thick ? 4|- inches thick ? 
 
 LESSON III. 
 Liquid or Wine Measure. 
 
 This measure is used in measuring all kinds of 
 liquids.* The wine pint-cup will hold 28^- cubic 
 inches. 
 
 TABLE. 
 
 4: gills (gi.) make 1 pint marked pt. 
 2 pints make 1 quart marked qt. 
 4 quarts make 1 gallon marked gal. 
 63 gallons make 1 hogshead marked hhd. 
 2 hogsheads make 1 pipe marked pi. 
 2 pipes make 1 tun marked tun. 
 
 I 
 
 1. How many gills in 2 pints ? 3 pints ? |- of a pint? 
 of a pint? in 2^ pints? in 6f pints? How many 
 pints in 8 gills? 9 gills? 15 gills? 60 gills? What 
 
 Kart of a pint is 1 gill ? 2 gills ? 3 gills ? 
 2. How many pints in 5 quarts ? 3 quarts ? 10-J- 
 quarts? 2^- quarts? in 12 quarts? How many gills 
 in 2 quarts? What part of a quart is 1 pint? How 
 many quarts in 12 pints? in 13 pints? 50 pints? 
 
124 DENOMINATE NUMBERS. 
 
 3. How many quarts in 3 gallons ? 2 gallons ? 5f 
 gallons? What part of a gallon is a quart? How 
 many gallons in 2 hogsheads ? 9 gallons is what part 
 of a hogshead? 21 gallons is what part of a hogshead? 
 
 4. If 3 quarts of oil cost 60 cents, what is the cost 
 of 5 gallons and 2 quarts ? 
 
 5. How many quart bottles can you fill from a 
 hogshead of wine ? How many gallons in a pipe ? 
 
 Dry Measure. 
 
 This measure is used in measuring grain, fruit, coal, 
 salt, &c. A pint-cup of this measure contains 
 cubic inches. 
 
 TABLE. 
 
 2 pints (pt.) make 1 quart marked qt. 
 8 quarts make 1 peck marked pk. 
 4 pecks make 1 bushel marked bu. 
 
 1. How many cubic inches larger is the fynt, Dry 
 Measure, than the pint, Liquid Measure ? How much 
 larger is the quart, Dry Measure, than the quart, 
 Liquid Measure? 
 
 2. How many quarts in 2 pecks ? in 4|- pecks ? in 
 3f pecks ? ^n ^ peck ? f of a peck ? -f- of a peck ? How 
 many pecks in 16 quarts? 18 quarts? 21 quarts? 40 
 quarts ? What part of a peck is a pint ? 
 
 3. How many pecks in 4 bushels ? in ^ of a bushel ? 
 in 5 bushels ? in 10 bushels ? in 100 bushels ? How 
 many bushels in 12 pecks? in 15 pecks? in 32 pecks? 
 in 45 pecks? 
 
 4. What is the cost of 1 bushel of corn at 5 cents a 
 quart-? What of 2 quarts of corn at 60 cents a bushel ? 
 
 5. How many quarts in 4 bushels ? in 3 bushels ? 
 in 2 bu. 3 pk. 3 qt. ? 
 
 6. If 2 pints of meal cost 5 cents, what is the cost 
 of a bushel ? What part of a bushel is a pint ? 
 
DENOMINATE NUMBERS. 125 
 
 LESSON IV. 
 Troy Weight. 
 
 Troy Weight is used in weighing gold, silver, and 
 jewels. The Troy pound is a piece of brass or other 
 metal, which weighs as much as 22 T 8 T cubic inches of 
 water, at a certain temperature. 
 
 TABLE. 
 
 24 grains (gr.) make 1 pennyweight marked pwt. 
 20 pennyweights make 1 ounce marked oz. 
 12 ounces make 1 pound marked Ib. 
 
 1. In 1 pennyweight how many grains ? in 2 ? in 3 ? 
 in 4 ? in 5 ? 
 
 2. How many pennyweights in 48 grains ? in 72 ? 
 in 75? 
 
 3. In 1 pound how many ounces ? in 2 ? in 4 ? in 5 ? 
 in |? in 2f? 
 
 4. In 24 oz. how many Ibs. ? in 36 ? 48 ? 28 ? 39 ? 
 
 Apothecaries' Weight. 
 
 The Troy pound is sometimes divided differently, 
 to weigh medicines in mixing them. The pound, 
 ounce, and grain are the same in Troy and Apotheca- 
 ries' weight. 
 
 20 grains (gr.) make 1 scruple marked sc. or 3. 
 3 scruples make 1 dram marked dr. or 3. 
 8 drams make 1 ounce marked oz. or 
 12 ounces make 1 pound marked Ib. or ft>. 
 
 1. In 1 scruple how many grains? in 2? in 3? in \ 
 of a scruple ? in 2|- scruples ? 
 
 2. In 2 drams how many scruples ? in 3 ? in 4 ? in 
 ^ of a dram ? in 2-f drams ? 
 
126 DENOMINATE NUMBERS. 
 
 t3?_4? I 
 
 many 
 
 3. In 2 ounces how many drams ? in 3 ? 4 ? 2f ? 
 
 4. In y^ of a pound how many grains ? How m 
 grains in 4 drams, 3 scruples, 5 grains ? 
 
 Avoirdupois Weight. 
 
 This weight is used in all ordinary weighing. The 
 Avoirdupois pound is heavier than the Troy pound, 
 144 Avoirdupois pounds being equal to 175 Troy 
 pounds. 
 
 TABLE. 
 
 16 drams make 1 ounce marked oz. 
 
 16 ounces make 1 pound marked Ib. 
 
 25 pounds make 1 quarter marked qr. 
 
 4 quarters, or 100 Ibs., make 1 hundred weight 
 marked cwt. 
 
 20 hundred weight make 1 ton, marked T. 
 
 1. In 2 pounds how many ounces ? in 3 ? in 4 ? in 
 % of a pound ? in 2f ? 
 
 2. In 32 ounces how many pounds ? in 48? 64? 80? 
 82? 
 
 3. How many pounds in 2 quarters? in 3? in 4^? 
 
 4. How many quarters in 50 pounds ? 100 pounds ? 
 200 pounds ? 76 pounds ? 
 
 5. How many cwt. in 8 quarters ? in 16 ? in 25 ? in 
 33? 
 
 6. How many cwt. in 2 tons? 4 tons? \ a ton ? -fa 
 of a ton ? in f of a ton ? in 5 tons ? What part of a 
 ton is 1 cwt. ? 
 
 7. How many tons in 40 cwt. ? in 80 ? in 60 ? in 45 ? 
 
 8. What is the cost of 4|- cwt. of sugar, at 9 cents a 
 pound? at 10 cents? 
 
 9. What is the cost of 5 tons of hay, at 20 dollars 
 a ton ? 
 
 10. How much will 2 cwt. 2 qrs. of sugar cost at 10 
 cents a pound ? 
 
 11. Bought 3 tons and 10 cwt, of hay at 12 dollars 
 a ton ? What was the cost of it ? 
 
DENOMINATE NUMBERS. 
 
 127 
 
 LESSON Y. 
 United States Money. 
 
 The units are coins of gold, silver, copper, and 
 nickel, of fixed weight, size, and shape. 
 
 TABLE. 
 
 Gold. 
 
 Gold. 
 
 Nickel. 
 
 Silver. 
 
 Silver. 
 
 10 mills make one cent. marked c. 
 
 10 cents make 1 dime marked d. 
 10 dimes make 1 dollar marked $. 
 10 dollars make 1 eagle marked E. 
 
 1. How many mills in 2 cents? 9 cents? 2} cents? 
 How many cents in 50 mills? 40 mills? 25 mills? 
 
 2. How many cents in 2 dimes ? in 3 ? in 4^ ? in 
 5|? How many dimes in 20 cents? 25 cents? 50 
 cents ? 75 cents ? 
 
128 DENOMINATE NUMBERS. 
 
 3. How many dimes in 2 dollars ? 4 dollars ? 
 dollars? 6^ dollars? How many dollars in 20 dimes"? 
 40 dimes? 45 dimes? 
 
 4. How many dollars in 2 eagles? in a half-eagle? 
 in 3^ eagles? in 5J eagles? How many eagles in 20 
 dollars ? 35 dollars ? 42 dollars ? 
 
 5 What part of a dollar is 1 cent ? is 1 mill ? is 50 
 cents ? 33| cents ? 25 cents ? 20 cents ? 1 dime ? 16| 
 cents? 12^ cents? 
 
 6. What part of an eagle is 1 djme ? 5 dimes ? 
 
 7. How many cents in |- of a dollar ? in J ? in ? in 
 -? in |? in |? -jV? ^-? ^-? ^- of a dollar? of a 
 dollar? 
 
 8. Bought six barrels of flour at $9.33^ cents a bar- 
 rel ; what was the cost of the whole ? 
 
 9. Paid 6J cents a pound for 48 pounds of nails ; 
 how many dollars did the whole cost ? 
 
 10. At 37J cents a yard, what is the cost of 6 yards 
 of ribbon ? 
 
 11. If 6 pounds of sugar can be bought for 60 cents, 
 how much can be bought for an eagle? 
 
 12. What is the cost of 100 articles at 1 cent each ? 
 2 cents? 3 cents? 12 cents? 
 
 Table of English Money. 
 
 4 farthings (far.) make 1 penny marked d. 
 
 12 pence make 1 shilling marked s. 
 .20 shillings make 1 pound marked . 
 
 The coins of this table are gold, silver, and copper. 
 A sovereign is a gold coin = 1 pound or 20 shillings. 
 The guinea is 21 shillings. A pound is 4 dollars and 
 84 cents. 
 
 1. How many farthings in 2 pence ? in 3 pence ? in 
 6J pence ? ^ a penny ? f of a penny ? in 4f pence ? 
 
 2. How many pence in 8 farthings ? in 10 ? in 20 ? 
 in 60? 
 
 
DENOMINATE NUMBEKS. 129 
 
 3. How many pence in 2 shillings ? in -J- of a shil- 
 ling ? in \ of a shilling ? in of a shilling ? -jj- of a 
 shilling ? in 5s. 6d. ? in 10s. 2d. ? in 10s. 6d. ? in 2s. 
 8<U 
 
 4. How many shillings in 24 pence ? 48 pence ? 32 
 pence ? 100 pence ? 
 
 5. How many shillings in 2? in -? in -J-? in 
 4? in T V? in 2? 2? 5? in 7? in 2 7s. ? in 
 10 6s. ? 
 
 6. How many pounds in 25 shillings? in 40 shil- 
 lings? in 68 shillings? How many guineas in 63 
 shillings? How many dollars in 10. 
 
 7. What part of a shilling is 2 pence ? 3 pence ? 4 
 pence ? 6 pence ? 18 pence ? 
 
 8. What part of a pound is 1 shilling ? 2 shillings ? 
 4 shillings? 2s. 6d. ? 3s. 5d. ? 
 
 9. What is the cost of 12 articles at 1 penny apiece? 
 2 pence apiece ? at 4d. ? at 3d. ? at lOd. ? 
 
 10. What is the cost of 24 or 2 x 12 articles, at 2d. 
 apiece? at 3 pence? 4d. ? 6d. ? 10d.? 
 
 11. What is the cost of 20 articles at 1 shilling 
 apiece? 3s. apiece? 4s.? 6 shillings? 10 shillings ? 
 
 12. What is the cost of 40 articles, or 2x20 at 3 
 shillings apiece? at 4s. ? 6s.? 10s.? 
 
 13. What is the cost of 8 bushels of corn at 2s. 8d. 
 a bushel ? What part of a pound is a penny ? 
 
 LESSON VI. 
 Measure of Time. 
 
