i^l^:..t-:'ate;:4^iu^:...M; :^:IuiS. IvIBRARV OF THK University of California. Received dA^i^iy . /<^9 ? • Accession No. ^7^ ^O . Class No. Digitized by the Internet Arcinive in 2007 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/elementsofwritteOOeatoricli THE ELEMENTS WRITTEN ARITHMETIC; COMBINING ANALYSIS AND SYNTHESIS; ADAPTED TO THE BEST MODE OF INSTRTICTION FOB BEGINNERS. JAMES S. EATON, M. A., INSTRUCTOB IIT PHILLIPS ACADEMY, ANDOVEE, Ain> AXTTHOB OF A SERIES OF ABITHMETICS. -^ BOSTON: 29 CORNHIIiL. 1868. J J Entered according to Act of Congress, in the year 1868, by JAMES H. EATON, M. A., In the Clerk's Office of the District Court of the District of Massachusetts. J. E. Fak>vell & Co., Stereo typers and Printers. 37 Congress Street, Boston. PREFACE During the few years Eaton's Series of Arithmetics has been before the Educational Public, it has been demonstrated, hy their cordial reception — by their circulation, already exten- sive and rapidly increasing — that a deep-felt want for books of their high character has been satisfactorily met. In the vast field at the South now being opened to educational advantages — in the extensive and rapidly growing settlements of the West, and in the cities and manufacturing districts of the older States, there is a large class of pupils whose school days are very limited. To these — next to reading and writing — a brief, practical course of Arithmetic must always form the most useful part of school training. It was to meet the wants of this class that the present work was projected by its author. Owing to an unfore- seen event, its completion has devolved upon others. This has been the occasion of some delay in its publication, which however has been taken advantage of to make it as pe'rfect as possible. This little work then is a short, practical course of Written Arithmetic, embracing the topics actually necessary to be mas- tered to enable one to pursue with intelligence the ordinary busi- ness avocations of life. Special attention has been bestowed upon the Fundamental Rules, United States Money and Percent- age, and a simple but full exposition of the New Metric System is also presented. 4 PREFACE. In its preparation no labor has been spared to adapt it to the end for which it is designed. It has been submitted to many prac- tical teachers, and thus embodies valuable suggestions from many sources. Especial credit is due Mr. J. P. Payson, Master of the Grammar School; Chelsea, Mass., and it makes its way to the public through the hands of Mr. James H. Eaton, son of the author of Eaton's Series of Arithmetics. It is hoped that it will prove worthy to stand beside the former works of the same well- known author. Boston, Sept. 30, 1867. -i?^^ CONTENTS SIMPLE NUMBERS. PACK Definitions 7 Notation and Numeration 7 Numeration Table 10 Exercises in Numeration 11 Exercises in Notation , . 12 Eoman Notation 14 Table of Koman Numerals 14 PAGE Exercises in Roman Notation ... 15 Addition 16 Subtraction 24 Multiplication 35 Division 50 General Principles of Division ... 66 Cancellation 69 DENOMINATE NUMBERS AND REDUCTION. Definitions 74 English Money 75 Troy Weight 78 Apothecaries' "Weight SO Avoirdupois Weight 81 Cloth Measure 82 Long Measure 83 Chain Measure 85 Square Measure 86 Solid Measure 88 Liquid Measure 92 Dry Measure 93 Time 94 Circular Measure 95 Examples in Reduction 97 GENERAL PRINCIPLES. Definitions 98 1 Greatest Common Divisor .... 100 Factoring Numbers , 98 Least Common Multiple 101 COMMON FRACTIONS. Definitions . . ^^^ General Principles 104 Mixed Numbers Reduced to Ira- proper Fractions 106 Improper Fractions Reduced to Whole or Mixed Numbers . . 106 Fractions Reduced to Lowest Terms 107 Fraction Multiplied by a Whole Number 108 Fraction Divided by a Whole Num- ber 109 Fraction Multiplied by a Fraction . Ill Fraction Divided by a Fraction . . 112 Common Denominator 113 Addition of Fractions 115 Subtraction of Fractions 116 Practical Examples 117 Miscellaneous Examples 118 Analysis 119 CONTENTS. DECIMAL FRACTIONS. PAGE Definitions 123 Numeration Table 123 Notation and Numeration 125 Addition 126 Subtraction 127 Multiplication 128 PAGK Division 131 Common Fractions Reduced to Decimals 133 Decimals Reduced to Common Fractions 133 Jliscellaneous Examples 134 UNITED STATES MONEY. Table 136 Definitions 136 Practical Examples 137 Table of Aliquot Parts 142 Barter 144 Bills 145 Miscellaneous Examples 147 COMPOUND NUMBERS. Addition 150 Subtraction 153 Multiplication 156 Division .158 Miscellaneous Examples 159 PERCENTAGE. Definitions 100 To find Percentage, the Base and Rate being given 162 To find the Rate, the Base and Per- centage being given 163 To find the Base, the Rate and Per- centage being given 164 Interest 165 To find Interest on any sum for any time at 6 per cent 166 To find Interest at other Rates than 6 per cent 169 Profit and Loss 170 To find the Absolute Gain or Loss . 171 To find Per Cent of Gain or Loss . 171 To find the Selling Price 172 To find the First Cost 173 MISCELLANEOUS. Miscellaneous Examples 1741 The Metric System 177 elemenIs^MjMetic. Article 1, A Unit is one, that is, any single tiling ; as, a horse, a dajy an apple, an inch. 3. A Number is a unit or a collection of units ; as, one, twOf six, three men, ten pints, 3. Arithmetic is tlic science of numbers, and tlic art of reckoning or computation. 4. There are six and only six different operations in Arithmetic, namely, Notation, Numeration, Addition, Sub. traction. Multiplication, and Division. NOTATION AND NUxMERATION. S, Notation is the art of writing or expressing numbers and their relations to each other by means oi figures and signs. O. Numeration is the art of reading numbers which have been written or expressed by figures. T. There are two methods of notation in common use, namely, the Arabic and the Roman. 8. Tho Arabic Notation employs ten figures to express numbers, namely: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine. Questions. 1. "What is a Unit ? 3. What is a Number? 3. What is Arithmetic ? 4. How many operations are there in Arithmetic ? What arc they called ? 5. What is Notation ? 6. Numeration ? 7. How many methods of Notation in common use ? What are they ? 8. How many and what figures in the Arabic Notation ? 8 NOTATION AND NUMERATION. 9. The first Arabio figure, 0, is called a cipher ^ naugJd, or zero, and when used without any other figure it stands for nothing ; thus, apples means no apples. Each of the other- nine figures stands for or signifies the same as the word written under it, and to distinguish them from 0, they are called signifijcant figures. 10. To express numbers greater than nine, these figures arc repeated and combined in various ways. Ten is expressed by writing the figure 1 at the left of the cipher; thus, 10. In like manner twenty, thirty, forty, etc., are expressed by placing 2, 3, 4, etc., at the left of ; thus, 20, 30, 40, 50, 60, 70, 80, 90. Twenty, Tliirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety. 11. The numbers from 10 to 20 are expressed by placing the figure 1 at the left of each significant figure ; thus, 11, 12, 13, 14, 15, 16, 17, etc. Kloven, Twelve, Thirteen, Fourteen, Fifteen, Sixteen, Seventeen, etc. In a similar manner all the numbers, up to one hundred, may be written ; thus, 21, 36, 66, 98, etc. Twenty-one, Thirty-six, Sixty-six, Ninety-eight, etc. 12. One hundred is expressed by placing the figure 1 at the left of two ciphers; thus, 100. In like manner two hundred, three hundred, etc., are written; thus, 200, 300, 600, 800, etc. Two hundred, Three hundred, Six hundred, Eight hundred, etc. 9. Names of the first figure, 0? Its value? What are the other figures called ? Why ? 10. How are numbers greater than 9 expressed i Illustrate. 11. Numbers from 10 to 20, how expressed? Other numbers to 100? la. One hundred, two hundred, etc., how expressed ? NOTATION AND NUMERATION. 9 13. The other numbers, up to one thousand, may be expressed by putting a significant figure in the place of one or each of the ciphers in the above numbers j thus, Two hundred and three, expressed in figures, is 203. Six hundred and eighty, •* " 680 Nine hundred and ninety-eight, " " ' 998 14. The simple name of any significant figure is always the same, but the numbor indicated by it depends upon the place the figure occupies ; for example, 6 is always sixy and never seven. So in each of the following numbers, 2, 20, and 200, the left hand figure is twOy but in the j^rs^ it is two units ; in the second, two tens or twenty ; and in the third, two hun- dreds. Thus each significant figure has a simple or name value, and a local or place value. 15. When two or more figures are used together they are said to express different orders of units. The right hand Jigure represents simple units, or units of the first order ; the second figure represents tens, or units of the second order ; the third represents hundreds, or units of the third order ; thus, in the number 426 the 6 is simply six, while the 2 is two tens or twenty, and the 4 is four hundreds; and the number expressed by the three figures taken together is four hundred and twenty-six. 16. The figures of large numbers, for convenience in reading, are often separated by commas into groups or periods of three figures each, beginning at the right. The first or right-hand group contains units, tens, and hundreds, 13. Other numbers to one thousand ? 14:. Does the name of a figure ever change? Does its value change? How many values has a figure? The names of those values ? 15. What is said of orders of units ? 16. What is said of groups or periods of figures ? 10 NOTATION AND NUMERATION. Viud is coMed the period of unifs ; the second period contains thousands, tens of thousands, and hundreds of thousands, and is called the period of thousands y etc., as in the following NUMERATION TABLE. 02 no J J. TO P O •^ .2 If |~^ «+-i ;^ «»H ^ u^ o ^ ^- reds of Th sands, 'S § 22^ s 2^ 1 ("Hund Tens Billio fHund Tens Millie fHund Tens [Thous 6 9, 5 4 0, 7 6, 4 7 6, 5th period, Trillions. 4th period, 3d period, 2d period, Billions. Millions. Thousands, 8 4 3, Ist period, Units. •IT". The value of the figures in this table, expressed in words, is sixty-nine trillion, five hundred and forty billion, seven hundred and six million, four hundred and seventy-six thousand, eight hundred and forty-three. Note. The reading of a number consists of two distinct processes : First, reading the order of the places, beginning at the right hand; thus, units, tens, hundreds, thousands, etc., as in the Numeration Table ; and, second, reading the value of the figures, beginning at the left, as in Article 17, above. To distinguish these processes, the first may be called numerating, and the second reading, the number. 18. The value of a figure is increased tenfold by removing Name the periods in the Numeration Table, beginninj? at the riglit. Name the figures in each group. 17. Read the value of the fibres in the Numer- ation Table. Note. How many processes in reading a nuniber? Describe them, and tell what they are called. 18. How is the value of a figure affected by changing its place ? Illustrate. What general law ? NOTATION AND NUMERATION. 