 The natural units in this table are the day and the 
 year. The smallest unit, the second, is the interval 
 between two ticks of a seconds clock, the pendulum 
 of the clock being 39 T V inches long. 
 
130 DENOMINATE NUMBERS. 
 
 TABLE. 
 
 60 seconds (sec.) make 1 minute marked min. 
 60 minutes make 1 hour marked lir. 
 24 hours make 1 day marked dy. 
 7 days make 1 week marked wk. 
 
 365 days make 1 common year marked yr. 
 
 366 days make 1 leap year. 
 
 100 years make 1 century marked c. 
 
 The year is divided into 12 months. 
 
 TABLE. 
 
 1st month, January, has 31 days. 
 
 2d month, February, lias 28 days in leap year, 29. 
 
 3d month, March, has 31 days. 
 
 4th month, April, has 30 days. 
 
 5th month, May, has 31 days. 
 
 6th month, June, has 30 days. 
 
 7th month, July, has 31 days. 
 
 8th month, August, has 31 days. 
 
 9th month, September, has 30 days. 
 
 10th month, October, has 31 days. 
 
 llth month, November, has 30 days. 
 
 12th month, December, has 31 days. 
 
 This table is usually given in the following rhyme : 
 
 Thirty days hath September, 
 April, June, and November ; 
 All the rest have thirty-one, 
 Except the second month alone, 
 And this has twenty-eight, in fine, 
 Till leap year gives it twenty-nine. 
 
 1. How many seconds in 2 minutes ? 3 minutes ? 
 4 minutes ? 10 minutes ? in ^ minute ? J minute ? -% 
 of a minute ? f of a minute ? in T 7 F of a minute ? 
 
 2. How many minutes in 120 seconds? in 180 
 
DENOMINATE NUMBERS. 131 
 
 seconds ? in 2 hours ? in 3^ hours ? in 5 hours ? in If 
 hour? an hour? of an hour? 
 
 3. How many hours in 2 days ? 3 days ? 4 days ? in 
 2 days? in If day? How many days in 48 hours? 
 72 hours ? 75 hours ? 36 hours ? 
 
 4. How many days in 4 weeks ? in 4 of a week ? in 
 4|- weeks ? in 10 weeks ? How many weeks in 28 
 days? 56 days? 31 days? 
 
 5. If October commences on Monday, on what day 
 of the week will November begin ? 
 
 6. How many months in 2 years ? 2-j- years ? 6 
 years ? 
 
 7. If a man earns 2|- cents a minute, how much 
 will he earn in 5 hours ? What part of an hour is 2 
 minutes ? 3 minutes ? 15 minutes ? 
 
 MISCELLANEOUS TABLE. 
 
 12 units make 1 dozen. 
 12 dozen make 1 gross. 
 20 units make 1 score. 
 24 sheets of paper 1 quire. 
 20 quires make 1 ream. 
 
 1. How many units in 3 dozen ? -J- dozen ? 5|- dozen ? 
 
 2. How many scores in 40 ? 50 ? 80 ? 60 ? 200 ? in 
 5 dozen ? 10 dozen ? in 4^- dozen ? 
 
 3. What is the cost of a ream of paper at 15 cents a 
 quire? at 10 cents ? 12 cents? at 5-^ cents ? What 
 part of a ream is half a quire ? 
 
132 METRIC SYSTEM. 
 
 CHAPTEK IX. 
 
 METRIC SYSTEM OF WEIGHTS AND MEASURES. 
 
 The adoption of the French or metric system of 
 weights and measures in this country, has been autho- 
 rized by the Government. The great advantage of 
 this system is the uniformity in the names and in the 
 division of the units in the different tables. The di- 
 vision is the successive division into tenths, after the 
 manner of the eagle, dollar, dime, cent, and mill of 
 the United States money. In addition to these ad- 
 vantages, this system has one table of weights instead 
 of three. One table for cubic, liquid, and dry meas- 
 ure. The names of the units indicate what multiple 
 or part of the principal unit they are. 
 
 Long Measure. 
 
 The Metre is the principal unit or basis of the table 
 of Long Measure. 
 
 The Metre is one yard 3 and T 4 T inches, or SOy 3 ^ 
 inches. 
 
 TABLE. 
 
 10 millimetres make 1 centimetre. 
 
 10 centimetres make 1 decimetre. 
 
 10 decimetres make 1 metre 39 T y in. marked M. 
 
 10 metres make 1 decametre- marked D . 
 
 10 decametres make 1 hectometre marked H m . 
 
 10 hectometres make 1 kilometre marked K m . 
 
 10 kilometres make 1 myriametre^O-^- miles, marked 
 
 M m . 
 
 The prefixes, milli, centi, deci, mean YoVo ? TST > an ^ 
 j^, respectively. The prefixes deca, hecto, kilo, 
 
METRIC SYSTEM. 133 
 
 myria, mean 10, 100, 1000, 10,000, respectively. 
 These are used in all the tables of this system. In 
 commerce the expressions 10 metres, 100 metres, 1000 
 metres, 10,000 metres, are invariably used instead of 
 decametre, hectometre, kilometre. 
 
 1. How many millimetres in 2 centimetres ? 4 cen- 
 timetres ? in 3^ centimetres ? What part of a centi- 
 metre is 1 millimetre ? 2 millimetres ? How many 
 centimetres in 50 millimetres ? in 65 millimetres ? 
 
 2. What part of a metre is a millimetre ? a centi- 
 metre ? What part of a myriametre is a metre ? How 
 many metres make a kilometre ? how many centi- 
 metres ? how many decimetres ? 
 
 3. Bought 5 decimetres of ribbon at 1 dollar a metre, 
 what was the cost ? 
 
 4. What is the cost of 30 metres of cloth at half a 
 dollar a decimetre ? at ^ of a dollar a decimetre ? at 
 1 dollar a decimetre ? 
 
 5. A tower is 55 metres high ; how many centi- 
 metres ? 
 
 6. How many miles in 100 myriametres ? 40 myria- 
 metres ? 
 
 Square Measure. 
 
 The natural unit is the centiare or square metre, 
 equal to 1-^- square yard, nearly. 
 
 100 centiares make 1 are = 100 square metres. 
 10 ares make 1 decare. 
 
 10 decares make 1 hectare = 2-^j- acres, 2 acres, 
 nearly. 
 
 1. How many square metres in 15 centiares? how 
 many square metres in 1 are ? in 1 hectare ? in 2 
 hectares? How many acres in 100 hectares? How 
 many square yards in 100 square metres ? 
 
 2. What is the cost of 15 hectares of land at 5 dol- 
 
134: METRIC SYSTEM. 
 
 lars the decare? "What is the cost of 15 acres of land, 
 at 50 dollars the hectare ? 
 
 NOTE. To make the table of square measure complete, we should 
 have the square decimetre, square centimetre, and square millimetre 
 but these are seldom used. 
 
 Cubic Measure. 
 
 The unit of this table is the cubic metre = 1.3 cub. 
 yard. This is called a stere when used in measuring 
 wood, or hectolitre when applied to liquid or dry 
 measure, of which the principal unit is the litre or 
 cubic decimetre = y 9 ^ of a quart, dry measure, and 
 1-rtSr f a quart, liquid measure. 
 
 TABLE. 
 
 10 millilitres make 1 centilitre = 10 cubic centi- 
 metres. 
 
 10 centilitres make 1 decilitre. 
 
 10 decilitres make 1 litre = 1 cubic decimetre = -fa 
 of a dry quart, D. M., lyf-o of a quart, L. M. 
 
 10 litres make 1 decalitre. 
 
 10 decalitres make 1 hectolitre. 
 
 10 hectolitres make 1 kilolitre or stere = 1 cubic 
 metre. 
 
 1. What part of a kilolitre is 1 litre ? 2 litres ? 1 
 decilitre? 1 decalitre? 5 hectolitres? 100 decalitres? 
 
 2. How many millilitres make one decalitre ? 
 
 3. How many quarts D. M. in 10 litres? How 
 many quarts, Liquid Measure, in 100 litres ? 
 
 4. What is the cost of 20 hectolitres of wine at 10 
 cents a litre ? 
 
 5. 1 hectolitre is what part of a cubic metre ? 1 
 decilitre is what part of a cubic decimetre? is how 
 many cubic centimetres ? 
 
METBIC SYSTEM. 135 
 
 Weights. 
 
 The principal unit is the gramme, or the weight of 
 a millilitre of pure water when it is heaviest. 
 The gramme is 15^^- grains Avoirdupois. 
 
 TABLE. 
 
 10 milligrammes make 1 ceirtigramme. 
 
 10 centigrammes make 1 decigramme. 
 
 10 decigrammes make 1 gramme = 15-^- grains 
 Avoirdupois. 
 
 10 grammes make 1 decagramme. 
 
 10 decagrammes make 1 hectogramme. 
 
 10 hectogrammes make one kilogramme = 2-^- Ibs. 
 Avoirdupois. 
 
 10 kilogrammes make 1 myriagramme. 
 
 10 myriagrammes make 1 quintal. 
 
 10 quintals make 1 inillier or tonneau = ly 1 ^- ton, 
 nearly. 
 
 1. How many kilogrammes in 1 quintal ? in 5 quin- 
 tals ? in 4 quintals ? How many grammes in 1 ton- 
 neau ? How many myriagrammes in 3 tonneaux ? 
 
 2. 1 milligramme is what part of two grammes ? of 
 3 decagrammes? 1 centigramme is what part of a 
 kilogramme ? How many grammes in 5 kilogram- 
 mes ? 
 
 3. What is the cost of two kilogrammes of hay at 
 1 dollar the quintal ? 
 
 4. At 2 dollars the kilogramme, what is the cost of 
 a hectogramme of tobacco? of a decagramme? of 15 
 grammes ? 
 
 5. How many pounds Avoirdupois in 1 myria- 
 gramme ? 
 
 6. One hundred pounds how many kilogrammes ? 
 One hundred kilogrammes how many pounds ? 
 
136 TABLE OF FRENCH MONEY. 
 
 7. What are some of the advantages of the metric 
 system of weights and measures ? 
 
 TABLE OF FRENCH MONEY. 
 
 10 millimes make 1 centime. 
 
 100 centimes make 1 franc = 19 cents ,,3 mills, U. 
 S. money. 
 
 20 francs make 1 louis. 
 
 1. 5 centimes make 1 sou. The franc is how many 
 sous ? 5 francs are nearly one dollar. 
 
 2. A franc in silver weighs 5 grammes. 40 five- 
 franc pieces in silver weigh one kilogramme. 
 
 3. What is the weight of 1000 silver francs? 
 
 4. What is the value of a sum of money weighing 
 two kilogrammes ? 
 
 5. What is the weight of a litre of pure water when 
 it is heaviest ? of a decilitre ? of a decalitre ? 
 
 6. Bought 500 kilogrammes of hay at 1 dollar the 
 quintal ; how much did it cost ? 
 
 7. A garden of 500 square metres is sold for $50 ; 
 how much is that for each are ? 
 
 8. A staircase of 12 steps descends into a cellar 2 
 metres and 76 centimetres deep j what is the height of 
 each step ? 
 