11 it one place toward the left ; a hundred fold by removing it two places, etc., that is, tea units of the first order make one ten, ten tens make one hundred, ten hundreds make one thousand, and, in short, ten units of any order make one unit of the next higher order, 10. The cipher, when used with other figures, fills a place that would otherwise be vacant; thus, in 206 the cipher occupies the place of tens, because there are no tens expressed in the given number. 30. From the foregoing, to numerate and read a number expressed by figures, we have the following EuLE 1. Beginning at the right, numerate, and point off the number into periods of three figures each. 2. Beginning at the left, read each period separately, giving the name of each period except that of units. Exercises in Numeration. SI. Let the learner read the followin g numbers : 1. 8 11. 4,683 21. 300,006 2. 13 12. 9,000 22. 5,634,872 28 13. 35,618 23*. 7,402,309 4. 346 14. 40,306 2t. 4.040,060 5. 701 15. 75,001: 25. 2,008,001: 6. 358 16. 97,400 26. 32,468,312 7. 490 17. 66,040 27. 461,084,307 8. 8,645 18. 345,284 28. 5,329,684,119 9. 3,059 19. 549,603 29. 42,382,000,000 10. 8,006 20. 203,940 30. 702,437,600,216 Note. The teacher should give examples similar to the above upon the bhickboard or slate, sometimes inserting and sometimes omitting the commas, until the pupil can readily group, numerate, 19. a numbe The cipher, for what used ? 30. Rule for numerating and reading i)er ? 12 NOTATION AND NUMERATION. and read all numbers likely to occur in his lessons or general reading. A like remark applies to all the following parts of tlie book. The teacher should give many original examples, varying in difficulty according to the abilities of his classes, and should encourage his pupils to make examples for themselves and for each other. 33, To write numbers, we have this KuLE 1. Beginning at the left, write the figures belonging to the highest period. 2. Write the figures of each successive period in their order, filling each vacant place with a cipher. Exercises in Notation. 33. Let the learner write the following numbers in figures, and read them : 1. Five units of the third order and six of the first. Ans. 50G. Note. A cipher is written in the second place, because no unit of the second order is given. 2. Three units of the fourth order, six of the second, and four of the first. ' Ans. 3,064. 3. Two units of the seventh order, one of the sixth, three 'of the third, and five of the second. Ans. 2,100,350. 4. Eight units of the fifth order, two of the third, and six of the first. 5. Six units of the eighth order, four of the sixth, two of the fourth, and five of the third. 6. Nine units of the sixth order, two of the fourth, and eight of the first. 33. Rule for writing a number. Note. In Notation where should be written .' NOTATION AND NUMERATION. 13 Y. What orders of units are there in the number 3,462,895 ? How many units in each order ? 8. What orders of units in the number 62,304,500 ? 'How many units in each order ? 9. How many tens in 46 ? How many units beside the tens ? How many units in the whole of the number ? 10. In 347 how many hundreds? How many tens in the tens' place ? How many units in the units' place ? How many tens in the number ? How many units in the number ? 24l, Write the following numbers in figures : 1. Two hundred and fifty-six. Ans. 256. 2. Fifty.four. Ans. 54. 3. Six thousand and nineteen. Ans. 6,019. * 4. One thousand eight hundred and sixty-five. 5. Four hundred and forty. 6. Twenty-five thousand two hundred and forty-nine. 7. Two hundred and forty-five thousand six hundred and fifty-four. 8. Five million six hundred thousand eight hundred and sixteen. 9. Twenty-two million two hundred and twenty-two thou- sand two hundred and twenty-two. 10. Five hundred and six million forty thousand two hun- dred and four. 11. Four billion eight million six thousand eight hundred and ten. 1 2. Thirty-five trillion four hundred and six billion eight hundred and twenty million two hundred and eighteen thou- sand four hundred and sixty- seven. 14 NOTATION AND NUMEKATION. 35. The Ko MAN Notation employs seven capital letters to express numbers, viz. : I, * V, X, L, C, D, M. One, Five, Ton, Fifty, One hundred. Five hundred, One thousand. All other numbers may be expressed by combining and repeating these letters. 36. The Koman Notation is based on the following prin- ciples : 1st. When two or more letters of equal value are united, or when one of less value follows one of greater, the sum of the values is indicated ; thus, XX stands for 20, XXX for 30, LXV for 65, DC for 600, MDCCLXVIII for 1768. 2d. When a letter of less value is placed before one of 4 greater, the difference of their values is indicated ; thus, IV stands for 4, IX for 9, XL for 40, XO for 90. 3d. When a letter of less value stands between two of greater value, the less is to bo taken from the sum of the other two ; thus, XIV stands for 14, XIX for 19, CXL for 140. TABLE OF KOMAN NUMERALS. I 1 X 10 XIX II 2 XI 11 XX III 3 XII 12 XXI IV 4 XIII 13 XXII V 5 XIV 14 XXIV VI 6 XV 15 XXV VII 7 XVI 16 XXIX VIII 8 XVII 17 XXX IX 9 XVIII 18 XL 19 20 21 22 24 25 29 30 40 85. How many and what characters are employed in the Roman Nota- tion? Value of each? How may other numbers be expressed? ae. What is the first principle in the Ilomau Notation ? Second ? Third ? NOTATION AND NUMERATION. 15 L 50 DC 600 MDCCXLIX 1749 LX CO DCCCG 000 MDCCCXVI 1816 XG 90 M 1000 MDCCCXLI 1841 C 100 MD 1500 MDCCCXLIX 1849 COCO 400 MDO 1600 MDCCCLVII 1857 D 500 MDCLXV 1665 MDCCCLXVI 1866 Exercises in Koman Notation. S7. Express the following numbers by letters: 1. Nine. Ans. IX. 2. Fifteen. ' Ans. XV. 3. Eighteen. 4. Twenty-four. 5. Twenty-six. 6. Thirty-nine. 7. Forty. 8. Sixty. 9. One hundred and eighty-four. 1 0. One hundred and ninety-six. 11. One thousand six hundred and forty-six. 1 2. The present year, A. D. . 38. Besides the Arabic and Roman figures, there are various marks used to indicate certain relations between num- bers and operations to be performed on them, as, for example, the sign of equality, ^=. ; the sign of addition, -|- ; the sign oj subtraction, — ; etc. These signs will be given, and their uses explained hereafter, when their aid is needed. 28. What characters are used in Arithmetic besides the Arabic and Koman figures? For what? 16 ADDITION. ADDITION. 39, Three apples and four apples are how many apples ? Ans. Three apples and four apples are seven apples. This is a question in addition. 30. Addition is the process of finding how many units there are in two or more numbers of the same kind taken to- gether. The result of the addition is called the sum or amount. ADDITION TABLE. 2 and 1 are 3 3 and 1 are 4 4 and 1 are 5 5 and 1 arc 6 2 " 2 " 4 3 " 2 " 5 4 " 2 " 6 5 «' 2 " 7 2 " 3 " 5 3 " 3 " 6 4 " 3 " 7 5 «* 3 " 8 2 " 4 " 6 3 " 4 " 7 4 " 4 " 8 5 " 4 '* 9 2 ♦' 5 " 7 3 " 5 " 8 4 " 5 " 9 5 " 5 " 10 2 " 6 " 8 3 " 6 " 9 4 " 6 " 10 5 " 6 " 11 2 *' 7 " 9 3 " 7 " 10 4 " 7 " 11 5 " 7 " 12 2 " 8 " 10 3 " 8 " 11 4 " 8 " 12 5 " 8 " 13 2 " 9 " 11 3 " 9 " 12 4 " 9 " 13 5 " 9 " 14 2 "10 " 12 3 " 10 " 13 4 " 10 " 14 5 "10 " 15 6 and 1 are 7 7 and 1 are 8 8 and 1 are 9 9 and 1 are 10 6 " 2 " 8 7 " 2 ♦♦ 9 8 " 2 " 10 9 " 2 " 11 6 " 3 " 9 7 " 3 " 10 8 " 3 " 11 9 " 3 " 12 6 " 4 " 10 7 " 4 " 11 8 " 4 " 12 9 " 4 " 13 6 " 5 " 11 7 " 5 " 12 8 " 5 " 13 9 " 5 " 14 6 '* 6 " 12 7 " 6 " 13 8 " 6 " 14 9 " 6 " 15 6 '• 7 " 13 7 " 7 " 14 8 " 7 " 15 9 " 7 " 16 6 " 8 " 14 7 ^' 8 " 15 8 " 8 " 16 9 " 8 " 17 6 " 9 •' 15 7 " 9 " 16 8 ^' 9 " 17 9 '• 9 " 18 6 "10 " 16 7 "10 " 17 8 " 10 " 18 9 "10 " 19 30. What is Addition f What is tlie reswZ^ called ? ADDITION. 17 Mental Exescises, Ex. 1. Robert has 5 cents in one hand, and 3 cents in the other ; how many cents has he in both hands ? Ans. 8. 2. John bought a pencil for 6 cents, and some paper for 5 cents ; how many cents did he pay for both ? 3. Greorge has 7 chickens and David has 8 ; how many have both? Ans. 15. 4. Mary has 6 tulips and 9 roses; how many blossoms has she? 5. Daniel caught 9 fishes, Abel caught 6, and James caught 5 ; how many did they all catch ? 6. A farmer had 6 cows in one pasture, 8 in another, and 7 in another ; how many cows had he in the three pastures ? 7. I paid 9 dollars for a barrel of flour, 8 dollars for a box of sugar, and 5 dollars for a cheese ; how many dollars did I pay for all? Ans. 22. 8. A man bought a ton of coal for 8 dollars, a cord of wood for 6 dollars, and a stove for 9 dollars ; what did he pay for all? 9. Charles has 5 marbles, Albert has 7, and Edward has 9 ; how many have they all ? 10. A farmer has 8 sheep in one pen, 9 in another, and as many in a third pen as in both the others ; how many has he in the third pen ? Ans. 1 7. 11. A gardener raised 3 bushels of cherries, 2 bushels of currants, 5 bushels of peaches, and 8 bushels of pears; how many bushels of fruit did he raise ? 12. George paid 10 cents for a writing-book, 8 cents for a pen-holder, 2 cents for pens, and 6 cents for ink ; how much did he pay for all ? 18 ADDITION. 31. A Sign is a mark which indicates an operation to be performed, or which is used to shorten some expression. 33. This mark, $, is often used as a sign of the word dollar or dollars ; thus, %\ stands for one dollar, $6 stands for six dollars. Note. It is customary to separate dollars and cents by a period; thus, f 4.25 stands for four dollars and twenty-five cents. 33. The sign of equality, =, signifies that the quantities between which it stands are equal to each other ; thus, ^ I ;= 100 cents; that is, one dollar equals one hundred cents. 34. The sign of addition, -(-, called j»^m5 or and, denotes that the quantities between which it stands are to be added to- gether ; thus, 3 + 2 = 5; that is, three plus two equal five, or, three and two are five. Ex. 12. How many are 3 + 6 + 4? Ans, 12. 13. How many are 2 + 6 + 5? 3 + 8 + 4? . 14. How many are 5 + 3 + 6? 9 + 2 + 6? 15. How many are 8 + 6 + 5? 9+3 + 7? 16. How many are 7 + 9 + 4? 6 + 9 + 8? 3^. Let the pupil frequently review the following Exercises in Addition. No. 1. No. 2. No. 3. No. 4. No. 5. 4 + 3 6 + 5 2-1-8 7 + 3 5 + 7 2 + 6 5 + 5 7 + 9 6 + 8 6 + 9 7 + 3 6+4 1 + 8 3 + 7 9+3 8 + 1 8 + 2 3 + 6 8 + 2 + 5 10 + 3 3 + 5 8+1 4 + 5 5 + 8 5 + 6 4 + 9 9 + 6 7 + 6 5 + 6 7 + 5 2 + 6 4 + 7 9 + 8 9+2 31. What is a sig^n? 3a. Make the sign of dollars on the black-board. How are dollars and cents separated ? Give an example. Another. 33. Make the sign of equality. What does it mean ? Illustrate. 34. Make' the sign of addition. What is it called .-' What does it mean ? ADDITION, 19 " No. 6. No. 7. No. 8. No. 9. No. 10. ( 9 + 4 1+4 10+ 3 2+ 1 H h 9 6--7 7 + 8 8 + 10 6 + 10 0- - '8 8 + 9 • 8 + 5 3+ 8 5 + 11 10- - ^ G --2 2 + 2+7 3 + 12 6 - -11 1 -- 5 5 + 4 9+ 7 9+0 1- - 2 2 -- 2 3 + 1 1+ 3 8+ 8 10- - 7 5 + 1 2 + 4 6+ 1 9+9 7H hio No. 11, No. 12. No. 13. No. 14. No. 15. 8+ 6 3+9 3+ 4 4+ 8 6+3 10+5 7+ 7 7+ 2 3+2 7 -- 12 11+2 6+6 12+5 3+3 12— 4 G + 12 10+9 10+6 12+8 11 -- 11 10+8 12+6 11+ 7 10 + 11 5 -- 12 9+ 1 9 + 10 9 + 12 12+7 11 -- 9 7 + 11 8-1-12 3 + 11 11+8 12+9 Written Exercises. 36. To add when the numbers are large, and the amount of each column is less than ten. Ex. 1. A farmer sold 234 bushels of corn, 423 bushels of oats, and 141 bushels of wheat; how many bushels of grain did he sell ? Having for convenience arranged the numbers OPERATION, so that uuits stand under units, tens under tens, 234 etc., add the units ; thus, 1 and 3 are 4, and 4 423 are 8, and set the 8 under the column of units. 