PERCENTAGE. 137 
 
 CHAPTER X. 
 
 PERCENTAGE. 
 LESSON I. 
 
 Per cent, is a contraction for per centum, which 
 means by the hundred, for each hundred, or in a hun- 
 dred. Thus 5 per cent, of any number means 5 parts 
 in every hundred parts of the number, or 5 hundredths 
 of ?he number. To take 6 per cent, of a number is 
 to take 6 hundredths of the number, or 6 times 1 per 
 cent., or 6 times 1 hundredth of the number. The 5 
 and 6 in 5 per cent, and 6 per cent, are called the rates 
 per cent. 5 per cent, and 6 per cent, are merely other 
 forms for the fractions y^ and T -| . 
 
 1. "What is meant by per cent. \ What is, then, 5 
 per cent, of any number \ What is meant by the rate 
 per cent. ? 
 
 2. How much is 1 per cent, of 40 ? 50 ? 60 ? 4 ? 5 ? 6 ? 
 
 3. How much is 1 per cent, of 20 ? 2 per cent, of 20 ? 
 
 3 per cent. ? 5 per cent. ? 4 per cent. ? -J of 1 per cent. ? 
 J of 1 per cent. ? 2|- per cent. ? 10 per cent. ? 
 
 PROCESS. 1 per cent, of 20 is y-^- of 20 = -gfa = \ ; 
 | 0/* 1 jp^/ 1 ?^. i ^ of ^ or T L- ; 4 jp^r cent, of 20 is 
 
 4 x -J-, 6>r |- ; 10 jt?^ tf^rf. is 10 x -J- = 2. 
 
 4. How much is 10 per cent, of 540 ? 320 ? 250 ? 
 100 ? How much is 20 per cent, of the same num- 
 bers? 
 
 5. How much is 100 per cent, of 5 ? of 6 ? of any 
 number? How much is 200 per cent, of 8? 40 ? of 
 any number? 
 
 6. Every per cent, fraction has what for numerator 
 and what for denominator ? 
 
138 PERCENTAGE. 
 
 ANS. The rate per cent, for the numerator, and one 
 hundred for the denominator. 
 
 7. Which per cent, fractions, or wliicli hundredths, 
 can be most readily reduced to simpler fractions ? 
 
 ANS. Those whose numerators are divisors of 100. 
 
 8. What fraction of a number having 1 for its nu- 
 merator is 2 per cent, of the number ? 
 
 PROCESS. 2 per cent, of a number is y-f-g- of the num- 
 ~ber, yf Q- in its simplest form = -^. 
 
 9. What is the simplest form of 4 per cent. ? 5 per 
 cent.? 10 per cent.? 20 per cent.? 25 per cent.? 50 
 per cent. ? 75 per cent. ? 
 
 10. What per cent, of a number is |- of the number ? 
 |- is how many hundredths ? 
 
 11. What per cent, of a number is ^ of it? -J- of it ? 
 TV of it? jV of it? ^ of it? T V of it? T^T of it? ^ 
 of it? of it ? yV of it ? y^ of it ? 
 
 12. Which fractions can be extended most easily to 
 hundredths ? 
 
 ANS. Those of whose denominators 100 is a mul- 
 tiple. 
 
 13. i of a number is what per cent, of the number ? 
 Extend ^ to hundredths. 
 
 PROCESS. 100-^-3 is 33^, hence % is -f- f^ ; -|- 6/*# n^ra- 
 Z>r is |-|^ ? 6>r 33|- jpr cent, of it. 
 
 11. What per cent, of a number is -J- of it ? |- of it? 
 T Vofit? T Vofit? f of it? | of it? -Jofit? f? -f- ? 
 
 15. What fraction of a number is 33^- per cent, of 
 the number? 16f per cent, of it? 12^ per cent, of it? 
 8^ per cent, of it ? 6^ per cent, of it ? 66f per cent. ? 
 87| per cent. ? 60 per cent. ? 80 per cent. ? 90 per 
 cent. ? 
 
 16. What part of 100 is 16f? 12J? 66f ? 6J ? 60? 
 80? 37? 90? 
 
 17. 5 is what per cent, of 60 ? 
 
 PROCESS. 5 is T ^- o/* 60, <m<$ y 1 ^- of a number is T 8 ^- 
 of it, or 8^- ^r cent. Hence 5 is 8-J- per cent, of 60. 
 
PERCENTAGE. 139 
 
 The question, What per cent? is simply, how many 
 hundredth* f 
 
 IS. What per cent, of 30 is 5 ? 10? 20? 25? 30? 
 15? 2? 
 
 19. 8 is what per cent, of 16? 32? 36? 40 ? 60? 
 72? 
 
 20. 2 is what per cent, of 10 ? 8 ? 20 ? 40 ? 50 ? 12 ? 
 100? 2? 200? 
 
 21. % is what per cent, of 5 ? 
 
 PROCESS. ^ is what part of 5 ? -^ of 5 is 10 per 
 cent, of it ; or, 1 is ^ of 5 3 or 20 per cent., hence ^ is 
 10 per cent. 
 
 22. f is what per cent, of 2 ? f is what per cent, of 
 10? 
 
 23. % is what per cent, of f ? 
 
 PROCESS. \ is -f of f ; -J- #/* & number is 16f 
 c'mtf., #716? f is 5 x 16f , <9/' 83-J- JP^ tftffitf. 
 
 24. y^ is what per cent, of f ? 
 
 25. 10 is 5 per cent, of what number ? 
 PROCESS. 5 per cent, is '^ 10 is ^ #/*200, 
 
 cent, of 200. 
 
 26. 18 is 6 per cent, of what number? 
 
 PROCESS. 18 is T f of what number? 6 in IS, 3 
 times, 3 x 100 300; 18 is 6 per cent, rt/300. 
 
 27. 20 is 5 per cent, of what number ? 4 per cent. ? 
 10 per cent. ? 2-J- per cent. ? 6 per cent, of what num- 
 ber ? 
 
 28. |- is 6 per cent, of what number? 8 per cent. ? 
 10 per cent. ? 
 
 29. 2|- is 5 per cent, of what number ? 12 per cent. ? 
 6 per cent. ? 33^ per cent. ? 40 per cent. ? 
 
 30. What is 2 per cent, of 5 per cent, of a number ? 
 PROCESS. 2 per cent, of 5 per cent. = -fa of 5 >6r 
 
 <?m. = -foper cent. = -^ per cent. 
 
 31. How much is 50 per cent, of 25 per cent, of 80? 
 PROCESS. First take 25 per cent, of 80 and then 50 
 
 per cent, of the result. 
 
140 PERCENTAGE. 
 
 32. 25 per cent, of 40 per cent, is what per cent, of 
 20 per cent, of a number ? 
 
 33. What fraction is 5 times 6 per cent. ? 12 is how 
 many times 5 per cent, of 40 ? 12 is 6 times what per 
 cent, of 50 $ 
 
 cent, of 50 
 
 LESSON II. 
 
 1. What is 2 per cent, of $10 ? $8 ? $20 ? $50 ? of 
 1 dime ? 
 
 2. What is 5 per cent, of $100? $300? $600? 
 $1000 ? 
 
 3. What is 6 per cent, of $1? $5? $50? $200? 
 $150? 
 
 4. What is % per cent, of $5 ? $10 ? 100 ? $500 ? 
 $1000 ? 
 
 5. What is 4 per cent, of $100? $200? $500? 
 $1000 ? 
 
 6. 6 cents is what per cent, of $1 ? 60 cents is what 
 per cent, of $10 ? $6 is what per cent, of $600 ? $60 
 is what per cent, of $1000 ? $600 is what per cent, of 
 $10,000? 
 
 7. 5 centimetres is what per cent, of a metre ? 5 
 metres is what per cent, of a hectometre ? 1 litre is 
 what per cent, of a decalitre ? 5 decilitres is what per 
 cent, of a litre? A gramme is what per cent, of a 
 kilogramme ? 
 
 8. How much is 10 per cent, of 60 metres ? 6 per 
 cent, of 20 litres ? 5 per cent, of 40 kilogrammes ? 
 
 9. What is 5 per cent, of 50? 100? 600? 
 1000 ? 
 
 10. How much is 4- per cent, of 1200 ? 4- per cent, 
 of 500? 
 
 11. 5 shillings is what per cent, of 1 ? 3d. is what 
 per cent, of a shilling? 
 
 12. What per cent, of a yard is 1 foot ? 6 inches ? 
 
 13. The number of inhabitants of a village was last 
 
PERCENTAGE. 
 
 year 650, this year it is 30 per cent, greater. What is 
 the increase for the year ? 
 
 14. A commission merchant sells $600 worth of 
 wheat jjpr a farme^ and deducts 4 per cent, commis- 
 sion ; how much does he pay over to the farmer ? 
 
 15. I bought 5 shares of railroad stock, which cost 
 $100 a share ; since the purchase each share has fallen 
 10 per cent. ; what are the 5 shares worth now? 
 
 16. A house valued at $1200 is insured by a com- 
 pany at a cost of 2 per cent, on this sum. How much 
 does the owner of the house pay the company ? 
 
 17. How much is 6 per cent, of $32 ? 80 bushels 
 of corn ? 64 barrels of flour ? 
 
 18. How much is 33|- per cent, of 60 gallons of 
 wine ? 
 
 19. A man owes $6000 and is able to pay only 66f 
 per cent, of it. How much do his creditors lose ? 
 
 20. Of a pound of gunpowder 75 per cent, is salt- 
 petre, 12|- per cent, is sulphur, and 12 is charcoal. 
 How many ounces of each form the pound ? 
 
 21. Cannon metal consists of copper and tin; it 
 contains 90 per cent, of copper. "What per cent, of 
 tin ? 
 
 22. Bought merchandise for $44, sold the same for 
 $22. What was the loss per cent. ? 
 
 23. Bought for $66, sold for $88. What was the 
 gain per cent. ? 
 
 24. Bought for $96, sold for $100. Gained what 
 per cent. ? 
 
 25. Bought for $72, sold for $63. Lost what per 
 cent. ? 
 
 26. Bought for $23, sold for $69. Gained what per 
 cent.? 
 
 27. Bought for $64, sold for $56. Lost what per 
 cent. ? 
 
 28. Bought for $48, sold for $44. Lost what per 
 cent. ? . 
 
14:2 PERCENTAGE. 
 
 29. Bought for $69, sold for $23. Lost what per 
 cent. ? 
 
 30. I bought merchandise for $75 and sold it at a 
 loss of. 6J per cent. For how much did I sell ? For 
 how much would I have to sell it to gain 6^ per cent. ? 
 
 31. Bought for $80, lost on sale 10 per cent. Sold 
 for how much ? 
 
 32. Bought for $72, gained on sale 33^- per cent. 
 Sold for how much ? 
 
 33. Bought for $88, gained 25 per cent. Sold for 
 how much 2 
 
 34. Bought for $57, gained 33^ per cent. Sold for 
 how much? 
 
 35. Bought for $57, lost 33^ per cent. Sold for how 
 much ? 
 
 36. Bought for $112, lost 12|- per cent. Sold for 
 how much? 
 
 37. A man has 200 bushels of wheat, he sells 75 per 
 cent, of it at $2 a bushel and the remainder at $3 a 
 bushel. How much does he get for the wheat ? 
 
 38. The rent of a house was last year $75, but this 
 year the owner increased the rent 20 per cent. What 
 is the rent this year? 
 
 39. Gunpow r der contains 75 per cent, of saltpetre. 
 How much saltpetre in 120 Ibs. of it? 
 
 40. It requires 16,000 bricks to build a certain house. 
 If we allow 6 J per cent, for loss of bricks by breakage, 
 how many must be purchased for the house ? 
 
 41. The rent of house was increased this year 10 per 
 cent., and is now $99. What was it before the in- 
 crease ? 99 is -^ of what number ? 
 
 42. A man gave $4000 for a house, and it costs him 
 for repairs each year 2 per cent, on the purchase money. 
 What is the yearly cost of repairs? 
 
 43. If 30 quarts of wine contain 6 quarts of water, 
 what per cent, of water does the wine contain ? 
 
 44. From $35.50 subtract 20 per cent. 
 
PERCENTAGE. 143 
 
 45. A merchant bought cloth at $4-J- a yard, and 
 wishes to sell it at 25 per cent, profit, ^hat must 
 be the selling price per yard ? 
 
 46. The gross weigJtt of an article of merchandise 
 is its weight including the box, barrel, or bag which 
 contains it. The net weight is the weight after the 
 weight of the box, &c., is deducted. The allowance 
 for the weight of the box or barrel, &c., is called the 
 tare. 
 
 47. The gross weight of some merchandise is 160 
 Ibs., and the tare is 6 per cent. "What is the net 
 weight ? 
 
 48. Gross weight 210 pounds, tare 10 per cent. 
 What is the net weight ? 
 
 49. Gross weight 250 pounds, net weight 230 pounds. 
 What per cent, of tare? 
 
 50. Gross weight 220 pounds, tare 5 per cent. 
 What is the net weight ? 
 
 51. A man bought a horse for $100, and sold him 
 for $300. What per cent, did he gain ? 
 
 52. The person who bought the horse at $300 sold 
 it for $100. What per cent did he lose ? 
 
 53. $5 is what per cent, of $6 ? $6 is what per 
 cent, of $5. 
 
 54. Bought an article for 2s. 6d., and sold it for 3s. 
 6d. What was the gain per cent. ? 
 
 55. Bought an article at $4.30, sold it for $5.30. 
 What was the gain per cent. ? 
 
 56. Bought 5 pounds*of sugar at Is. 6d., and sold it 
 at 5d. a pound. What per cent, did I gain ? 
 
 57. Sold a horse for $03, and lost 12J per cent, of 
 what he cost me. What did he cost ? 
 
 SOLUTION. Lost -J-. Hence $63 is^of his cost. 
 
 58. If $150 in currency are worth only $100 in 
 gold, what percentage is lost in receiving in currency 
 a debt payable in gold? $50 is what part of $150 '} 
 % of a number is what per cent, of it ? 
 
144 PERCENTAGE. 
 
 59. If $100 in gold are worth $150 in currency, 
 at what per cent, premium is gold ? 
 
 NOTE. Let the pupil frame questions similar to the above, and 
 give the ablutions. 
 
 LESSON III. 
 SIMPLE INTEREST. 
 
 Interest on money is the sum which the borrower 
 pays the lender for the use of the borrowed money. 
 The money loaned is called the Principal. The sum 
 of the principal and interest is called the Amount. 
 
 Simple interest is computed at so much for a 
 given time usually a year for every hundred dollars, 
 pounds, or francs of the loan / in other words, at a 
 given rate per cent, per year on the principal. 
 
 In computing for shorter periods than a year, thirty 
 days are usually considered a month, and twelve 
 months a year. 
 
 1. At 6 per cent., what is the interest on $1 for a 
 year? $5 for a year? $10? $50? $500? $20? 
 
 2. At 7 per cent., what is the interest on $1 for one 
 year ? on $2 for one year ? on $5 ? $10 ? $20 ? $100 ? 
 $1000? $300? 
 
 3. At 8 per cent., what is the interest on $1 for one 
 year ? on $10 for one year ? on $100 ? on $1000 ? on 
 $5? $50? $500? $5000? 
 
 4. At 10 per cent., what id the interest on $1 for 
 one year ? on $10 ? on $5 ? $100 ? $1000 ? $500 ? 
 $3000 ? 
 
 5. At 3 per cent., what is the interest on $1 for 
 one year? on $10? on $100? on $1000? on $60 1 
 on $600 ? 
 
 6. What is the interest on $1 for 2 years at 6 per 
 cent. ? 7 per cent. ? 8 per cent. ? 10 per cent. ? 3|- per 
 cent. ? 4J per cent. ? 
 
SIMPLE INTEREST. 14:5 
 
 7. What is the interest on $20 for 3 years at 5 per 
 cent. ? 6 per cent. ? 7 per cent. ? 8 per cent. ? 10 per 
 cent. ? 3^ per cent. ? 
 
 8. What is the interest on $200 for 3|- years at 5 
 per cent. ? 7 per cent. ? 10 per cent. ? 3J per cent. ? 6 
 per cent. ? 
 
 9. What is the interest on 25 for one year at 6 
 per cent. ? 
 
 10. What is the interest on 20 for 10 years at 5 per 
 cent. ? 
 
 11. What is the interest on $1 for 1 month at 6 per 
 cent. ? 
 
 ANS. r V of 6 cents, or -J- cent = 5 mills. 
 
 12. T V of 6 per cent, of any number is what per cent, 
 of that number? 6 per cent, per year on a sum of 
 money is what per cent, per month on it ? 
 
 13. What is the interest on $10 for 1 month at 6 per 
 cent.? on $100? on $1000? on $3 ? $30? $25? 
 
 14. What is the interest on $1 at 6 per cent, per 
 year for 2 months? 3 months? 5 months? 9 months? 
 7 months ? On $10 for 2,V months ? on $50 for 3 months ? 
 on $100 for 6 months? *8 months ? 10 months? 
 
 15. What is the interest on $1 for 1 month at 8 per 
 cent. ? 7 per cent. ? 10 per cent. ? 5 per cent. ? 8 per 
 cent, for 12 months is what per cent, for 1 month ? 
 
 16. How much is f of 1 per cent, of $1 ? T 7 j of 1 
 per cent, of $1? |- of 1 per cent, of $1 ? T 5 T of 1 per 
 cent, of $1 ? 
 
 17. What is the interest on $100 for 7 months at 6 
 per cent. ? 7 per cent. ? 10 per cent. ? 8 per cent. ? 
 
 18. What is the interest on $84: at 7 per cent, for 2 
 months ? 3 months ? 6 months ? 8 months ? 10 months ? 
 9 months? 2, 3, 6, 8, 9, and 10 form what parts, re- 
 spectively, of 12 ? 
 
 19. At 6 per cent., what is the interest on $50 for 4 
 years and 3 months ? for 3 years and 2 months ? for 3 
 years and 6 months ? 2 years and 4 months ? 1 year 
 and 7 months? years and 9 months? 
 
14:0 SIMPLE INTEREST. 
 
 20. What is the interest on $200 for 3 years and 5 
 months at 7 per cent. ? 8 per cent. ? 10 per cent. ? 3 
 per cent. ? 5 per cent. ? 
 
 21. The interest on any sum for a month is what 
 part of the interest for one year ? The interest on any 
 sum for 1 day is what part of the interest on it for 1 
 month ? on any sum for 2 days ? 3 days ? 5 days ? 6 
 days ? 15 days ? 10 days ? 9 days ? 12 days ? 18 days ? 
 24 days ? 
 
 22. * At 6 per cent., what is the interest on $1 for 6 
 days ? on $5 for 6 days ? on $50 for 6 days ? on $500 
 for 6 days ? on $5000 for 6 days ? on $1 for 12 days ? 
 24 days ? 60 days ? 3 days ? 5 days ? 7 days ? 
 
 23. AVhat is the interest on $100 for 4 months and 
 15 days at 7 per cent. ? 8 per cent. ? 6 per cent. ? 5 
 per cent. ? 10 per cent. ? 3|- per cent. ? 
 
 24. What is the interest on $50 at 8 per cent, for 3 
 days? 5 days? 10 days? 12 days? 24 days? 8 days? 
 
 25. At 6 per cent., what is the interest on $1 for 60 
 days? on $10 for 60 days? on $50? on $250? on 
 $144? 
 
 26. At 6 per cent., what is the interest on $120 for 
 3 years 2 months and 12 days ? for 4 years 5 months 
 and 10 days? 
 
 27. What is the interest on $300 for 2 years 4 
 months and 15 days at 7 per cent. ? 8 per cent. ? 10 
 per cent. ? 5 per cent. ? 
 
 28. The interest on a certain sum for a given time 
 at 6 per cent, is $18. What is it at 5 per cent. ? at 1 
 per cent. ? What part of 6 per cent, is 5 per cent. ? 
 
 29. The interest on any principal at 1 per cent, for 
 one year is what part of the principal ? at 4 per cent., 
 is what part of the principal ? at 12 J per cent. ? 
 
SIMPLE ESTEREST. 
 
 LESSON IV. 
 
 1. What is the amount of $1 at 6 per cent, for 1 
 year ? What is the amount of $10 at 5 per cent, for 
 1 year ? 
 
 ANS. 1. $1.06; 2. $10.50. 
 
 2. What is the interest on $150 for 2 years at 6 per 
 cent. ? What is the Amount ? 
 
 3. At 6 per cent, what is the amount of $200 for 
 60 days ? What is the amount of $80 for 90 days ? 
 
 4. What is the amount of $50 for 2 years and 6 
 months at 4 per cent. ? 7 per cent. ? 8 per cent. ? 10 
 per cent. ? 5 per cent. ? 
 
 5. What is the amount of $200 for 2 years 6 months 
 and 15 days at 6 per cent. ? at 10 per cent. ? 5 per 
 cent. ? 
 
 6. What is the amount of $300 for 9 years at 8 per 
 cent. ? at 7 per cent. ? 
 
 7. At what rate per cent, will $100 yield $5 interest 
 in 1 year ? At what rate per cent, will $20 yield $1 
 in one year? $5 is what per cent, of $100 ? $1 is 
 what per cent, of $20 ? 
 
 8. The interest on $200 for 2 years is $14. What 
 is the rate per cent. ? 
 
 SOLUTION. What is the interest for 1 year f $7 is 
 what per cent, of $200 ? 
 
 9. If the interest on $60 for 1 year and 4 months is 
 $4, what is the rate per cent. ? 
 
 10. If the interest on $90 for 3 years and 4 months 
 is $30, what is the rate per cent. ? 
 
 11. At what per cent, will $20 gain $6 in 1 year and 
 3 months ? 
 
 12. At what per cent, will $100 gain $100 interest, 
 or double itself, in 20 years ? in 10 years ? in 5 years ? 
 
 16 J years ? in 12-|- years ? in 14f years ? At what 
 
 '" 
 
148 SIMPLE INTEREST. 
 
 rate per cent, will any sum double itself in the times 
 mentioned ? 
 
 SOLUTION. Tjf $100 gain $100 in 20 years, how much 
 must it gain in one year? If $100 gain $100 in 20 
 years, liow many dollars interest will $500 gain ? 
 