141 Then, add the tens; thus, 4 and 2 are 6, and 3 are 9, and set the 9 under the column of tens, Sum 798 and so proceed till all the columns are added. Thus we find that the entire sum is 7 hundreds, 9 tens, and 8 units, or t98 bushels the answer. 36. How are numbers arranged for addition ? added first ? "What is done with the Bum ? Why ? Which column is 20 ADDITION. In like manner add the numbers in the following examples : Ex. 2. 3. 4. 5. 6. 7. 8. $1.90 242 143 $26.01 324 1240 51234 2.47 126 421 12.31 23 2036 2130 3.11 211 235 41.32 241 3712 513 Ans. $7. 48 799 6988 9. 10. 11. 12. 13. 14. 15. Miles. Bushels. Men. Apples. Sheep. Birds. Days. 1310 3241 4120 4160 203 1321 3122 3247 1302 312 1306 6120 3200 2231 2131 2144 2103 2012 62 2134 2101 6687 6655 16. In 1850 the population of Virginia was 1,421,661, and that of Vermont was 314,120 ; what was the total population of Virginia and Vermont in 1850 ? Ans. 1,735,781. 17. In 1860 the population of Massachusetts was 1,231,065 and that of Kentucky was 1,155,713 ; what was the total popu- lation of Massachusetts and Kentucky in 1860? Ans. 2,386,778. 18. A gentleman paid $135 for a horse, $243 for a chaise, and $121 for a harness ; what did he pay for all ? 19. Add 2316, 3120, 1201, and 2002. Ans. 8669. 20. Add $35.41, $21.24, $1.32, and $2.01. Ans. $59.98. 21. Add 43216, 20431, 14030. Ans. 77,677. 22. AVhat is the sum of 3241 + 2312 + 1203 + 3120? 28. What is the sum of 1325 -f 2312 -f 1321 + 4031 ? 24. What is the sum of 1242 + 2123 + 1312 + 2112? Ans. 6789. 25. What is the sum of 3124 + 1232 + 2113 + 1220? 26. What is the sum of 23102 + 52454 + 24342 ? 27. What is the sum of 15323 + 32354 + 41302 ? ADDITION. 21 37, To add when the amount of any column Is ten or more. 28. A farmer raised 473 bushels of potatoes, 285 bushels of onions, 568 bushels of carrots, and 359 bushels of turnips; how many bushels of vegetables did he raise ? Ans. 1685. Having arranged the numbers so that units OPERATION, stand under units, tens under tens, etc., as 473 . in example 1, add the numbers m the column 285 of units; thus, 9 and 8 are 17, and 5 arc 668 22, and 3 are 25 units, (= 2 tens and 5 359 units). The 5 units are set under the column of units and the 2 tens are added to the tens Ans. 1685 given in the example; thus, 2 and 5 are 7 and 6 are 13, and 8 are 21, and 7 are 28 tens (= 2 hundreds and 8 tens). The 8 tens are set under the tens, and the 2 hundreds are added to the hundreds in the example, giving 16 hundreds, or 1 thousand and 6 hundreds, which, written in their proper places, give 1685 for the answer. 38. In the same manner, add the numbers in the follow- ing short columns f and also add across the page, as suggested by the signs. 29. 3846 + 2843 + 63542 + 35842 + 91326 + 73241 30. 8305 3654 82735 12600 82145 38642 31. 9160 5003 230G4 81264 34208 26341 21311 lin05 129706 138224 32. 3462 -f 1538 + 56421 + 36245 + 35496 + 82437 33. 1354 6242 91367 24687 23549 43621 34. 1534 6215 13579 21683 35462 10820 35. 5104 3160 20013 61000 301Q4r^ 28006 11454 181380 1246U 37. Explain the operation in Ex. 28., Hiirm^ 22 ADDITION. 36. 4006+3567 + 41323 + 30000 + 5436 37. 5143 2G4 346 3812 46 38. 5274 3S0G 5148 346 876 39. 8463 88 63405 87420 45362 + 284 3864 29 389 22886 110227 40. 8716 + 501 + 432167 + 41. 4822 9 9S721 42. 1920 2001 702 43. 1861 92 96 51720 67958 + 8957351 2780 2761852 8765 8578127 83217 101 531686 20297431 30. In solving the foregoing examples, the learner has already become familiar with all the operations in addition; but to enable him readily to tell how to add, we give the following KuLE. Write the numbers in order, units under units, tens under tens, etc. Draw a line beneath, add together the figures in the units column, and if the sum be less than ten set it und^r the column ; but if the sum bs teii or more, write the units as before, and add the tens to the next column. Thus proceed till all the columns are added. 40* Proof. The usual mode of proof is to begin at the top and add downward. If the work is right, the two sums will be alike. Note 1. By this process, we combine the figures differently, and hence shall probably detect any mistake which may have been made in adding upward. 39. Why is a Rule for addition given ? Repeat the Rule. If the amount of any column is 10 or more, where is the right-hand figure of the amount written i Why .'' What is done with the left-hand figure or figures .-' Why ? 40. How is addition proved? Why not a,M upward a second timet In addition is it desirable to name the figures as we add themf Why not? ILLUSTRATION. ADDITION . 23 Ex. 44. In adding upward we say 4 and 6 are 10, and 5 are 15, and 8 are 23, etc. ; but in adding doxmiwardy we say 8 and 5 are 13, and 6 53468 are 19, and 4 are 23, etc. ; thus obtaining 72635 the same residt, but by different comhina- 24376 tions.- 27594 If we do not obtain the same result by Sum, 178073 the two methods, one operation or the other is wrong, perhaps both, and the work must Iroof, 178073 \fQ carefully performed again. Note. In adding it is not usually desirable to name the figures tliat we add ; thus, in Ex. 44, instead of saying 4 and 6 are 10, and 5 are 15, and 8 are 23, it is shorter and therefore better to say, 4, 10, 15, 23; and then setting down the 3, say 2, 11, 18, 21, 27, etc. . 45. A grain dealer bought 3756 bushels of wheat of A, 2347 bushels of B, 1346 bushels of C, and 5468 bushels of D ; how many bushels of wheat did he buy? Ans. 12917. 46. I paid 8 3465 for a farm, S15000 for a mill, $ 6795 for a lot of wool, and $ 4620 for 40 shares of railroad stock ; how much did I pay for all this property? Ans. $29880. 47. Bought 3 city lots for $15345, and sold them so as to gain $ 3639 ; what sum did I receive for them? Ans. $18984. 48. A man commenced trade with $ 5345, and in one year he gained $ 3462 ; what was he worth at the end of the year ? 49. Add three hundred and twenty. five ; two thousand one hundred and fifty- four ; two hundred and fourteen ; twenty- three thousand five hundred and forty-one ; and three hundred and seventy-five. Ans. 26609. 60. What is the sum of thirty-four thousand five hun- 24 SUBTRACTION. dred and forty-six ; five million, two hundred and seventy-six thousand, four hundred and nineteen ; and forty-two million , six hundred and twenty-four thousand, five hundred and eighty seven? Ans. 47,935,552. 51. England and Wales contain about 55,100 square miles ; Scotland, 29,G00 ; and Ireland, 32,000 ; what is the area of the British Islands ? 52. The population of England in 1851 was 10,921,888 ; of Scotland, 2,888,742 ; of Wales, 1,005,721 ; and of Ireland, 6,515,794; what was the population of Great Britain and Ireland? Ans. 27,332,145. 53. In 1850 the population of New York was 515,547; of Philadelphia, 340,045; of Baltimore, 169,054 ; of Boston, 136,881; of New Orleans, 116,375; and of Cincinnati, 115, 436 ; what was the number of inhabitants in these six cities in 1850? Ans. 1,393,338. SUBTKACTION. 4:1. Three apples taken from seven apples leave how many apples ? Ans. Three apples from seven apples leave four apples. This is a question in Subtraction. 4^. Subtraction is taking a less number from a greater number of the same kind, to find their difference,. The greater number is called the minuend ; the less number, the SUBTRAHEND ; and the difference, the eemainder. 4a. WhatisSubtxactiou.' What the Minuend .' Subtrahend? Remainder? SUBTRACTION, 25 SUBTRACTION TABLE. 1 from 2 leaves 1 2 from 3 leaves 1 3 from 4 leaves 1 1 " 3 " 2 2 " 4 * * 2 3 ^ ' 5 " 2 1 " 4 '' 3 2 " 5 * ' 3 3 • ' 6 " 3 1 '^ 5 " 4 2 - 6 ' ' 4 3 ' ' 7 " 4 1 " 6 " 5 2 '^ 7' ' ' 5 3 ' ' 8 '• 5 1 " 7 " 6 2 " 8 ' ' C 3 ' ' 9 " 6 1 " 8 *,' 7 2-9 ' * 7 3 * ' 10 " *7 1 ** 9 *• 8 2 "10 ' ^ 8 3 ' Ml "8 1 " 10 " 9 2 " 11 ' ' 9 3 ' ' 12 " 9 1 « 11 *♦ 10 2 "12 ' ' 1^ 3 ' ' 13 " 10 4 from 5 leaves 1 5 from 6 leaves 1 6 from 7 leaves 1 4 " 6 " 2 " 7 ' ' 2 6 ' ' 8 " 2 4 " 7 " 3 5 " 8 ' ' 3 6 « ' 9 " 3 4 '« 8 " 4 5 " 9 ' ' 4 6 ' ' 10 " 4 4 " 9 " 5 5 "10 ' 5 6 « ' 11 " 5 4 ♦* 10 ♦* 6 5 ♦* 11 ' ' G 6 ' ' 12 " 6 4 " 11 " 7 5 "12 ♦ * 7 6 ♦ ' 13 " 7 4 " 12 " 8 5 " 13 ♦ ♦ 8 6 ' * 14 " 8 4 " 13 « 9 5 "14 ' 9 6 ' * 15 " 9 4 " 14 ♦• 10 5- " 15 ' * 10 6 ' ' 16 " 10 7 from 8 leaves 1 8 from 9 lea ves 1 9 from 10 leaves 1 7 " 9 " 2 8 "10 ' ' 2 9 ' ' 11 " 2 7 " 10 '* 3 8 "11 ' ' 3 9 ' M2 " 3 7 " 11 " 4 8 "12 ' ' 4 9 ' '13 "4 7 " 12 " 5 8 "13 ' ' 5 9 ' ' 14 " 5 7 " 13 " 6 8 "14 ' ' G 9 ' ' 15 " 6 7 " 14 '* 7 8 "15 ' * 7 9 ' ' 16 " 7 7 " 15 " 8 8 " IG * ' 8 9 • * 17 " 8. 7 ^' IG •' 9 8 " 17 ' ' 9 9 ' ' 18 " 9 7 " 17 " 10 8 "18 * ' 10 9 ' • 19 " 10 26 SUBTRACTION^. Mental Exercises. Ex. 1. Joseph has 8 marbles in his right hand, and 5 in his left hand ; how many more marbles has he in his right hand than in his left? Ans. 3. 2. Thomas paid 10 cents for a melon, and 4 cents for an orange ; how much more did the melon cost than the orange ? 3. Daniel paid $ 1 2 for a colt and $ 5 for a lamb ; how much leas did the lamb cost than the colt ? 4. A boy having 15 peaches gave away 8 of them ; how many had he remaining? 5. A man owing $ 17 paid $ 9 ; how much did he then owe? 6. Bought goods for $9 and sold them for $.13 ; how much did I gain ? Ans. $ 4. 7. Sold goods for $15, which was $6 more than they cost me ; what did I pay for them ? 8. William is 18 years old and George is 9 years younger ; how old is George? 9. John had 1 7 cents and spent 9 of them ; how many cents had he then ? 10. A tailor had 15 yards of cloth, from which he sold 9 yards ; how many yards remained ? Ans. 6. 11. Samuel is 16 years old and David is 9 ; how much older is Samuel than David? 12. Isaac had 12 marbles, but has lost 7 of them; how many marbles has he now ? 4:3. The sign of subtraction, — , called minus or less, signifies that the number after it is to be taken from the num- ber before it ; thus, 7 — 4=3; that is, seven minus four, or seven less four, equals three. Ex. 13. How many are 9 — 5? Ans. 4. 43. Make the sign of Subtraction on the black-board. What is it called ? What docs it mean i Illustrate. SUBTRACTION, 27 14. How many are 8 — 6? 12 — 3? 10 — 7? 15. How many are 12 — 5? 9 — 6? 11 — 5? 16. How many are 16 — 7? 15 — 9? 13 — 8? 17. How many arc 17 — 6? 12 — 8? 18 — 9? 18. How many are 18 — 7? 16—9? 14—9? 44:. Lef: the pupil frequently review the following Exercises in Subtraction. No. 1. No. 2. No. 3. 1 No. 4. No. 5. 6 — 2 9 — 5 7 — 4 8 — 6 13—6 8 — 5 6 — 3 9 — 6 7 — 7 6—4 . 3 — 1 10 — 4 12 — 5 7-0 8— 3 9 — 7 7 — 5 8 — 7 9 — 3 10—5 7 — 3 3 — 3 4—1 2 — 2 7— 1 5 — 4 3 — 5 — 3 6 — 5 11—3 4 — 2 7 — 2 7 — 6 15 — 9 12—4 2— 1 8 — 4 9 — 8 12 — 8 15 — 10 No. 6. No. 7. No. 8. No. 9. No. 10. 9 — 4 10 — 6 12— 9 15—7 14—8 8 — 2 12 — 3 14— 6 17—9 16—6 12 — 6 16 — 4 11—7 14—5 10—9 10 — 7 11 — 5 4— 3 10—8 13—4 12 — 7 18 — 2 9— 2 9— 1 15—5 14 — 2 14 — 7 11— 6 10—0 18— 7 16 — 7 8 — 1 12— 10 10—10 17—8 15 — 8 11 — 4 15 — -6 18—8 14—4 No. IL No. 12. No. 13. No. 14. No. 15. 17— 6 15 — 11 18— 6 17 — 11 16 — 12 16— 9 17 — 10 16— 10 18—5 18— 4 14— 10 14—9 14—12 16—8 17—5 18— 9 11—8 12— 1 13—9 14—3 12—11 16—5 15—4 15—3 17— 7 15 — 12 10—3 13—8 18 — 16 16 — 12 13—7 13—5 18—12 16—2 13 — 11 18—10 18—11 15 — 13 14 — 11 11— 9 28 SUBTRACTION, Writtejj Exercises. 45. To subtract when no figure in the subtrahend is fjreater than the fio^ure above it. Ex. 1. From 837 take 523. Ins. 314. Operation. Minuend, 837 Subtrahend, 523 Kemainder, 314 Having written the less number under the greater, units under units, tens under tens, etc., we say 3 from 7 leaves 4, 2 from 3 leaves 1, and 5 from 8 leaves 3 ; therefore the remainder is 314. In like manner solve the following examples : Ex. 2. 3. 4. 5. Erom $53.68 Take $21.43 $ 736.45 $325.13 38697 13543 Ans. $32.25 $411.32 7. 8. Hours. Men. Erom 9368 65439 Take 3215 25316 Women. 63548 21410 386495 243345 143150 10. Children. 390642 180321 6. 836942' 314241 11. Horses. 897436 135223 40123 210321 12. By the census of 1860, there were 326072 inhabitants in New Hampshire, and 628276 in Maine; how much less was the population of New Hampshire than of Maine ? 13. By the census of 1860, the population of Mississippi was 791396, and that of the United States Territories was 220143 ; how many more people were there in Mississippi than in the Territories ? Ans. 5 7 1 25 3. 14. A farmer bought a farm for $ 3465, and sold it for $ 4689 ; how much did he gain? Ans. $ 1224. 15. How many are 29 less 16 ? 876 less 346 ? 45. How are numbers arranged for Subtraction r" Why? Which figure is subtracted first ? Where is the Kemainder written ? SUBTRACTION. 29 16. How many are 89 less 74 ? 963 less 241 ? 17. Howmany are 836— -215? 8360—6320? 18. Howmanyare 869—349? 9386 — 2150? 40. To subtract when any figure in the subtrahend is greater than the fio^ure above it. 19. From 863 take 249. Ans. 614. Two methods for explaining this'opera- Operation. ^Jqjj ^j.g jj^ common use. Minuend, 863 1st. As we cannot take 9 units from 3 Subtrahend, 249 units, 07ie of the 6 tens is put with the 3 ^ . , units, making 13 units, and then, 9 units Remainder, 614 ^ ,o -1.1 a -j. i.- v • t. from 13 units leave 4 units, which is set under the units. Now, as one of the 6 ietis has been used, only 5 tens remain in the minuend, and 4 tens from 5 tens leave 1 ten, and, finally, 2 hundreds from 8 hundreds leave 6 hundreds; therefore the entire remainder is 614. 2d. We may add 10 units (equal to I ten) to the three units, making 13 unit?. From this sum we subtract the 9 units. In subtracting the next column, instead of taking away 1 of the 6 tens in the minuend, we may add 1 ten to the 4 tens in the subtrahend, and then take the sum (5 tens) from the 6 tens, and the result is 1 ten as by the former process. The second mode depends on the principle, that if two num- bers are equally increased, the difference between them remains unchanged. Now, in solving Ex. 19 by the second method, we add 10 units to the minuend, and 1 ten (the same as 10 units) to the subtrahend^ and therefore find the same remainder as by the first method. 46. How many methods of subtracting when a figure of the subtrahend is greater than the figure over it ' Explain the first method. Explain the second. The second depends on what principle .'' Is the same number added to mimtend and subtrahend ? Mow 1 30 SUBTRACTION. 47. In the same manner solve the following examples, taking each lower number from the one over it in each exam- ple ; also, subtract in the manner indicated by the signs. 20. 21. 22. 23. ("86326 — 43710 53684—36146 74668 i3462 |34613 — 23620 21392 — 19324 43158 — 25319 61713 f 58327 32292 16822 28143 36-118 66888 — 43682 83621 — 34261 \43618 — 18294 51364 — 35176 52S42 — 21638 18124 (73926 — 53614 83654 ■[61498—39182 8263 8506 46839 93654 3642 8432 12623 9342 584 (99594 14432 74660 46832 (81940—50706 36481 43197 39481 22814 8758 73162 — 68243 61928 — 34821 23954 11234 48. The pupil having become familiar with the modes of subtracting, we aid him by giving the following : EuLE 1. Write the less number under the greater, units under units, tens under tens, etc., and draw a line beneath. 2. Beginning at the right hand, take each figure in the sub- trahend from the figure above it, and set tJie remai7ider under the line. 3. Jf any figure in the subtrahend is greater than the figure above it, add ten to the upper figure and take the lower figure from the sum ; set down the remainder, and considering the next figure in the minuend one less, or the next figure in the subtrahend one greater, proceed as before. 48. What is the Rule for Subtraction ? SUBTRACTION. 31 49; Proof. Add the subtrahend and the remainder together, and the sum should be the minuend. Note 1. This proof rests on the self-evident truth, that the whole of a thing is equal to the sum of all its parts ; thus, the min- tiendis separated into the two parts, subtrahend and remainder; hence the sum of tliose parts must be the minuend. ILLUSTRATION. Minuend, 8264: Subtrahend, 3692 Eemainder, 4572 Ex. 24. As the sum of the subtrahend and remainder is the minuend, the work is supposed to be right. Proof, 8264 25. Inches. From 8365 Take 4879 26. Men. 636554 482732 Rem., 3486 27. Gallons. 96G482 3S1779 584703 28. Apples. 835670 482984 • Proof, 8365 (7) (9) (13) From 8 3 Take 2 6 7 Ans. 29. Here, we cannot take 7 from 3, nor can we borrow from the tens' place, as that place is occupied by ; but we can borrow one of the 8 hundreds, and separate the one hundred into 9 tens and 10 units ; then, putting the 9 tens in the place of tens, and adding the 10 units to the 3 units in the minuend, we can subtract 7 from 13, 6 from 9, and 2 from 7. Note 2. This process will probably be more readily understood by the young learner than the second method given in the rule, tliougli the latter, for convenience, is usually adopted. 49. What the proof? On what principle does the proof rest f Illustrate. Explain Ex. 29. Which mode of subtracting is more readily understood.'' Which more convenient ? 32 SUliiK AUTIUJX. 30. 31. 32. 33. 34. From $5304: Days. 6403 Sheep. 6030 Miles. 9084 Bushels, 8005 Take $2457 3846 2684 7692 3689 $2847 2346 4316 • 35. Washington was born in 1732, and died in 1799; at what age did he die ? Ans. 67, 36. How many years have passed since the discovery of America in 1492 ? 37. Jamestown, in Virginia, was settled in 1607; how many years from that date was the Declaration of Independence in 1776 ? Ans. 169. 38. Queen Victoria was born in 1819 ; how old was she in 1865? Ans. 46. 39. A merchant bought goods for $ 3846, and sold the same for $ 5050 ; what was his gain ? Ans. $ 1204. 40. A merchant paid $ 8004 for goods, and sold the same for $ 684G ; what was his loss? 41. How many years from the discovery of America by Columbus, in 1492, to the settlement of Plymouth by the Puritans, in 1620? Ans. 128. 42. In 1864 a man died at the age of 87 years ; in what year was he born ? 43. The sum of two numbers is 80304, and the greater number is 54836 ; what is the less? Ans. 25468. 44. The less of two numbers is 34685, and their sum is 90304 ; what is the greater? Ans. 55619. 45. The difference between two numbers is 3684, and the greater number is 8002 ; what is the less ? 46. From one thousand eight hundred and sixty-fivo one thousand four hundred and ninety-two. SUBTRACTION. 33 47. From two million, three hundred and sixty-one thou- sand, four huudred and seventeen, take one million, five hun- dred and forty- six thousand, two hundred and eighty-nine. Ans. 815128. 48. Suppose the distance from the earth to the sun is 94879956 miles, and that from the earth to the moon is 240000 miles ; how much farther is the sun than the moon from the earth ? 49. The population of the United States was 31,443,790 in 1860, and 23,191,876 in 1850 ; what was the increase in ten years? Ans. 8,251,914. 60. Suppose the outstanding public debt of the United States to be $ 2,800,000,000, and that $ 125,375,287 now in the treasury be applied to its payment, what would then be their indebtedness ? EXAMPLES IN ADDITION AND SUBTRACTION. 1. From the sum of 94 and 86, take 117. Ans. 63. 2. From the sum of the three numbers, 629, 493, and 896, take the sum of 968 and 563. Ans. 487. 3. I owe three notes, whose sum is $ 3895 ; one of these notes is for S 1348, another for $ 863 ; for how much is the third? Ans. $1684. 4. A farmer having 1275 acres of land, sold 318 acres at one time, 227 at another, and 175 at another ; how many acres has he remaining ? 5. If a man's income is $1865 a year, and he pays $200 for rent, $468 for food, $278 for clothing, and $712 for other expenses, how much will he save in the year ? Ans. $ 207. 7. How many are 876 + 392 + 847 — 963 ? 8. How many are 986 + 389 -f- 549 — 846 ? 34 SUBTRACTION. 9. Two men start from the same place and travel in the same direction, one goes 125 miles, the other 876, how far apart are they ? How far if they had travelled in opposite directions ? 10. From T,000,000 subtract 8901 + 101. Ans. 6990998. 11. A man purchased a farm for $ 6890, and having paid $ 575 for an additional piece of land, he sells the whole for $ 7500 ; does he gain or lose, and how much? 12. In a Union school there are four departments ; in the first there are 125 scholars, in the second 379, in the third 437, and in the fourth 487 ; how many scholars does it con- tain ? If 692 are boys, how many are girls? First Ans. 1428. Second Ans. 736. 13. A general started oat on a campaign with three regi- ments of soldiers, the first numbered 1025 men, the second 975, the third 875 ; after a battle he finds but 2575 in all reported fit for duty ; how many men has he lost ? 14. How many are 687 + 594 + 369 — 918? 15. In 1850, the population of New York was 615547; that of Philadelphia, 340045; of Baltimore, 169054; and of Boston, 136881. At the same time, the population of London was about 2363241 ; what was the difference between the population of London and the aggregate population of the four cities named in the United States ? Ans. 1201714. 16. A merchant bought some flour for $ 347, some rye for $ 236, and some oats for $ 563 ; he sold the whole for $ 1275. Did he gain or lose ? How much ? 17. Mr. Jones gives $ 2376.43 to his four sons, as follows : to Daniel, $ 534.68 ; to James, $ 354.68 ; to Thomas, $ 486.39 ; and the rest to David. What does David receive? Ans. 81000.68. MULTirLICATIOX. 35 MULTIPLICATION. 50. In 1 bushel there are 32 quarts ; how many quarts are there in 8 bushels? 1st Method, 2d Method, This example may be solved BY ADDITION. BY MULTIPLICATION, by addition, as by the 1st 3 2 3 2 method ; but as there will 3 2 8 evidently be 8 times as many 3 2 quarts in 8 bushels as there 3 2 Product, 2 5 6 are in 1 bushel, it may be 3 2 more briefly solved as by the 3 2 2d method ; thus, 8 times 2 units are 1 6 units, 3 2 = 1 ten and 6 units ; write the 6 units in the 3 2 place of units, and then say 8 times 3 tens are — I — 24 tens, which, increased by the 1 ten previ- Sum, 2 5 6 Q^giy obtained, make 25 tens, = 2 hundreds and 5 tens, and when these are written in their proper places we have 256 quarts for the true result. This, when solved by the 2d method, is a question in multiplication. 51. Multiplication is a short method of adding equal numbers; or, it is a short method of finding how many units there are in any number of times a given number. The number repeated is called the multiplicand ; the num- her showing how many times the multiplicand is taken is the multiplier ; the sum, or result of the multiplication, is the product. The Multiplicand and Multiplier are called Fac- tors. 50. Explain the two methods of solving the above example. "Which is best? 51. What is Multiplication ? Another definition. What is the Multipli- cand? Multiplier? Product? Factors? 