 13. If the interest on $60 at 5 per cent, is $6, how 
 long has it been on interest ? 
 
 SOLUTION. The interest on $60 for one year at 5 per 
 cent, is how much f $3 is contained in $6 how many 
 times f 
 
 14. In what time, at 6 per cent., will $10 gain $6 
 interest ? In what time, at 8 per cent., will $50 gain 
 $12 interest? 
 
 15. In what time, at 7 per cent., will $40 gain $6 
 interest ? 
 
 16. In what time, at 10 per cent., will $50 gain 
 $1241 
 
 17. In what time will $1 gain $1 interest, at 6 per 
 cent. ? 
 
 SOLUTION. At 1 per cent. $1 will double itself, or 
 make WO per cent., in 100 years. Hence, at 6 per cent., 
 it will double itself in -i-g- - years, or 1 6f years. 
 
 18. In what time will a given principal double 
 itself at 5 per cent. ? 7 per cent. ? 8 per cent. ? 10 per 
 cent. ? 3| per cent. ? 25 per cent. ? 
 
 19. In what time will $100 gain $200 interest, at 6 
 per cent. ? at 7 per cent. ? at 8 per cent. ? at 10 per 
 cent. ? 
 
 20. What principal, at 6 per cent., will, in 1 year, 
 gain $48 interest ? 48 is yf^ of what number ? 
 
 21. What principal, at 6 per cent., will, in 4 years, 
 yield $48 interest ? $48 in 4 years is how much in 
 1 year ? 12 is y^ of what number ? 
 
 22. What principal, in 1 year and 4 months, at 8 
 per cent., will yield $32 interest ? 
 
 23. What principal, in 2 years and 3 months, at 7 
 per cent., will yield $21 interest? 
 
SIMPLE INTEREST. 149 
 
 24. What principal, in 7 months and 15 days, at 8 
 per cent., will yield $40 interest ? 
 
 25. What principal, in 7 months and 15 days, at 4 
 per cent., will yield $40 interest ? 
 
 26. What principal, at 10 per cent., will in 3 years 
 and 6 months yield $60 interest ? 
 
 27. What principal, at 10 per cent., will amount to 
 $55 in one year ? 
 
 SOLUTION. The interest is ^- of^ file principal. 
 The amount is therefore ^ of the principal, and $55 
 is 1. of $50. 
 
 28. What principal will, in 12 years, at 5 per cent., 
 amount to $120 ? 
 
 29. What principal will, in 5 years, at 7 per cent., 
 amount to $85 ? 
 
 30. What principal, at 8 per cent., will amount to 
 $116 in 4 years? 
 
 31. What principal, at 10 per cent., will amount to 
 $44 in 2 years ? 
 
 32. What principal, at 10 per cent., will amount to 
 $200 in 10 years ? What at 5 per cent, will amount 
 to $200 in 20 years ? What at 6 per cent, will amount 
 to $200 in 16| years ? What at 8 per cent, to $200 in 
 
 33. The present worth of a debt due at some future 
 time without interest is equal to a principal which put 
 at interest will amount to the debt when it is due. 
 
 The discount on a debt is the difference between the 
 debt and its present worth. 
 
 34. What is the present worth and the discount of 
 $45 due live years hence without interest, money 
 'being at 10 per cent. ? 
 
 SOLUTION. What principal at W per cent, will in 
 5 years amount to $45 ? What is the difference be- 
 tween $45 and $30 ? 
 
 35. What is the present worth of $100 due in 90 
 clays, money being at 6 per cent. ? at 7 per cent. ? at 
 
150 SIMPLE INTEREST. 
 
 8 per cent. ? at 10 per cent. ? at 5 per cent. ? What 
 is the discount in each case ? 
 
 36. What is the discount of $100 due in 10 years, 
 money being at 10 per cent. ? due in 12 years at 8 per 
 cent. ? in 14| years at 7 per cent. ? 
 
 37. What is the discount of $600 dollars due in 5 
 years at 4 per cent. ? due in 4 years at 5 per cent. ? 
 due in 2 years at 10 pe^ cent. ? due in 1 year at 20 
 per cent. ? 10 years at 2 per cent. ? 
 
 38. What is the discount on $160, due in 4 months, 
 at 6J per cent. ? 
 
 39. What is the discount on $144, due in 15 days, at 
 8 per cent. ? 
 
 40. What sum placed at interest will amount to 
 $960 in 6 years at 10 per cent. ? in 12 years at 5 per 
 cent. ? in 10 years at 6 per cent. ? 
 
 41. When banks lend money, they usually 'discount 
 or take off from the sum for which the borrower gives 
 his note, the interest for the time, and lend the bor- 
 rower the difference. 
 
 42. What is the bank discount on $100 for 6 months, 
 at 6 per cent. 
 
 SOLUTION. The bank discount is the interest on $100 
 for 6 months at per cent., $3. The 'borrower gives 
 his note for $100 without interest, and gets $97. 
 
 43. What is the bank discount on $5,000 for 60 
 days, at 6 per cent. ? at 10 per cent ? 
 
 44. I wish to borrow $990 from a bank for 60 days, 
 at 6 per cent. For what sum must I give my note 
 to the bank ? 
 
 45. What is the bank discount on $100 for 90 days, 
 at 6 per cent. ? 7 per cent. ? 8 per cent. ? 10 pel 1 
 cent. 3 
 
MISCELLANEOUS. 151 
 
 CHAPTEE XI. 
 
 MISCELLANEOUS. 
 L E S S 1ST I .Proportional Parts. 
 
 1. Divide 30 into two numbers proportional to the 
 numbers 1 and 2 ; or parts which shall be to each 
 other as 2 is to 1 ; or which divided one by the other 
 shall give the same quotient as 2 -f- 1. 
 
 SOLUTION. 2 and 1 make 3. Divide 30 into 3 equal 
 parts, and 2 of tJiese parts will make one of the num- 
 berSj and the remaining part will be the other. We 
 thus get 20 and 10, and 20 -^ 10 = 2-f-l. 
 
 2. Divide 35 into two parts proportional to 3 and 4. 
 
 3. Divide 50 into two parts proportional to 4 and 5. 
 
 4. Divide 40 into two such parts that the first shall 
 be 4 times the second. 
 
 5. I divided 60 oranges between Charles and Frank, 
 giving Charles 7 for each 5 I gave to Frank ; how 
 manj did each receive ? 
 
 6. I wish to divide $190 between two men, giving 
 $10 to one for each $9 which I give to the other ; how 
 much will each receive ? 
 
 7. Divide 36 into 3 parts wliich shall be to each 
 other as the numbers 2, 3, and 4. 
 
 8. Three men worked together for a certain time at 
 diiferent wages. The first at $3 a day, the second at 
 $2 a day, and the third at $1 a day. At the end of 
 the time they received $60 together. What is the 
 share of each ? 
 
 9. Divide 80 into two parts which shall be to each 
 other as ^ is to |-. 
 
152 MISCELLANEOUS. 
 
 SOLUTION. J- is to % means -J- -~- -J-, or 3 -4- 2. 
 divide 80 -info 2 jpa/'te proportional to 3 a?irf 2 ; 
 is to -J- a-9 f is to f as 3 &s to 2. 
 
 10. Divide 34 into two parts proportional to 
 and f . 
 
 11. Divide 51 into three parts proportional to f , , 
 and J. 
 
 NOTE Fractions reduced to a common denominator are to each other 
 as their numerators. 
 
 12. In an orchard containing 96 trees there are 5 
 apple trees for every 4 peach trees, and 4 peach trees 
 for every 3 pear trees ; how many trees of each kind 
 in the orchard ? 
 
 13. Divide $200 among A, B, and C, so that B 
 may get twice as much as A, and C three times as 
 much as A. 
 
 14. Two men cut a certain amount of wood for $55 ; 
 one cuts 6 cords while the other cuts 5 ; what should 
 be the share of each one ? 
 
 15. Two men engage in trade together ; one puts in 
 $8,000 for a year and the other $8,000 for a year. At 
 the end of the year the profits are $2,200 ; how much 
 should each receive ? 
 
 16. Three men engage in trade ; A puts in $2,000 
 for one year, B $3,000 for the same time, and C 
 $5,000 for the same time also. They gain together 
 $1,200 ; what is the share of each in the profits ? 
 
 17. Three men hire a pasture for a given time for 
 $60. One of them keeps 3 cows in it for the whole 
 time, the second keeps 5 cows, and the third keeps 4 
 cows. "What part of the $60 should each one pay ? 
 
 REMARK. When partners put sums of money in 
 business together for different times, their respective 
 shares of tie profit or loss are sometimes by agree- 
 ment proportional to the amounts put in multiplied by 
 the times respectively. 
 
 18. Two men form a partnership ; A puts in $200 
 
MISCELLANEOUS. 153 
 
 for 5 months, and B puts in $300 for 2 months. They 
 gain $96 ; what is each man's share of the profits ? 
 
 19. In trading together, A puts in $100 for 6 
 months, and B $50 for 4 months. They lose $64 ; 
 what is each one's share of the loss? 
 
 20. Three men rent a pasture for $60. In it A 
 pastures 5 cows for 2 weeks, B pastures two cows for 
 4 weeks, and C 4 cows for 3 weeks ; how much must 
 each one pay for the pasturage? 
 
 21. Eeduce the ratio of 7,000 ~ 5,000 to its simplest 
 form. 
 
 22. Reduce the ratio of *jj- -f- f to the ratio of two 
 whole numbers. 
 
 23. Change the terms of the ratios 25 -f- 5, 40 -f- 8, 
 120 -f- 12, 4 3, 11 -r- 5, in such manner that each 
 divisor becomes 1 and each quotient remains un- 
 changed. 
 
 24. What number is that to which if its half and 
 fourth be added the sum will be 28 ? 
 
 25. A and B are 50 miles apart, traveling toward 
 each other. When they meet it is found that of the 
 distance traveled by A is -| of that traveled by B ; 
 what distance had each traveled ? 
 
 26. ^ of the time past noon = \ of the time before 
 midnight ; what o'clock is it ? 
 
 SOLUTION. Divide 12 into two parts, which bear 
 to each other the ratio of \ to -J, i. <?., 3 to 5. 
 
 27. What o'clock is it when the time past noon is 
 -| of the time before midnight ? 
 
 28. What number added to twice itself gives 24 ? 
 
 LESSON II. 
 
 1. The sum of two numbers is 25, and the greater 
 exceeds the less by 11 what are the fwo numbers? 
 
 SOLUTION. Twice the less number added to 11 is 
 equal to 25. Hence, twice the less number is equal to 
 
154 MISCELLANEOUS. 
 
 the difference 'between 25 and 11, or the less number 
 == |- /us difference = 7. -7% greater is then 7+11 
 = 18. ^ 
 
 2. Divide 40 into two such parts that the greater 
 shall exceed the less by 22. 
 
 3. Of a pole 50 feet long, -J- as much is in the mud 
 as in the water, and 3 times as much in the air as in 
 the mud and water ; how much in the mud, air, and 
 water, respectively ? 
 
 4. A boy being asked how many apples he had, re- 
 plied, if besides what he then had, he also had {- and -J- 
 as many more he would hafre 57. How many had he? 
 
 5. One man can do a piece of work in 5 days, and 
 another in 20 days ; in what time can both do the 
 work ? 
 
 SOLUTION. The first does -J- of the work in one day, 
 and the second $. Together they do ^ -f -fa = -^ = J 
 of the work in one day or the whole work in 4 days. 
 