53. Repeat the Multiplication Table. 36 MULTIPLICATION. 53. The pupil, before advancing further, should learn the following MULTIPLICATION TABLE. Once 1 is 1 2 " 2 3 4 5 6 7 Twice 1 are 2] 2 <( 4 3 (( 6 4 (( 8 5 <« 10 6 (( 12 7 (< 14 8 (( 16 9 <« 18 10 << 20 11 <( 22 12 •' 2\: Three times 1 are 3 2 <( 6 3 <( 9 4 (« 12 5 8. Similar reasoning applies however many figures there may be in the multiplier. Hence, EuLE 1. Set the multiplier under the multiplicand, and draw a line beneath. 2. Beginning at the right hand of the multiplicand, multiply the midtiplicand by each Jig ure of themidtiplier, setting tJie first figure of each partial product directly under the figure of the multiplier which produces it. 3. The Sum of the partial products willbe the true product. 57, Explain the solution of Example 30. Where is the first figure of each partial product written ? "Why? 58. Kepeat the JRule. 42 MULTIPLICATION. 50, Proof. Multiply the multiplier hy the multiplicand, and, if correct, the result will be like the first product. Note. This proof rests on the principle, that tlie order of tlie factors is immaterial ; 3X4=4X3; 6X2X7=2X7X5=2X5X7, etc. ' Ex. 28. Multiply 669 by 418. OPERATION. PROOF. MultiplicanJ, 5 69 47 8 Multiplier, 4 7 8 , * 5 6 9 4552 4302 3983 2868 2276 2390 Product, 271982 = 271982 60. In the same manner solve the following examples, multiplying each upper number by the one under it in each example ; also multiply in the manner indicated by the signs. oq(8X9 24 X n 634X376 4362X3264 ^•|6X7 21X14 428X245 2468X1357 g^f7X6 36X24 568X492 5486X3698 4^ X8 27X32 634X346 2534x4368 Q, (9X9 46X54 648X396 8682X3842 "^^'18X6 35X43 827X423 6342x4362 „^f9X5 88X77 986X684 9999X6843 "^"'18X7 64X72 379X793 4682X7953 33. If a steamboat sails 12 miles per hour, how far will she sail in 24 hours ; that is in 1 day? 34. If a steamboat sails 288 miles per day, how far will she sail in 28 days ? 59. Proof? Principle?' MULTIPLICATION. 43 « ;7 acres of land, at $ 13.36 per acre? Ans. $494.32. 36. How long will it take 1 man to do as much work as 24 men can do in 75 days? Ans. 1800 days. 37. How far will a horse travel in 27 days, if he travels 37 miles per day ? 38. How many yards of cloth in 33 pieces, if each piece contains 54 yards? Ans. 1782. 39. Multiply two hundred and fifteen thousand eight hun- dred and forty-seven by six hundred and fifty-nine. Ans. 142,243,173. 40. "What is the cost of building 243 miles of railroad, at $48,750 per mile? 41. if a garrison of soldiers eat 5876 pounds of bread per day, how much will they eat in 365 days? 01. Ciphers between the significant figures of the multi- plier may be neglected, taking care to set the first figure of each partial product directly under the figure of the multiplier which gives that product. Ex. 42. Multiply 7543 by 2005. This is only carrying out the principle OPERATION. ^.^ addition) of setting units under 7543 units, tens under tens, etc. The 2 of the 2005 multiplier is 2000, and 2000 times 3 is "TZTTT 6000, therefore the 6 of the partial 15086 product should be written in the place of thousands ; that is, directly under the 2 Product, 15123715 of the multiplier. 43. Multiply 3642 by 3008. Ans. 10955136. 61. What may be done with ciphers between the significant figures of the multiplier ? What care is required ? Principle ? 44 MULTIPLIOATIOIT. * 44. What cost 507 miles of railroad at $ 3G4-8 per mile? Ans. $1,849,536. 45. How many lemons in 806 boxes, if each box contains 309 lemons? 46. How many pounds of cotton in 3004 bales, each bale containing 537 pounds? Ans. 1,613.148. Contractions. G2, The rules already given are sufficient for all examples that can arise in multiplication, but there are various devices for shortening the process in particular cases. 63. To multiply by a composite number. A Composite Xumber is i\iQ product of two or more numbers; 15 is a composite number, whose factors are 3 and 5 ; 12 is a composite number, whose factors are 2 and 6, or 3 and 4, or 2, 2, and 3. It will be observed that a composite number may hmie several sets of factors. 47. How many dollars have 35 men, if each man has $ 43 ? The 35 men may be sepa- rated into 7 groups of 5 men each. Now 1 group of 5 men will have 5 times S 43, := $215, and if 1 group has $215, evidently 7 groups will have 7 times $ 215 z= $1505, Ans. That is, 7 times 5 times a number are Product, $ 1505 35 times that number. 63. What ifl said of the rules already given for Multiplication ? What of shorter modes ? 63. What is a composite number ? May a composite number have more than one set of factors ? Rule for multiplying by a composite number ? OPERATION. 35 r= 5 X 7. Multiplicand, 1st Factor of Multiplier, $43 5 2d Factor of Multiplier, $215 7 MULTIPLICATION. 45 48. Multiply 367 by 168. Ans. 61656. FIRST OPERA TIOX. SE0OND OPERATION. 168 = 8 X 7 X 3. 168 = 4 X 7 X 6. Multiplicand, 3 6 7 3 6 7 1st Factor of Multiplier, 8 4 2936 1468 2(i Factor of Multiplier, 7 7 * 20552 10276 8d Factor of Multiplier, 3 6 Product, 61656 = 61656 Several other sets of factors of 168 may be used, and give tbe same product. Every similar example may be solved in like manner. Hence, KuLE, Multiply the multiplicand hy one of tJie factors of the multiplier t and that product hy another factor, and so on until all the factors in the set have been taken ; the last prodiLct will he the true product. 49. Multiply % 8.37 by 3G. Ans. % 301.32. 50. Multiply % 659 by 56. 53. Multiply 8356 by 81. 51. Multiply 737 by 72. 54. Multiply 6753 by 49. 62. Multiply 967 by 96. 55. Multiply 7045 by 54. 64. To multiply by 10, 100, 1000, or 1 with any number of ciphers annexed. Rule. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the numher so formed will he the product. 64:. How is a number multiplied by 10 ? By 100 ? Why ? 46 MULTIPLICATION. Note. The reason of the rule is obvious-. Annexing a cipher re- moves each figure in the multiplicand one place toward the left, and thus its value is increased ten fold (Art. 18). 56. Multiply 74 by 10. Ans. 740. 57. Multiply 357 by 1000. Ans. 357000. 58. What is 8769 X 100? 59. What is 3568 X 10,000? 60. What is 9806 X 100,000 ? 65. To multiply by 20, 50, 500, 25000, or any number, with ciphers at the riglit : , EuLE. Multiply by the significant figures, and to the prod- uct annex as many ciphers os there are ciphers at the right of the significant figures of the inultipUer. 61. Multiply 756 by 30. Ans. 22680. OPERATION. 7 5 6 This is upon the principle of Art. 63. 3 The factors of 30 are 3 and 10. Having multiplied by 3, the product is multiplied by 2 2 6 8 10 by annexing (Art. 64). Q2, Multiply 743 by 3500, using factors. OPERATION. ;f 7 4 3 *7 The factors of 3500 are 7, 5, and 100, therefore multiply fiirst 5 2 1 by 7, then by 5, then annex two 5 ciphers. Product, 2 6 005 00. 63. Multiply 5386 by 42000. Ans. 226212000. 64. Multiply 6539 by 240000. Ans. 1569360000. 65. Multiply 0743 by 630000. 65. How is a number multiplied by 20? Why? MULTIPLICATION. 47 06. To multiply when tliere are ciphers at the right of both multiplicand and multiplier : EuLE. Multiply the significant figures of the multiplicand by those of the multiplier, and then annex as many ciphers to the product as there are ciphers at theright of both factors. Q>Q, Multiply 8000 by 900. Ans 7200000. The factors of 8000 are 8 and 1000, OPERATION. and those of 900 arc 9 and 100. Now, 8 as it is immaterial in what order the fac- 9 * tors are taken (Art. 59, Note), first mul- tiply 8 by 9, then multiply this* product Ans. 7 2 by 1000 (Art. 64), and this product by • lOX 67. Multiply 730000 by 2900. OPERATION. 730000 2900 657 14G Product, 2 117 0. 68. Multiply 37000 by 29000. Ans. 1073000000. 69. 730000 by 47000. Ans. 34310000000. 70. 17000 by 79000000. 71. 4500 by 720000. Ans. 3240000000. 67. To multiply by 9, 99, or any number of 9's. Rule. Annex as many O's to the multiplicand as there are 9's in the multiplier ^ and from the number so formed subtract the multiplicand ; the remainder will be the product sought. 66. Rule when there are ciphers at the right of both factors? Reason 1 6T. Rule for multiplying by 9, 99, 999, etc. ? Reason? 48 MULTIPLICATION. 72. Multiply 234: by 99. OPERATION. 2 3 4 = 100 times the multiplicand. 2 3 4= 1 time the multiplicand. 23 166 = 99 times the multiplicand, Ans. 73. Multiply 5379 by 999. Ans. 5373621. 74. Multiply 638 by 9999. Ans. 6379362. 75. Multiply 739 by 99. By 999. Examples in the Foregoing Principles. 1. A merchant bought 156 barrels of flour at $ 9 per bar- rel, and 75 barrels at $1 2 per barrel. He also sold 987 bushels of wheat at $ 2 per bushel ; how much more did he pay for the flour than he received for the wheat ? Ans. $ 330. 2. Two men start from the same place, and travel in the same direction, one at the rate of 48 miles, and the other 36 miles per day ; how far apart are they at the end of 17 days ? Ans. 204 miles. 3. Had the men named in Ex. 2 travelled in opposite direc- tions, how far apart would they have been in 31 days ? Ans. 2604 miles. 4. A farmer killed 2 oxen weighing 975 pounds each, 3 cows weighing 462 pounds each, and 5 swine weighing 456 pounds each ; how much more beef than pork had he ? Ans. 1056 pounds. 5. The President of the United States receives a salary of $ 25000 a year ; what will he save in a year of 365 days, if his expenses are $ 60 a day ? Ans. $ 3 100. 6. A man having a journey of 287 miles to perform in 5 days, travels 62 miles a day for 4 days ; how far must he go on the fifth day ? MULTIPLICATION. 49 7. Bouglat a herd of 25 cows, paying as many 'dollars for each cow as there were cows in the herd. Paid $ 500 in money, and gave my note for the balance ; what was the amount for which the note was given? Ans. $ 125. 8. Bought 13 cows at $42 each, and 21 pair of oxen at $ 87 a pair ; what did I pay for all ? Ans. $ 2373. 9. Sold 3 farms ; for the first I received $ 2345, for the second, $ 364' less than for the first, and for the third, twice as much as for the other 2 ; how much did I receive for the 3 farms? Ans. $ 12978. 10. The factors of one number are 31 and 43, and of another 29 and 17 ; what is the difference of the two numbers? 11. A teacher receives 8 1200 a year, and pays $ 364 a year for board, 8 96 for clothe?, $75 for books, and $ 356 for other expenses ; how much will he save in 5 years ? 12. A manufacturer receives $37950 in one year for the products of a certain factory. For materials he pays out $ 15675, for labor $ 10369, for repairs of machinery $2006; how much profit remains to him? Ans. $ 9900. 13. If the above manufacturer, after paying out of his income $ 5 on every hundred dollars for United States tax, $ 3 on every hundred for other taxes, expends $4875 for the support of his family, how much remains? Ans. $ 4233. 14. A drover bought 280 head of cattle for an average cost of $ 75 per head, 10 horses for $ 210 each; after deducting the expenses of transporting them to the market he found he had made $15 per head on the cattle, $50 apiece on the horses, what was the amount of his profits? 15. A steamboat makes 300 trips in one season ; she carries an average of 225 passengers each trip, and 75 tons of mer- chandize. If the average receipts are $2 for each passenger and $ 1 for each ton of freight, how much money does she receive ? Ans. $ 167500. 50 DIVISION. 16. A country merchant went to the city to purchase goods, carrying with him $3000. He bought 20 barrels of flour, at $ 12 per barrel, 275 gallons of molasses, at $ 1 per gallon, a box of sugar for $ 178, two pieces of broadcloth at $ 56 a piece ; other dry goods to the amount of $ 525, and other groceries to the amount of $ 118, and a variety of small goods to the amount of $375. After paying for these how much money had he left? DIVISION. 68. How many peaches, at 2 cents each, can I buy for 6 cents ? Ans. As many as 2 cents is contained times in 6 cents ; therefore I can buy 3 peaehss^ hecaicse 2 cents is contained 3 times in 6 cents. If 12 apples are divided equally among 3 boys, how many apples will each boy have ? Ans. JSach boy will have 4 apples, because if 12 apples are divided into 3 equal parts each part is 4 apples. These are questions in Division. 60. Division is the process of finding how many times one number is contained in another ; or. Division is the pro- cess of separating one number into as many equal parts as there are units in another number. The number to be divided is called the Dividend ; the num- ber by which to divide is the Divisor ; the number of times the dividend contains the divisor is the Quotient ; and, if the dividend does not contain the divisor an exact number of times, the joar^ of the dividend that is left is the Kemainder. Note. The remainder, heing a part of the dividend, is always of the same kind as the dividend. 68. Explain the Examples in Art. 68. 69. What is Division ? Another definition ? What is the Dividend ? Divisor ? Quotient ^ Kemamrler ? Of what kind is the remainder ? Why ? DIVISION. DIVISION TABLE. 51 1 in I Once. 2 in 2 Once. 3 in 3 Once. 1 iu 2 Twice. 2 in 4 Twice. 3 in 6 Twice. 1 1 in 3 3 times. 2 in 6 3 times. 3 in 9 3 times. 1 in 4: 4 " 2 in 8 4 " 3 in 12 4 •< 1 in 5 5 " 2 in 10 5 " 3 in 15 5 - i 1 in 6 6 " 2 in 12 6 " 3 in 18 6 " 1 in 7 7 - 2 in 14 7 " 3 in 21 7 *' 1 in 8 8 " 2 in 16 8 " 3 in 2t 8 " 1 in 9 9 " 2 in 18 9 " 3 in 27 9 " 1 4 in 4 Once. 5 in 5 Once, 6 in 6 Once. 4 in 8 Twice. 5 in 10 Twice. 6 in 12 Twice. 4 in 12 3 times. 5 in 15 3 times. 6 in 18 3 times 4 in 16 4 " 5 in 20 4 " 6 in 24 4 " 4 in 20 5 " 5 in 25 5 - 6 in 30 5 " 4 in 24 6 " 5 in 30 6 *' 6 in 36 6 " 4 in 28 7 " 5 in 35 7 '* 6 in 42 7 " 4 in 32 8 " 5 in 40 8 •« 6 in 48 8 " 4 in 36 9 - 5 in 45 9 " ' in 54 9 " 7 in 7 Once. 8 in 8 Once. 9 in 9 Once. 7 in 14 Twice. 8 in 16 Twice. 9 in 18 Twice. 1 in 21 3 times. 8 in 24 3 times. 9 in 27 3 times 7 in 28 4 " 8 in 32 4 *' 9 in 36 4 " 7 in 35 5 " 8 in 40 5 " 9 in 45 5 " 7 in 42 6 " 8 in 48 6 - 9 in 51 6 " 7 in 49 7 " 8 in 56 7 " 9 in 63 7 " 7 in 56 8 - 8 in 64 8 '' 9 in 72 8 " 7 in 63 9 •' 8 in 72 9 '» 9 in 81 9 " 52 DIVISION. Mental Exercises. Ex. 1. How many oranges, at 5 cents apiece, can be bought for 15 cents? Ans. As many as 5 cents is contained times in 15 centSf namely, 3. 2. At 5 cents an ounce, how many ounces of cloves can be bought for 30 cents? Ans. 6. 3. At $ 6 a cord, how many cords of wood can I buy for 8 24. 4. At $ 8 a ton, how many tons of coal can I buy for $ 24 ? For $40? For $56? For $ 32 ? Last Ans. 4. 5. In how many weeks, at $ 9 a week, will a man earn $ 27 ? $54? $36? $63? ,6. At $ 9 a barrel, how many barrels of flour can I buy for $45? For $81? For $63? 7. In how many hours will a horse travel 36 miles if he travels 6 miles per hour ? If 9 miles ? If 4 miles ? 8. When blueberries cost 10 cents a quart, how many quarts can be bought for 40 cents? For 70 cents? 90 cents? 9. How many sheep, at $ 1 1 apiece, can I buy for $ 55 ? For $ 44 ? For $ 66 ? Last Ans. 6. 10. How many pounds of coffee, at 12 cents a pound, can I buy for 36 cents ? For 48 cents ? For 72 cents ? 11. Two men, 72 miles apart, approach each other at the rate of 9 miles per hour; in how many hours will they meet? 12. I divided 15 cents equally among 5 boys; how many cents did each boy receive? Ans. If l^ cents are divided into 5 equal parts, each part is 3 cents, therefore each boy received 3 cents. 13. A farmer sold 5 sheep for $ 45 ; what was their average price ? DIVISION. 53 14. If 9 men can cut 54 cords of wood in a week, how many cords can 1 man cut in the same time ? 15. A pile of 48 barrels of apples will exactly fill 8 equal bins; how many barrels will eacJh bin hold ? 16. I divided 55 cents equally among 11 boys; how many cents did each receive ? 17. A dairy woman has 84 pounds of butter which she wishes to divide equally among her 1 2 customers ; how many pounds can she furnish each ? 18. If a workman earns $48 in one month of 4 weeks, how much does he earn in one week ? 1 9. How much will the above workman earn in one day ? 20. How many bushels of wheat, at $ 3 per bushel, will it take to pay for 15 bushels of rye, at $ 2 per bushel? 21. If 7 suits of clothes can be made from 64 yards of cloth, how many yards does it take for one suit? 22. If the cloth for one of the above suits costs S 21 , how much is that per yard ? 70, The sign of division, -^, indicates that the number before it is to be divided by the number after it; thus, 8 -^ 2 r= 4 ; that is, 8 divided by 2 equals 4 ; or, 2 in 8, 4 times. Ex. 23. How many are 10 -i 24. How many are 9 25. How many are 15 26. How many arc 16 27. How many are 49 28. How many are 72 29. How many are 84 -i- 7 ? 30. How many are 8 1 3? 5? 8? 7? 9? 70. Make the sign of Division on the blackboard 54 DIVISION. 71,. Keview frequently the following Exercises in Division. No. 1. No. 2. No. 3. No. 4. No. 5. 8- r4 16- r 4 6- ^3 35- f-5 so- ^-6 18- rQ 27- i-3 12- ^-4 42- ^6 le - r 8 15- -3 45- r 9 24- '-Q 72- ^•8 36- r9 49- -7 16- r2 42- '-1 24- ^4 14- -7 30- -6 32- -8 40- h5 9- ^9 20- r2 8- -1 35- -7 48- r 8 18- ^3 28- r4 24- -8 36- -6 54- r 9 14- ^2 15- r-5 18- -2 25- -5 18- r2 56- f-7 21- r3 No. 6. No. 7. No. 8. No. 9. No. 10. 30- -10 44- -11 56-^ -8 48- -12 1 -. -7 63- -9 72- -9 36-: -12 84- -7 40-: -10 45- -5 54- -9 70-: -7 12- -3 60-: - 12 36- -4 40- -8 12-^ -6 77- -11 81 -: -9 48- -6 4- -4 20 H -4 6- -6 40-^ -4 63- -7 30- -3 27-: -9 60- -4 55^ - 11 64- -8 21- -7 50-: -10 20- -5 80-^ -8 12- -2 65- -5 60-: -5 8 - -8 50^ -5 No. 11. No. 12. No. 13. No. 14. No. 15. 72- -12 88-^ - 11 60- ^10 22- - 11 132- ^-12 80- -10 70-^ -10 96- r 12 108- -12 50- r^ 77- -7 99^ -9 90- ^-9 100- - 10 66 - '-& 60- -6 96- -8 88- :-8 48- -4 110- '- 10 36- r3 33- -3 24- :-i2 28- -7 132 - '- 11 22- -2 24- -2 99- :-ii 120- -12 110- '- 11 12- r 1 84- -12 90- '- 10 121- -11 20- MO 66- r 11 33- - 11 108- ^9 120- -10 144- '- 12 DIVISION. 55 72. Division is indicated not only by the sign 4-, given in Art. 70, but also by the coloUj thus, 8:2; also, by writing the divisor before the dividend, with a curved line between them, thus, 2)8; also, by writing the divisor under the divi- dend, with a line between them, thus, | ; each of which expressions means that 8 is to be divided by 2. 73. The last mode of indicating division, sometimes called the fractional sigyi, gives the following compact DIVISION TABLE. i = l 1 = 1 f = l t=l 1 = 1 1=1 f = 2 1 = 2 1 = 2 1 = 2 1^0=2 ¥=2 ? = 3 J = 3 1 = 3 V = 3 ¥ = 3 ¥ = 3 \ = i 1 = 4 '# = 4 '/=4 ^=4 ¥=* \ = 5 ^ = 5 ¥ = 5 ^ = 5 ¥ = 5 ¥ = 5 1=6 L2 — 6 ¥ = 6 2/ =6 3^0 — 6 ¥=6 1 = 7 ^=7 V=7 ^ = 7 ¥=7 V = 7 f = 8 .J =8 ¥ =8 ¥=8 V=8 V=8 f=9 ■^3=9 V=9 \« = 9 V=9 ¥ = 9 f=l 1 = 1 1 = 1 i« = l « = 1 if = l V = 2 V=2 ¥ = 2 *a = 2 H = 2 ^J = 2 V=3 ¥ = 3 ¥ = 3 fj = 3 3|=3 11=3 y = 4 ¥ = * ¥=4 ^§ = 4 H = 4 f| = 4 V=5 4/ = 5 V = 5 iJ = 5 H = 5 |»=5 V =6 V = 6 V=6 53- = 6 ff = 6 if = 6 V = 7 ¥ = 7 V = 7 U = 7 ii=7 41 = 7 y =8 V=8 V=8 ?J = 8 ff=8 11 = 8 V = 9 y = 9 V=9 fj = 9 f^=9 %'=9 73. Second sign of Division, wliat is it ? Third mode of indicating Divi- sion, what is it? Fourth method, what? T3, How are the dividcud and divisor written in the second Division Table ? 5Q DIVISION. Ex. 31. How many are ^S or 24 -^- 6 ? Ans. 4. 32. How many are -3/, or 35 -f- 5 ? 3^2.^ or 32 ~ 8 ? 33. How many are ^, or 18 H- 2 ? \% or 28 -^ 7 ? 34. How many are ^2.^ or 42 4- 6 ? 4^-, or 49 -^ 7 ? 35. How many are %3-, or 63 -^ 9 ? -7/, or 72 -f- 8 ? Written Exercises. 74, To perform Short Division. Ex. 1 . If 7 days make a week, how many weeks are there in 364 days? Having written the 'divisor OPERATION. ^^^ dividend as in the margin, Divisor, 7)364 Dividend, we first say, 7 in 36, 5 times and 1 remainder ; set the quotient, 5, Quotient, 5 2 under the 6 of the dividend, and then imagine the remainder, 1, placed before the 4, and say, 7 in 14, 2 times; set the 2 under the 4, and thus we find the quotient to be, 52, 75, This process, called Short Division, usually employed when the divisor is small, may be performed by the following KuLE. Having set the divisor at ths left of the dividend with a line between them, divide the fewest figures in the left of the dividend that will contain the divisor, and set the quotient under the right hand figure taken in tJie divideyid ; if anything remains, prefix it mentally to the next figure in the dividend, and divide the number thus formed as before, and so -proceed till all the figures of tJie dividend have been employed. 75. "When is Short Division usually employed? Rule? How are di- visor and dividend written ? Which figures of the dividend are used first ? How many? Quotient, where set ? Remainder, to what is il prefixed? How? What is done with the number so formed ? How far is the process carried ? DIVISION. 57 Ex. 2. 3. Divisor, 6 ) S 35 1.54 Dividend. 5 ) 875 Quotient, $58.59 U2 5. 6. 7. 8. 9. Divide 7218. 8127. 6345. 3528. 2576. By 8. 3. 5. 9. 7. TG. When there is no remainder ^ as in the first nine exam- ples, the division is complete. The dividend is then said to be divisible by the divisor, and the divisor is called an exact divisor. When there is a remainder, as in Ex. 10, the division is incom- plete, and the dividend is said to be indivisible by the divisor. 10. Divide 325 by 7. Ans. 46f . OPERATION. Divisor, 7)325 Dividend. Quotient, 4 6 . . 3 Remainder. Note 1. The remainder is often written over the divisor in the fractional form (Art. 73), and the fraction is annexed to the quo- tient; thus, the answer in Ex. 10 is written 4G|^ which, expressed in words, is forty-six and three-sevenths. Other fractions are read in a similar manner; thus, ^ means one-half; J one-third; | twO' thirds; ^Jive-ninths ; etc. Note 2. The remainder, when written in a fractional form as a part of the quotient, becomes like the quotient. 11. Divide 6276 by 5. Ans. 1255, and 1 remainder. 12. Divide 8765 by 5. 18. Divide 7358 by 7. 13. Divide 4823 by 8. 19. Divide 8454 by 9. 14. Divide 6358 by 6. 20. Divide 8684 by 4. 15. Divide 7296 by 2. 21. Divide $6.84 by 4. 16. Divide 2594 by 3. 22. Divide $985 by 5. 17. Divide 7828 by 4. 23. Divide $9.85 by 5. 76. "When is the division complete ? When is one number divisible by an- other ? What is aa exact divisor ? When is one number indivisible by another ? How is the remainder often written ? The fraction where pla -cd ? 58 DIVISION. 24. How many pounds of sugar, at 9 cents per pound, can be bought for $35.64 ? Ans. 396. 25. How many barrels of flour, at $ 8 a barrel, can be bought for $5368? 26. If 6 shillings make a dollar, how many dollars are there in 7416 shillings? Ans. 1236. 