 6. A can reap a field of wheat in 10 days, B can 
 reap the same field in 90 days ; in what time can they 
 both do the work together ? 
 
 7. A can do a piece of work in 4 days, B can do it 
 in 6 days, and C in 3 days ; in what time can they all 
 do it together ? 
 
 8. A barrel of flour lasts a certain man and his wife 
 12 weeks, but when the man is absent it lasts his w.ife 
 20 weeks ; how long would a barrel last the man alone ? 
 
 9. A and B can do a piece of work in 12 days, but 
 when B is absent A requires 20 days to do the same 
 work ; how long would it require B to do the work 
 alone ? 
 
 10. A and B can do a piece of work in 3 days, A 
 and C in 4 days, B and C in 5 days ; in what time 
 can each one, working alone, do it? In what time can 
 they all do the work together ? 
 
 11. A cistern is discharged by three pipes ; the first 
 and second together will discharge it in one hour, the 
 
MISCELLANEOUS. 155 
 
 first and third together in 2 hours, the second and third 
 in 3 hours; in what time can they all three discharge 
 it ? In what time can each discharge it ? 
 
 12. A, B, and C can perform a certain work in 4 
 days, C working alone can do the same work in 12 
 days, and B alone in 10 days ; in what time can C do 
 the work ? In what time can B and C together ? 
 
 13. One pipe leading into a cistern can fill it in 3 
 hours, and another pipe leading from a cistern can 
 empty it in 5 hours. If the water is running through 
 both pipes at the same time, in what time will the cis- 
 tern be filled ? 
 
 14. A man and his wife would eat one cwt. of flour 
 in 18 days ; after living together for 10 days, the 
 woman could eat the remainder in 30 days ; in how 
 many days does the man eat one cwt. of flour ? 
 
 15. A man wishes to. give 10 cents apiece to some 
 beggars, but finds he has not money enough by 9 
 cents : he gives them, however, 8 cents apiece, and* 
 has 5 cents remaining; how many beggars were there ? 
 
 16. A can reap f of a certain field in 2 days, and 
 Bean reap T 9 Q- of the same field in 3 days; in what 
 time can they both together reap the whole field? 
 
 17. A can reap 1 acre in 2 hours, B can reap 2 
 acres in 3 hours ; in what time will they together 
 reap 14 acres ? And if the payment is half a dollar an 
 acre, how much will each one receive ? 
 
 18. A man hired a laborer, giving him one dollar 
 a day for each day that he labored, and exacting 25 
 cents for each day that he was idle. At the end of 
 18 days he received 8 dollars ; how many days did 
 he labor and how many was he idle ? 
 
 19. The sum of two numbers is 20, and three times 
 the less is less by 15 than twice the greater part ; what 
 are the two parts ? 
 
 20. The sum of three numbers is 20. The sum of 
 the first and second is equal to the third, and half the 
 
156 MISCELLANEOUS. 
 
 sum of the first and third is equal to twice the 
 second. 
 
 21. If f of a number is increased by 10 the sum is 
 40. What is the number ? 
 
 22. If |- of a number be diminished 10 the remainder 
 is 25. What is the number? 
 
 23. A can do a piece of work in -J- of a day and B 
 in J of a day. How long will it take both to do it ? 
 
 24. Three men, A, B, and C, can do a piece of 
 work in 6 days ; A can do it alone in 10 days, and B 
 in 12 days ; in how many days can C do the work ? 
 
 25. A cistern of 200 gallons is emptied in 30 minutes 
 by 3 pipes. The first pipe discharges -J- gallon in a 
 minute, the second l gallon ; how much does the 
 third pipe discharge per minute ? 
 
 26. A can do a piece of work in 2 days, B in 3 
 days, C in 4 days, and D in 6 days. In what time 
 can they all together do the w r ork ? 
 
 LESSON III. 
 
 1. Eggs are sold at the rate of 5 for 3 pence. At 
 what rate must they have been bought in order to 
 make this a gain of 20 per cent. ? 
 
 2. A man mixed 20 gallons of ale worth 11 pence a 
 gallon with 30 gallons of beer worth 8 pence a gallon ; 
 what is the price per gallon of the mixture ? What 
 must be the selling price per gallon in order to gain 
 20 per cent. ? 
 
 3. A merchant sold his flour at various prices one 
 barrel for $9, a second for $10, a third barrel for $12, 
 a fourth barrel for $11, a fifth barrel for $8. What 
 was the average price per barrel ? 
 
 4. What is the average of the numbers 7, 8, 12, 7J 
 and 5iJ-? 
 
 5. If a pole 30 feet high at a given time of day casts 
 a shadow 15 feet long, how long a shadow will be cast 
 
MISCELLANEOUS. 157 
 
 at the same time by a pole 13 feet high ? How high 
 a pole will at the same time cast a shadow 20 feet 
 long ? 
 
 6. If ten oxen eat 2 tons of hay in 30 days, how 
 many oxen will it take to eat 16 tons in 50 days ? in 
 60 days ? in 75 days ? 
 
 7. If 6 men can dig a ditch of a certain depth and 
 width 8 rods long in 4 days, how long will it take 12 
 men to dig a ditch of the same depth and width, 40 
 rods long ? 72 rods long ? 
 
 8. The posts of an electric telegraph by the side of 
 a railway are placed at intervals of 60 yards ; at what 
 rate must a train move to pass 10 of these in 20 
 seconds ? 
 
 9. The work of 54 men in 8 days is equal to that 
 of 16 workmen in how many days '{ 
 
 10. If 15 workmen in 30 days finish 100 rods of a 
 wall of certain thickness and height, how many rods 
 can 6 workmen finish in 81 days ? How many rods 
 60 workmen in 9 days ? 9 workmen in 90 days ? 12 
 workmen in 54 days? 
 
 11. If 12 workmen can dig 48 rods of a ditch of 
 certain width and depth in 6 days, in how many days 
 can 50 workmen dig 600 rods ? 30 workmen 180 rods? 
 
 12. A small steam-engine of one horse-power burns 
 in an hour 20 pounds of coal ; how many hundred- 
 weight will it require in 5 days? 
 
 13. When $100 in gold is worth $148 in currency, 
 how much is $5 in gold worth? 
 
 14. What is the price of 240 articles at 3 pence 
 apiece ? at 8 pence apiece ? What is the price of 
 300 articles at 5 shillings each ? of 240 articles at 3 
 cents each ? of 300 articles at $5 each ? 
 
 15. A fox is 50 rods ahead of a hound, and is run- 
 ning 70 rods a minute ; the hound is pursuing at the 
 rate of 75 rods a minute ; in how many minutes will 
 the hound overtake the fox ? 
 
158 MISCELLANEOUS. 
 
 10. If an army marches 130 miles in 6|- days, 
 marching 9 hours a day, how far can it march in 4 
 days by marching 10 hours a day? 
 
 LESSON IV. 
 
 1. 5 is ^-. Hence to multiply a number by 5", an- 
 nex and divide by 2. 42 x 5 = 420 -=- 2 =* 210. 
 
 Multiply 94, 48, 63, 75, and 448 by 5. 
 
 2. 25 = J-jp-. Hence to multiply a number by 25, 
 annex 00 and divide by 4. 36 x 25 = 3600 -^- 4 900. 
 
 Multiply 36, 72, 68, 84, and 79 by 25. 
 
 3. 125 = ^-. Hence to multiply a number by 
 125 we annex 000 and divide by 8. 
 
 4. Multiply 56, 72, 88, and 91 by 125. 
 
 5. 99 is 100 1. Hence to multiply a number by 
 99 we multiply it by 100 and then subtract the num- 
 ber itself from the product. 67 x 99 = 6700 67 = 
 6633. 
 
 Multiply 34, 76, 29, and 34 by 99. 
 
 6. Multiply 45, 63, 72, and 51 by 999. 
 
 7. Multiply 32, 24, 48, and 52 by 98; also by 102. 
 
 8. What change is made in the product of 28 multi- 
 plied by 32 by adding 10 to the multiplier ? What 
 change by adding 10 to the multiplicand ? What 
 change by multiplying the multiplicand by 10 ? the 
 multiplier by 10? by multiplying the multiplier and 
 multiplicand both by 10 ? by multiplying the multi- 
 plicand by 10 and dividing the multiplier by 10 at 
 the same time ? 
 
 9. What change is made in the quotient of 80 16 
 by dividing the dividend by 5 ? by dividing the divisor 
 by 4? by dividing both by 4? What change is made 
 in the quotient by multiplying both the divisor and 
 dividend by 4 ? 
 
 10. What is the square of 24 ? 
 
 SOLUTION. 24 = 20 + 4; (20 + 4) x (20 + 4) gives 
 
MISCELLANEOUS. 159 
 
 20 x20 + 4x20+4 x 20 + 4x4 = 28x20 + 16= 560 + 
 16 = 576. Hence, to square a number of two figures 
 we add the number of units to the number, multiply 
 this l>y the tens, and add to it the square of the units. 
 Thus, to square 46, add 6 to 46, giving 52. Multiply 
 52 l>y 4 tens or 40, giving 2080, and to this add 36. 
 
 11. What is the square of 13? of 14 ? of 15 ? 16 ? 
 IT? 18? 19? 
 
 12. What is the square of 20 ? of 30 ? of 40 ? of 90 ? 
 
 13. Find the square of 31; of 37; of 52; of 56 ; 27. 
 In all the preceding chapters of miscellaneous 
 
 questions the pupil should be required to frame 
 several questions of each sort presented, and the 
 teacher should add questions in each recitation. 
 
160 NUMERATION AND NOTATION. 
 
 CHAPTER XII. 
 
 ELEMENTARY LESSOXS IN WRITTEN 
 ARITHMETIC. 
 
 LESSON I. 
 Numeration and Notation. 
 
 1. IN the preceding chapters the pupil has learned 
 orally the numeration and notation of numbers as 
 high as thousands. The numbers from one to one 
 thousand form a class of three orders of numbers. 
 The first order is units; second, tens of units; and 
 third, hundreds of units. In the same manner there 
 are three orders of the class thousands: first, thou- 
 sands ; second, tens of thousands third, hundreds of 
 thousands. As to ten hundred we gave a new name, 
 thousand so to ten hundred thousand it is conve- 
 nient to give a new name, and thus form a third class. 
 This we call a 'million. 
 
 2. We have also learned in a former chapter how 
 to express numbers as high as thousands by means of 
 the nine Arabic or Indian figures, together with the 
 symbol of place, 0. These nine figures are called sig- 
 nificant figures^ and is called a cipher, or naught, or 
 zero. We have learned that a figure, as 7, standing 
 by itself expresses 7 simple units ; that a or another 
 figure placed to the right of it, thus, 70 or 72, makes 
 it express seven tens, that is, seven units of the next 
 higher order; that two O's, or two figures, or a cipher 
 and a significant figure, placed to the right of it, as 
 700, 708, 718, or 780, make it express seven units of 
 
NUMERATION AND NOTATION. 161 
 
 the next higher order, or seven hundreds ; and that 
 three significant figures or ciphers placed to the right 
 of it cause the seven to express seven thousands. 
 