27. If 4 weeks make a month, how many months are there in 3716 weeks? 28. How many oranges, at 6 cents apiece, can you buy for $35.64? Ans. 594. 29. If 7328 marbles are divided equally among 8 boys, how many marbles will each boy receive ? Ans. 916. 30. If an estate, worth $16,492 dollars, is divided equally among 7 children, how many dollars does each child receive ? 31. Divide two thousand one hundred and forty-two by six. 32. A mile is equal to 5280 feet ; how many steps of 3 feet each will a man take in walking a mile ? 77. To perform Long Division : 33. Divide 4654 by 13. Ans. 358. OPERATION. Having set the divisor and dividend 13')4654C3 58 ^^^° Short Division, draw a curve at the 3 9 right of the dividend, and then say, 13 in 46, 3 times, and set the 3 at the right 7 5 of the dividend. Then multiply the di- ".*^ visor by the quotient, 3, and set the pro- 7TT duct, 39, under the 46 of the dividend, J Q ^ and subtract the 39 from the 46. To the remainder, 7, annex 5, the next figure of the dividend, so forming a new partial dividend, 75, and then say, 13 in 75, 5 times, and set the 5 as the next figure of the quotient. Multiply the divisor by 77. Explain £x. 33. Of what order is any quotient figure ? Illustrate. DIVISION. 59 tliis new quotient-figure, and subtract the product from the partial dividend. Proceed ia this manner until the whole dividend has been divided; the entire quotient is 358. Every quotient-jig are is of the same orcler as the right-hand figure of the dividend used in obtaining that quotient-figure ; thus in Ex. 33, the 46 of the dividend is hundreds, and the 3 of the quotient is also hundreds ; the 75 is tens and the 5 of quotient is also tens ; etc. 78. This process, called Long Division, usually employed when the divisor is large, may be performed by the following EuLE 1. Write the divisor and dividend as in Short Division. 2. Divide the smallest number of figures in the left of the dividend that will contain the divisor, and set the result as the first figure of tJie quotient at the right of the dividend. 3. Multiply the divisor by the quotient figure, and set the product under that part of the dividend taken. 4. Subtract the product from the figures over it, and to tlie remxdnder annex the next figure of the dividend for a new partial dividend. 5. Divide, and proceed as before, until the whole dividend has been divided. Note 1. It will be seen that the process of dividing consists of four distinct steps, viz. : first, to seek' a quotient figure ; second, multiply ; third, subtract ; and, fourth, form a new partial dividend by annexing the next figure of the dividend to the remainder. Note 2. If any partial dividend will not contain the divisor, must be placed in the quotient, and another figure brought down and annexed to the dividend. 78. When is Long Division employed? Give the rule for Long Division. How many steps in dividing ? What are they ? Repeat Note 2. 60 DIVISION. Note 3. If the product of the divisor multiplied by the quotient figure is greater than the partial dividend, the quotient figure is too large, and must be diminished. Note 4. If the remainder equals or exceeds the divisor, the quotient is too small, and must be increased. 79. In the same manner solve the following examples, dividing each upper number by the one under it in each ex- ample ; also, in the same manner, as suggested by the signs, o, /Divide 18564-^156. Ans. 119. "^•(By 1092 -^ 12. o. (Divide 24453-^143. Ans. 171. "^^•■[By. 1287 -^ 11. Oft j Divide 20995 -^ 221. Ans. 95. '^^•(By 1105^ 13. 07 j Divide 143405 -^ 989. Ans. 145. "^^•(By 4945 -^ 23. 80. Division is the reverse of multiplication. In mul- tiplication, the two factors are given, and the product is required ; in division the product and one factor are given, and the other factor is required. The dividend is the product, and the divisor and quotient are the factors ; thus, IX MULTIPLICATION. IN DIVISION. Factors, Product. Dividend, Divisor, Quotient. 5 X 4 = 20 20 -^ 5 z= 4 Or, 20 -f- 4 = 5 Hence the following Proof. Multiply the divisor hy the quotient, and to the prod- uct add the remainder ; the sum should be the dividend. 78. Repeat Note 3. Note 4. 80. What is said of Division and Multipli- cation ? In Multiplication what is given ? What required 'i In Division what i s given ? Required ? How is Division proved ? DIVISION. 61 38. Divide 2537 by 53. OPERATION. 53)2537(4 7 2 1 2 PROOF. 5 3 Divisor. 4 7 Quotient 4 1 7 3 7 1 3 7 1 2 1 2 4 6 Remainder. T?^Tn'lTT>'^'i^ A A X.W w 1 1 lkK JH-T IVAUX, i yj 2 5 3 7 Dividend. 39. 40. 43] »87349(2031 4 7) 9 43 4 54 ( ;20073 8 6 94 1 34 3 4 5 1 29 32 9 6 9 1 6 4 43 1 4 1 Remainder, 1 6 Eemainder, 2 3 Quotients. Rem, 41. Divide 6384 by 79. 80. 64. 42. Divide 7639 by 83. 92, 3. 43. Divide 18805 by 37. 44. Divide 116092 by 29. 4003, 5. 45. Divide 47065 by 231. 46. Divide 29768 by 123. 242, 2. 47. Divide 17693 by 149. 48. • Divide 98074 by 483. 203, 25. 49. Divide 69847 by 348. 50. A farm containing 327 acres, was bought for $ 37605 ; what was the price per acre ? Ans. $115. 62 DIVISION. 5 1. Divide six thousand eight hundred and forty- four acres of land into twenty-nine equal parts. Ans. 236 acres. 52. A drover paid $2331 for 37 oxen; what was the average price per ox ? Ans. $63. 53. The product of two numbers is 35068765, and one of the numbers is 8765; what is the other number? Ans. 4001. 54. In how many days will a steamboat sail 11352 miles, if she sails 264 miles per day ? 55. If a railroad 359 miles long cost $ 3545484, what was the average cost per mile ? Ans. $ 9876. Contractions. 81. To divide by a composite number. 66. Divide $ 1855 equally among 35 men. OPERATION. The 35 men may '35 = 7 X 5. ^® separated into 7 1st Factor, 7 ) $ 1 8 5 5 Dividend. groups of 5 men each. Then dividing by 7 2d Factor, 5 ) $ 2 6 5 1st Quotient, gives $265 for each TTT rx, ^ . group, and dividing $ 5 3 True Quotient. ^ ^ ^ the $ 265 by 5 gives $ 53 for each man. Note. When a composite number is made up of different sets of factors, as in Ex. 67, it is immaterial which set is taken. It is also immaterial in what order the factors are taken. 81. Rule for dividing by a composite number ? Is it material which factor of the divisor is used first, or which set of factors is employed ? divisio:n. 63 57. Divide 10656 by 288. 288 = 4X6X12 = 6X6X8 = 8X3X12, etc. FIRST OPERATION. SECOND OPERATION. 4)10656 6)10656 6)2664 6)1776 12)444 8)296 3 7 Ans = 3 7 Ans. From these examples we have the following KuLE. jyivide the dividend hy one factor of the divisor, and the quotient so obtained by another factor^ and so on till all the factors of the set have been used. The last quotient will be the true quotient. 58. Divide 20088 by 24 ; = 6 X 4. Ans. 837. 59. Divide 8445 by 15. 60. Divide 23296 by 32. Ans. T28. 61. Divide 26568 by 12. 62. Divide 22720 by 64. Ans. 355. 63. Divide 33696 by 144; = 12 X 12. 8S. In dividing by the factors of the divisor, there may be a remainder, after either or each of the divisions. Should the learner find a difficulty in determining the true remainder, he has but to remember that it is always of the same kind as the dividend. (Art. 69, Note). 64. Divide 86 by 21 ;= 7 X 3. OPERATION. 7)86 In this example, as 86 is the true dividend, 2 is the true re- 3)^..2Eem. ^^^^^^ Quotient, 4 sa. Rule for finding the true remainder when the factors of the divisor are used separately ? The reason ? 64 DIVISION. 65. Divide 92 by 28 ; nz 4 X 7. OPERATION. In this example, as 23 is only 4)9 2 one-fourth of the true dividend, so the remainder, 2, is only one 7)23 fourth of the true remainder ; therefore the true remainder is Quotient, 3 .. 2 Eem. 2x4 = 8. C6. Divide 527 by 42 ; z= 6 X 7. OPERATION. From the previous explanations 6)527 we see that 5 our first remainder here is one part of the true re- 7)87 . . 5 Kern, mainder, and that 3, the second ^ . ~~" „ ^ remainder, multiplied by 6, the Quotient, 1 2 . . 3 Eem. ^ , ,. . . Ji .i . nist divisor, is the other part ; that is, 5 -|- 3 X 6 = 23 ; is the true remainder. Similar rea- soning applies when there are more than two divisors. Hence, To obtain the true remainder when division is per- formed by using the factors of the divisor : EuLB. 3fultiply each remainder ^ except that left hy tJie first dwision, hy the continued product of the divisors preceding that which gave the remainders severally, and the sum of the prod- ucts, together with the remainder left hy the first division, will he the true remainder. KoTE. When there are but two divisors and two remainders, it only requires the addition of the first remainder to tlie product of the first divisor and second remainder. 67. Divide 1834 by 35 ;= 5 X 7. Ans. Quo. 52, Eem. 14. 08. Divide 15008 by 315 ; =5X7X9. Ans. Quo. 47, Eem. 203. 69. Divide 7704 by 105 ; =3X5X7. 70. Divide 45621 by 405 ; =5X9X9. Ans. Quo. 112, Eem. 261. 71. Divide 55242 by 25. DIVISION. 65 83. To divide by 10, 100, 1000,^etc. Rule. Gat off by a point, as many figures from the right hand of tJie dividend as there are ciphers in the divisor. The .figures at the left of the point are tJie quotient, and those at the right are the remainder. 72. Divide 756 by 10. Ans. 75.6, == 75 Quo. and 6 Eem. Note. The reason of the rule is obvious. By taking away the right-hand figure, each of the other figures is brought one place nearer to units, and its value is only one-tenth as great as before (Art. 18), and therefore the whole is divided by 10. For like rea- sons, cutting of[ two figures divides by 100; cutting oflf three figures divides by 1000, etc. 73. Divide 4867 by 100. Ans. Quo. 48, Rem. 67. 74. Divide 37692 by 1000. 75. Divide 5367842 by 1000000. •76. Divide 20687432004 by 1000000000. 84. To divide by 20, 50, 700, or aoy like number : Rule. Gut off as many figures from the right of the divi- dend as there are ciphers at the right of the significant figures of the divisor, and then divide the remaining figures of the dividend by the significant figures of the divisor. Note 1. This is on the principle of dividing by the factors of the divisor ; therefore the true remainder will be found by the rule in Art. 82. 77. Divide 74689 by 8000. Ans. 9 Quo. and 2689 Rem. OPERATiox. We divide by 1000 by cut- 8) 7 4.6 8 9 ting off 689, which gives 74 for a quotient and 689 for a Quotient, 9 ... 2 Rem. remainder ; then divide 74 by 8, and obtain the quotient, 9, and remainder, 2. This remain- der, 2, is 2000, which, increased by 689, gives 2689 for the true remainder (Art. 82). 83. Rule for dividing by 10 ? By 100 ? Reason of rule ? 84. Rule for divid- ing by 20? By 500? Reason? How is the true remainder found ? 66 DIVISION. Note 2. It will be observed that the true remainder, in all ex- amples like the 77th, is obtained by annexing the 1st to the 2d re- mainder. 78. Divide 3764 by 200. Ans. Quo., 18, Rem., 164. 79. Divide 4547 by 400. 80. Divide 3876423 by YOOO. Ans. Quo. 553, Kern. 5423. 81. Divide 7943862 by 210000. General Principles of Division. 85. The value of a quotient depends upon the rela- tive values of the divisor and dividend, and not upon their absolute values. This will be seen by the follovv^- ing propositions. (1st). If the divisor remains unaltered^ multiplying the dividend hy any number is, in effect, multiplying the quotient by the same number ; thus, 15 -^ 3 irr 5 4 4 60-1-3 = 20 that is, multiplying the dividend by 4 multiplies the quotient by 4. (2d), Dividing the dividend by any number is dividing the quotient by the same number ; thus, 24-^-2=12 3)24 ^-^2= 4=nl2-^-3; that is, dividing the dividend by 3 divides the quotient by 3. 