 3. The law here is, that each additional figure to the 
 right of a figure increases tenfold the value of the 
 original figure. We can now extend this law to all. 
 higher numbers. To make the figure 7 express seven 
 tens of thousands, we annex four figures to it ; to make 
 it express seven hundreds of thousands, we annex five 
 figures ; to make it express seven millions, we annex 
 six figures. If a number is expressed in seven figures, 
 the first figure on the right expresses units, the next 
 tens, the next hundreds, the next thousands, the next 
 tens of thousands, and so on to millions. That is, 
 moving a figure one place to the left, makes its value 
 ten times as great as before. 
 
 Table. 
 
 
 
 PI 
 
 
 
 
 c3 
 
 
 
 
 00 
 
 
 
 
 g 
 
 ^ 
 
 
 
 s 
 
 
 
 
 
 w 
 
 2 
 
 
 
 o 
 
 o 
 
 
 
 . -3 
 1 | 
 
 
 
 "o 
 
 1 
 
 CO 
 
 CO 
 
 3 s 
 
 r3 d 
 
 ce 
 
 d 
 
 n 
 
 o 
 
 3 
 
 ^ M 
 
 H 
 
 EH 
 
 PH 
 
 
 
 I, 
 
 3 
 
 
 3 
 
 5, 
 
 6 
 
 2 
 
 9 
 
 o, 
 
 1 
 
 4, 9 
 
 
 
 2, 
 
 6 
 
 to 
 
 7 4 
 
 4 1 (2' 
 
 6 2 (3) 
 
 (4) 
 
 (1) iri read, One thousand three hundred and seventy- 
 four. 
 
162 NUMERATION AND NOTATION. 
 
 (2) is read, Thirty-five thousand six hundred and 
 forty-one. 
 
 (3) is read, Two hundred and ninety thousand one 
 hundred and sixty-two. 
 
 (4) is read, Four million nine hundred and two 
 thousand six hundred. 
 
 4. The is put in the place of any order of units 
 which is wanting in the number to be expressed. 
 Thus five thousand and six has no tens nor hundreds. 
 Hence it is written 5006, because in order that 5 may 
 express five thousand, it must have three figures after 
 it, and as we have only one significant figure to write, 
 we therefore use two O's. 
 
 NOTE. When the number of tens is more than 1, we use the ter- 
 mination -ty. 74 is seven- ty-four, instead of seven-ten s-and-four. 
 Twenty, thirty, forty, fifty, are slightly changed from twain-ty, three-ty, 
 four-ty, five-ty. In 62,354, instead of saying six-tens-and-two thou- 
 sand three hundred five-tens-and-four, we say sixty-two thousand 
 three hundred and fifty-four. 
 
 5. EXERCISES. 
 
 Let the pupil read, or write in words, the following 
 numbers : 
 
 1. 684, 596, 4000, 24000, 1000, 1001, 1021. 
 
 2. 50, 500, 5000, 50000, 50500, 50505. 
 
 3. 403, 430, 4308, 4764, 40764. 
 
 4. 10000, 10094, 10904, 10094. 
 
 5. 60, 604, 1604, 21604, 421640. 
 
 6. 123, 321, 312, 444, 4444, 44444, 444444. 
 
 7. 2000, 20000, 200200, 2200200, 2220202. 
 Express in figures the following numbers : 
 
 1. Six hundred; six hundred and forty; six hun- 
 dred and forty-six. 
 
 2. Four hundred and four ; four hundred and four- 
 teen ; four hundred and forty. v 
 
 3. Three hundred and thirty ; three thousand ; three 
 thousand and three hundred. 
 
ADDITION. 163 
 
 4. Six thousand; six thousand four hundred and 
 forty ; six thousand four hundred and four. 
 
 5. Nine hundred and one ; one hundred and nine ; 
 one hundred and ninety. 
 
 6. Eight hundred ; eight Hundred thousand ; eight 
 hundred thousand four nundred and fifty ; four hun- 
 dred and fifty-five. 
 
 7. One thousand ; one million ; six million ; six 
 million six thousand and six hundred ; six million six 
 hundred and fifty-four thousand three hundred and 
 twelve. 
 
 LESSON II. 
 Addition. 
 
 6. By ADDITION we unite two or more numbers in 
 one number. 
 
 The result of the addition is called the SUM or 
 AMOUNT. 
 
 In adding large numbers, we add the units, the 
 tens, the hundreds, &c., separately. Two cases occur. 
 
 7. I. When the sum of each column is not more 
 than 9. 
 
 EXAMPLE 1. Add 641, 132, 213. 
 
 Operation. 
 641 
 132 
 213 
 
 986 
 
 We write the numbers so that units are in the same 
 column, tens in the same column, hundreds in the 
 same column. Beginning with units, we add : 3 and 
 2 and 1 are 6.; 1 and 3 and 4 are 8 ; 2 and 1 and 6 
 are 9. The sum is 986. 
 
164: ADDITION. 
 
 Add the following numbers : 
 
 2. 234 2222 3333 2000 5001 4210 
 
 312 3132 2141 1423 2111 1434 
 
 421 2321 2203 1004 422 1244 
 
 22 2314 1221 4352 1241 3101 
 
 3. What is the sum of 3241, 1320, 2103, 1025 ? 
 
 4. Add together 2345, 1201, 4321, 1102. 
 
 5. Add together 1234, 4321, 2222, 2222. 
 
 6. A man bought a house for 2300 dollars, a horse 
 for 250 dollars, and a wagon for 120 dollars; what 
 did he pay for them all ? 
 
 8. II. When the amount in any column is more 
 than 9. 
 
 EXAMPLE. If a merchant has 1276 dollars in bank 
 bills, 479 dollars in gold, 784 dollars in silver, and 4 
 dollars in cents, how much money has he in all ? 
 
 Operation. 
 1276 
 479 
 
 784 
 4 
 
 2543 
 
 We write, as before, units under units, tens under 
 tens, &c. 4, 4, 9 and 6 units are 23 units, or 2 tens 
 and 3 units. We write 3 units tinder units, and add 
 the 2 tens with the column of tens, thus : 2, 8, 7 and 
 7 are 24 tens, or 2 hundreds and 4 tens. Writing 4 
 tens under tens, we add 2 hundreds with the column 
 of hundreds : 2, 7, 4 and 2 are 15 (hundred). We write 
 5 under hundreds, and add 1 with thousands : 1 and 1 
 are 2. The sum is 2543. The merchant has 2543 
 dollars. 
 
ADDITION. 165 
 
 EXAMPLES. 
 
 1. 2704 4623 66 704 358 
 
 48502 598 749 8506 4854 
 
 79508 1942 8515 9402 987 
 
 706 749 15652 56708 74749 
 
 2.490 777 333 888 888 123 
 
 244 444 330 22 99 221 
 
 130 999 444 3 11 456 
 
 355 442 440 44 2 654 
 
 240 111 505 555 33 789 
 
 166 222 606 6 590 978 
 
 3. 8649 9876 5436 6 6789 
 
 5842 5432 4563 66 9876 
 
 3876 1234 3654 6836 7890 
 
 9204 5678 5463 6956 9876 
 
 7587 1875 6345 6796 1234 
 
 4. How many are 902010 + 178393 + 643421 + 
 366780 + 703020 + 683463 + 604050 + 221894? 
 
 5. What is the sum of forty-four thousand three 
 hundred and twelve ; six thousand eight hundred and 
 twenty-seven ; nine hundred and fifty -.two ; thirty 
 thousand and thirty ? 
 
 6. Find the sum of ninety ; eighty thousand seven 
 hundred and one ; two thousand seven hundred and 
 fifty-four; one thousand five hundred and seventy; 
 forty-two thousand and four. 
 
 7. In a nursery of trees there are 1250 apple trees, 
 2050 pear trees, 3500 peach trees, 1750 plum trees, 
 850 apricot trees; how many trees in all? 
 
166 SUBTRACTION. 
 
 LESSON III. 
 
 Subtraction. 
 
 9. By SUBTRACTION we find the difference of two 
 numbers, by taking the less number from the greater. 
 
 The greater number is the MINUEND. 
 
 The less number is the SUBTRAHEND. 
 
 The difference is also called the REMAINDER. 
 
 To subtract, we place the subtrahend under the 
 minuend ; units under units, tens under tens, &c. 
 Two cases may arise. 
 
 10, I. Each figure in the subtrahend may be less 
 than the figure above it in the minuend. 
 
 EXAMPLE 1. From 576 subtract 324. 
 
 Operation. 
 576 
 
 252 
 
 Subtracting each figure in the subtrahend from the 
 figure above it, beginning with units, we have 4 from 
 6, 2 ; 2 from 7, 5 ; 3 from 5, 2. The remainder is 252. 
 
 PROOF. Add the units, tens, &c. of the subtrahend 
 and remainder 2 and 4 are 6; 5 and 2 are 7; 2 and 
 3 are 5 and we have the minuend, 576. 
 
 EXAMPLES. 
 
 2. From 540 take 200. From 860 take 300. 
 
 3. 800 5000 60000 9000 790 80000 
 200 2000 20000 3000 400 3000 
 
 4. 8760 9630 6320 5670 4810 5945 
 3000 7000 2000 3000 1000 3000 
 
SUBTRACTION. 167 
 
 5. 937 877 697 867 633 857 6848 
 122 352 245 423 121 307 3007 
 
 6. 85674 4789 62476 34897 
 1352 3254 231 12244 
 
 11. II. One or more figures in the subtrahend may 
 be greater than the figures above them in the minu- 
 end. 
 
 EXAMPLE. From 468 take 349. 
 
 Operation. 
 
 10 
 
 468 
 349 
 
 119 
 
 Write the less number under the greater. As we 
 cannot take 9 units from 8 units, we mentally add 10 
 to 8, and then say, 9 from 18 leaves 9, which 'we write 
 down. Since we added 10 to the minuend, we add 1 
 ten, to the 4 tens of the subtrahend, and say 5 from 6 
 leaves 1, which we write down. Finally, 3 hundred 
 from four hundred leaves 1 hundred. 119 is the dif- 
 ference between the numbers, because we have taken 
 all the parts of the less from the greater, and we have 
 balanced the ten added to the greater by adding ten 
 to the less. 
 
 EXAMPLE. From 9434 take 3537. 
 
 Operation. 
 
 101010 
 
 9434 
 3537 
 
 5897 
 
168 
 
 SUBTRACTION. 
 
 We say, 7 from 14, 7 ; 4 from 13, 9 ; G from 14, 8 ; 
 4 from 9, 5. The remainder is 5897. 
 
 If there are fewer figures in the lower line than in 
 the upper, proceed as if O's were written on the left of 
 the lower number until the number of figures is the 
 same in both. 
 
 732 
 13 
 
 807 
 718 
 
 8403 
 2390 
 
 EXAMPLES. 
 
 
 10 
 
 
 
 
 
 1. 
 
 670 
 
 860 490 
 
 540 
 
 650 880 
 
 
 25 
 
 43 38 
 
 26 
 
 39 32 
 
 
 10 
 
 
 
 10 
 
 
 2. 
 
 7860 
 
 1780 
 
 2690 
 
 806 
 
 709 
 
 
 243 
 
 437 
 
 272 
 
 243 
 
 333 
 
 
 10 
 
 
 
 
 
 3. 
 
 6072 
 
 7043 
 
 4068 
 
 
 5037 
 
 
 2342 
 
 2520 
 
 3660 
 
 
 1717 
 
 10 10 101010- 
 
 4. 4372 782 965 585 7184 4360 7343 
 2657 355 428 247 6526 3608 6740 
 
 1715 
 
 5. 2814 375 5006 4176 7583 8064 
 1033 185 2818 3648- 3065 3975 
 
 6. 6004 60010 10000 8200102 815240 
 3716 528 3196 4866769 379468 
 
 7. 7802008 10000000 951315 9555555 
 3357564 1 947688 2888889 
 
MULTIPLICATION. 169 
 
 8. How much smaller is 38567 than 60600 ? 
 
 9. How much is 4009000 less 5678 ? 
 
 10. How much greater is 700000 than 599328 ? 
 
 11. Take from 43518 the number 7253, and from 
 the remainder 7253, and so on, and give the remainder 
 after the sixth subtraction. 
 
 12. How much is 3820 + 16488 + 347 + 5688 + 
 2433 - 31210? 
 
 13. A man owns a farm which cost him 5500 dol- 
 lars ; he has paid 2875 dollars ; how much more has 
 he to pay. 
 
 14. A merchant sold for 2825 dollars goods which 
 cost 2138 dollars ; how much did he gain ? 
 
 15. A man bought a house and lot for 2750 dollars, 
 and sold it for 3125 dollars ; how much did he gain ? 
 
 16. How old would a man be in 1868 who was 
 born in 1839? 
 
 LESSON IV. 
 
 Jfuttiplicat ion. 
 
 12. If a man can earn 8 dollars in one week, in 4 
 weeks he can earn 4 times 8 dollars, or 32 dollars. 
 
 This is multiplication: 8, the number to be re- 
 peated, is called the MULTIPLICAND : 4, which shows 
 how many times the multiplicand is to be taken, is 
 called the MULTIPLIER, and 32, the result of the multi- 
 plication, is the PRODUCT. 
 
 In multiplying large numbers, there are two cases : 
 
 13. I. When the multiplier is a single figure. 
 EXAMPLE 1. Multiply 467 by 5. 
 
 Operation. 
 
 467 
 5 
 
 2335 
 
170 MULTIPLICATION. 
 
 . We write the multiplier under the multiplicand, 
 units under units, and say, 5 times 7 units are 35 
 units, or three tens and 5 units. We write 5 units 
 below, and reserve 3 tens. 5 times 6 tens are 30 tens, 
 which with 3 tens make 33 tens, or 3 hundreds and 
 3 tens. Writing 3 tens in tens' place, we reserve 3 
 hundred. 5 times 4 hundred are 20 hundred, which 
 with 3 hundred mate 23 hundred. This we write 
 down, and have for the product, 2335. 
 
 2. 356 825 6745 4567 Multiplicand. 
 
 4536 Multiplier. 
 
 1424 27402 Product. 
 
 3. 8474 9876 5432 12345 
 
 2789 
 
 4. Multiply 4825 by 2; by 3 ; by 4; by 5 ; by 6. 
 
 5. Multiply 3775 by 5 ; by 6 ; by 7 ; by 8 ; by 9. 
 
 6. Multiply 5709 by 4; 'by 5; by 6; by 7; by 8; 
 by 9. 
 
 7. If one horse costs 275 dollars, what is the cost 
 of 8 horses ? 
 
 8. If one house is worth 3275 dollars, what is the 
 value of 7 similar houses ? 
 
 9. If the multiplicand is 3917 and the multiplier 
 is 6, what is the product? 
 
 10. Multiply the sum of 1847 and 2043 by 8. 
 
 11. Multiply the difference between 3194 and 1536 
 by 9. 
 
 14. II. When the multiplier contains more than 
 one figure. 
 
 EXAMPLE 1. Multiply 843 by 26. 
 
MULTIPLICATION. 171 
 
 Operation,, 
 
 843 
 
 26 
 
 5058 
 1686 
 
 21918 
 
 Write the multiplier under the multiplicand, units 
 under units, tens under tens, &c. 
 
 Multiply by 6, the units figure, as in Case I. In 
 multiplying by 2 tens, place the first figure of this 
 product under 2, the multiplier. For, 2 tens times 3 
 are 6 tens; therefore, the 6 should be in the second, 
 or tens', place. 
 
 Add the two partial products, and find the entire 
 product, 21918. 
 
 EXAMPLE 2. Multiply 4357 by 345. 
 
 Operation. 
 
 4357 
 345 
 
 21785 
 17428 
 13071 
 
 1503165 
 
 7 _ 35 ; write 5 and " carry" 3. 
 
 = 25, + 3 28 ; write 8. 
 = 15, + 2 = 17 ; write 7. 
 = 20, + 1 = 21 ; write 21. 
 
 7 = 28 ; write 8 under 4. 
 5 = 20, + 2 22 ; write 2. 
 
 3 _ 12, + 2 = 14 ; write 4. 
 
 4 = 16, + 1 17 ; write 17. 
 
1 72 MULTIPLICATION. 
 
 3 x 7 = 21 ; write 1 under 3. 
 3 x 5 = 15, + 2 = 17 ; write 7. 
 3x3 = 9, + 1 = 10; write 0. 
 3 x 4 12, + 1 13 ; write 13. 
 
 The sum of the partial products is the entire pro- 
 duct, 1,503,165. 
 
 3. 3458 8431 1357 2465 5438 
 
 24 35 54 89 146 
 
 4. 5417 5104 3547 . 1596 2705 
 123 456 258 357 403 
 
 5. Multiply 1047 by 284; by 207; by 615. 
 
 6. Multiply 3579 by 123 ; by 612 ; by 246. 
 
 7. Multiply 4680 by 340 ; by 360 ; by 801. 
 
 8. Multiply 6143 by 200; by 4200. 
 
 6143 6143 
 
 200 4200 
 
 1228600 1228600 
 
 24572 
 
 25800600 
 
 9. Multiply 3434 by 300 ; by 5300. 
 
 10. Multiply 1234 by 600 ; by 1054. 
 
 11. 1417 x 34; 6417 x 27; 6946 x 123. 
 
 12. 340 x 20 x 16 ; 853 x 300 x 4 ; 803 x 101. 
 
 NOTE. The teacher will readily supply additional examples for 
 practice under each rule. They can be written on the blackboard, 
 and copied by the pupils. 
 
DIVISION. 173 
 
 LESSON V. 
 Division. 
 
 15. By DIVISION we find how many times one num- 
 ber is contained in another larger number. 
 
 The DIVIDEND is the number to be divided. 
 
 The DIVISOR is the number by which we divide. 
 
 The number of times the divisor is contained in the 
 dividend is the QUOTIENT. 
 
 The REMAINDER is what is sometimes left after 
 dividing. 
 
 16. There are two methods of performing division. 
 Ex. 1. How many times is 4 contained in 948 ? 
 
 1st Operation. 
 
 4)948(237 
 8 
 
 14 
 12 
 
 28 
 
 28 
 
 
 
 We begin at the left of the dividend to divide. 4 
 is contained in 9, 2 times. Place 2 at the right of the 
 dividend, as the first figure of the quotient. 2 times 
 4 are 8, which we set under 9, and subtract. To the 
 remainder, 1, we bring down 4, the next figure of the 
 dividend, making 14. 4 in 14, 3 times. Place 3 in 
 the quotient and multiply the divisor, and proceed as 
 before, till all the figures are brought down and 
 divided. The entire quotient is 237. 
 
174 DIVISION. 
 
 2d Operation. 
 
 4)948 
 
 237 
 
 In this method, we do in the mind a part of what 
 we wrote out in the 1st operation, and write the quo- 
 tient under the dividend. 
 
 4 is contained in 9, 2 times and 1 over. We place 
 2 under 9 as the first figure of the quotient, and 
 imagine the remainder 1 placed before 4, the next 
 figure of the dividend, making 14. 4 in 14, 3 times 
 and 2 over. We place 3 under 4, and imagine 2 set 
 before 8, making 28. 4 in 28, 7 times. The entire 
 quotient is 237, as before. 
 
 1 7. This second method is sometimes called SHORT 
 DIVISION, and is used when the divisor is not more 
 than 12. 
 
 The first method is called LONG DIVISION, and is 
 used when the divisor is more than 12. 
 
 2. 9)1251 9)1251(139 7)28345(4049 
 
 9 28 
 
 139 
 
 35 34 
 
 27 28 
 
 81 65 
 
 3. 7)28345 81 63 
 
 4049 2 Eem. 2 Eem. 
 
 Work the following examples by Short Division : 
 4. 4)57432 6)31740 8)17344^ 9)5562 
 
 618 
 
DIVISION. 175 
 
 5. 5)74035 4)67434 6)10746 8)84755 
 
 105943 Kern. 
 
 Prove all the examples in Division, by multiplying 
 the quotient and divisor together and adding the re- 
 mainder, if there is any, to the product. The result 
 must equal the dividend. 
 
 "Work the following by both methods : 
 
 6. 6)74789 7)78743 8)97145 9)23474 
 
 EXAMPLE 7. Divide 9274 by 35. Here, of course, 
 we use Long Division. 
 
 Operation. Proof. 
 
 35)9274(264 264 
 
 70 35 
 
 227 1320 
 210 792 
 34 
 
 174 
 
 140 9274 
 
 34 Kem. 
 
 How many times is 35 contained in 92? 3 in 9, 3 
 times; but, multiplying 35 by 3, we have 105, which 
 is more than 92; we therefore take a quotient figure 
 one less, 2, and find 2 times 35 = 70, which is less than 
 92. So, to find the second quotient figure : 3 in 22, 7 
 times. Finding 7 too large, we try 6. 
 
 8. Divide 3754 by 24. Quo. 156. Kem. 10. 
 
 9. Divide 17473 by 56. Quo. 312* Rem. 1. 
 
176 DIVISION. 
 
 10. Divide 3746 by 36. How can you know whe- 
 ther your quotient is correct ? 
 
 11. Divide 2756 by 48. Operation. 
 The divisor, 48, is nearly 50. To 48)2756(57 
 
 find the quotient figures we say, 5 in 240 
 
 27, 5 times ; multiplying the divisor 
 
 by 5, we have 240, less than 275 ; 356 
 
 subtracting, we find 35 for remain- 336 
 
 der, a number less than the divisor. 
 
 So, 5 in 35, 7 times, &c. 20 Rem. 
 
 12. Divide 9437 by 27; by 31 ; by 35 ; by 14. 
 
 13. Divide 10743 by 45 ; by 450 ; by 412. 
 
 14. Divide 12345 by 103 ; by 301 ; by 345. 
 
 15. Since there are 365 days in one year, how many 
 years are there in 2000 days ? 
 
 16. If I pay 9744 dollars for 112 acres of land, how 
 much is it an acre ? 
 
 17. How many cows can I buy for 581 dollars, if 
 each cow is worth 83 dollars ? 
 
 18. If the dividend is 7839, and the divisor is 214, 
 what are the quotient and remainder \ 
 
 19. Add together 359, 895, and 1054; subtract 
 1343 from the sum ; multiply the remainder by 8, 
 and divide the product by 4. What is the quotient ? 
 
 NOTE. The same example expressed by means of signs is 359 -f 
 895 + 1054 1343 x 8 -j- 4 = ? 
 
 20. 874 + 917-1056 x 6 ~ 5 ? Quotient 
 882. 
 
 21. 3846 + 1904 + 3841 3489 x 100 ~ 75 = ? 
 
 22. 804 x 200 -f- 25 + 3568 - 10000 = 2 
 

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