85. Does the size of the quotient depend upon the absolute size of the divisor and dividend ? Upon what does it depend ? VThat is the first propo- sition? Second? Third? Fourth? } DIVISION. 67 (Sd). Multiplying the divisor hy any numher divides the quotient hy the same numher ; thus, 3 -^ 2 = 15 3 30-^6== 5 = 15-^3; that is, multiplying the divisor by 3 divides the quotient by 3. (4 th). Dividing the divisor hy any numher multiplies the quotient hy the same numh3r ; thus, 40 -I- 10=r4 5) 10 40 -^ 2 = 20 = 4 X 5; that is, dividing the divisor by 5 multiplies the quotient by 5. (5th). It follows, from (1st) and (2d), that the greater the dividend the greater is the quotient; and the less the dividend the less the quotient. (6th). Also, from, (3d) and (4th), that the greater tJie divisor, the less is the quotient ; and the less the divisor the greater the quotient. 86. From the illustrations in Art. 85 we see that any change in the dividend causes a similar change In the quotient, and that any change in the divisor causes an OPPOSITE change in the quotient. Hence, (1st), Multiplying hoth dividend and divisor hy the same numher does not affect the quotient ; thus, 12-^-3 = 2 2 2 4-^6 = 4, Quotient unchanged. 83. What follows from (Ist) and (2d) ? What follows from (3d), (4th)? From (5th), (6th) ? 86. Any change in the dividend, how affects the quotient ? Any change in the divisor, how? First inference ? Second? Third? Illustrate. 68 DIVISION. (2d). Dividing both dividend and divisor hy the same number does not affect the quotient; thus, 20 -^ 10:^2 5)20 6)10 4 ~ 2 z= 2, Quotient unchanged. (3d). It follows, from (1st) and (2dJ, that the operations of multiplying and dividing hy the same number cancel^ that is destroy, each other ; for example, If a number be multiplied by any number, and the product be divided by the multiplier, the (juotient will be the multipli- cand; thus, 8 X 7 = 56 and 56 -^ 7 == 8, the multiplicand. Also, if a number be divided by any number, and the quo- tient be multiplied by the divisor, the product will be the dividend; thus, 15 -h 3 — 5, and 5 X 3 = 15, the dividend. 87, These general principles may be naore briefly stated as follows ; (1st). Multiplying the dividend multiplies the quotient; and dividing the dividend divides the quotient (Art. 85, 1*^ and 2nd). (2d) . Multiplying the divisor divides the quotient ; and dividing the divisor multiplies the quotient (Art. 85, 3c? and ^th) . (3d). Multiplying both dividend and divisor by the same number ; or dividing both by the same number, does not affect the quotient (Art. 86, 1st and 2d). 8T, A more brief statement of these principles : First ? Second ? Third ? DIVISION, 69 CANCELLATION. 88. How many are 6 times 7 divided by 6 ? OPERATION. Indicating the multiplication and 0X7 division (Art. 73), we may cancel or = 7, Ans. strike out the equal factors, 6 and 6, from the divisor and dividend, and so shorten the process without changing the result. How many are 7 times 1 2 divided by 6 ? OPERATION. Separating the 12 into 7 X 12 7X2X0 the two factors, 2 and 6, -^-^^ or, := 14 Ans. cancel the G from divisor 6 and dividend, and there is left 7 times 2, equal to 14, for the quotient. '2 This process is, in ef- 7 X t ^ feet, the same as the other. = 14, Ans. Instead of resolving 12 into the factors, 2 and G, we merely divide 12 by 6, setting the quotient, 2, over the 12 ; then, cancelling the 6 and 12, the result is 7 times 2, equal to 14, as before. How many are 8 times 15 divided by 12 ? First, reject or cancel the factor 4 OPERATION. fj.^^ I^J^^^ 8 and 12, giving 2, which 2 5 is placed over 8 and 3, placed under $ X 3^5 12 ; thien cancel 3 in 3, and in 15, — — =10, Ans. giving 5 to place over 1 5 , and we have rt 2 times 5, equal to 10, for the result. These examples arc solved, in part, by cancelling. Hence, 89. Cancellatiox is a process for shortening the solution 89. What is Cancellation? On what principle does it depend? 70 DIVISION. of an example, by rejecting, or cancelling the same factors from both divisor and dividend. It depends on the principle (Art. 86, 2c?), that dividing both dividend and divisor hy the same number does not affect the quotieiit. Ex. 1. Divide 8 X 3 X 10 X 63 by 4 X 5 X 7. OPERATION. ' We cancel 4 in 8, giv- 2 2 9 iiig 2 ; 5 in 10 giving 2, § X 3 X 3:0 X 03 and 7 in 63, giving 9. .^ . ^ry == 10^' ^°3- Then, 2X3X2X9 ^^^^f^ ^ 108, the Ans. 2. Divide 6 X 21 X 15 X 11 by 18 X 7 X 5. Ans. 33. 3. Divide 9 X 14 X 26 X 8 by 3 X 21 X 13 X 4. 4. How many cords of wood, at $6 a cord, will pay for 5 tons of hay, at §12 a ton? 6. How many tons of hay, at $15 a ton, will pay for 4 acres of land, at $ 45 an acre?" 6. How many pounds of butter, at 33 cents a pound, must be given for 3 boxes of raisins, each containing 22 pounds, at 15 cents a pound? Ans. 30. 7. How many pieces of cloth, containing 32 yards each, at $3 per yard, will pay for 48 barrels of flour, at $12 per barrel? • Ans. 6. 8. How much sugar, at 15 cents a pound, will be required to pay for 3 boxes of lemons, containing 305 lemons each, at 4 cents apiece. Ans. 244. 9. How many boxes of tea, each containing 45 pounds, at 66 cents a pound, must be given for 15 boxes of sugar, each containing 220 pounds, at 18 cents a pound? 10. How many bags of corn, each containing 2 bushels, at 96 cents a bushel, will pay for 128 bags of oats, each con- taining 3 bushels, at 47 cents a bushel? Ans. 94. DIVISION. 71 Eeyiew and Test Questions. 90. Let the pupil answer the following questions, illustrating them with his own original examples : 1. How will you find the sum of several given numbers? 2. How will you find the difierence between two given numbers ? 3. How will you find the subtrahend when the minuend and remainder are given ? 4. How will you find the minuend when the subtrahend and remainder are given ? 6. How will you find the remainder, when the minuend and subtrahend are given ? 6. When the sum of several numbers and all of the numbers except one are given, how do you find that one ? 7. When the multiplicand and multiplier are given, how can you fiiid the product ? 8. When the multiplicand and product are given, how can you fiud the multiplier ? 9. When the multiplier and product are given, how can you find the multiplicand ? 10. How do you find the quotient, when the dividend and divisor are given ? 11. How do you find the divisor, when the dividend and quo- tient are given ? 12. How do you find the dividend, when the divisor and quotient are given ? 13. How do you find the dividend, when the quotient, divi- sor, and remainder are given ? 14. How do you find the divisor, when the dividend, quotient and remainder are given ? 72 DIVISION* Examples in the Foregoing Principles. 1. A boy sold a sled for $2.00 and in payment received 50 cents in money, 5 pineapples at 20 cents each, and the re- mainder in oranges at 5 cents each ; how many oranges did he receive? Ans. 10. 2. If 2 barrels of flour will pay for 5 yards of broadcloth, how many barrels of flour will pay for 3 times 5 yards of broad- cloth? 3. How many barrels of apples, at $ 3 a barrel, must be given for 6 yards of flannel, at $ 2 a yard ? 4. A speculator bought 80 acres of land at $ Vo per acre, and sold the whole for S 6720 ; how much did he gain by the transactions ? How much per acre ? First Ans. $ 720. 5. Bought 160 acres of land for $4000, and sold it at $20 per acre; did I gain or lose? How much? How much per acre ? 6. If 2 men build 16 rods of wall in 4 days, in how many days will 5 men build 50 rods ? Ans. 5. t. How many miles per hour must a steamboat sail to cross the Atlantic, 2880 miles, in 10 days of 24 hours each? Ans. 12. 8. The product of two factors is 595 ; one of the factors is 17 ; what is the other ? 9. The product of three factors is 9177; two of the factors arc 21 and 19 ; what is the other? Ans. 23. 10. The divisor is 18, and the quotient 13 ; what is the divi- dend? ' 11. The divisor is 23, the quotient 37, and the remainder 19 ; what is the dividend ? Ans. 870. DIVISION. 73 12. The first of three numbers is 8, the second is 4 times the first, and the third is 3 times the sum of the other two ; what is the difi"erence between the first and third ? Ans. 112. 13. In how many days of 24 hours each, will a ship cross the Atlantic', 2880 miles, if she sails 12 miles per hour ? 14. If I receive $ 80 and spend S 55 por month, in how many years of 12 months each shall I save $ 1800 ? 15. Bought 87 yards of cloth, at $ 4 per yard, and paid S 200 in money and the rest in wheat at S 2 per bushel ; how many bushels of wheat did it take ? Ans. 74. 16. The subtrahend is 3762, and the remainder is 2657 ; what is the minuend ? 17. The minuend is 8063, and the remainder is 5604 ; what is the subtrahend? Ans. 2459. 18. The greater of two numbers is 8327, and the diflfcrence is 5364 ; what is the less number ? Ans. 2963. 19. The sum of two numbers is 5836, and the less number is 2467 ; what is the difference between the two numbers ? 20. A man having engaged to work 12 months for S 432, left his employer at the end of 9 months ; at the rate agreed upon, what should he receive ? Ans. $ 324. 21 A merchant received $376 on Monday, $567 on Tues- day, $487 on Wednesday, $684 on Thursday, $293 on Friday, and $ 857 on Saturday ; what were his average receipts per day? 22. If 732 is multiplied by 27 and the product divided by 36, what is the quotient ? Ans. 549. 23. Bought 175 barrels of flour for $ 1750, and sold 86 bar- rels of it at $ 12 a barrel, and the remainder at $ 9 a barrel ; did I gain or lose ? How much ? 24. How many are 376 + 874 + 563 — 937? 25. How many are 384 + 562 -j- 1728 -^ 191 ? 74 REDUCTION . DENOMINATE NUMBEKS AND EEDUCTION. 91, All numbers are either concrete or abstract. A Concrete Number is one that is applied to a particular object ; as 6 books, 4 men, 7 days, 3 rods. A concrete num- ber is often called a Denominate Number, because it denomi- nates or names the thing to which it is applied. An Abstract Number is one that is not applied to any par- ticular object; as 6, 9, 23. 93, All numbers are either simple or compound. A Simple Number consists of but one kind, and may be abstract or concrete j as 2, $4, 10 miles, 3 pounds. A Compound Number consists of two or more denominations, and is necessarily concrete ; as 4 days and 7 hours ; 3 pecks, 5 quarts, and 1 pint ; 8 rods, 4 yards, 2 feet, and 10 inches. Note 1. The several parts of a compound number, though of dif' ferent denominations, are yet of the same general nature ; thus, 2 weeks, 3 days, and 6 hours are similar quantities, and constitute a compound number; but 2 weeks, 3 miles, and 6 quarts are unlike IN THEIR NATURE, and do NOT Constitute a compound number. Note 2. The first division of each of the following tables should be thorovghly committed to memory. The second division is designed for reference. 91. What is a Concrete Number ? What is it often called ? Why? What is an Abstract Number ? 95i. What is a Simple Number ? May it be abstract ? Concrete ? Illustrate. What is a Compound Number ? May it be abstract ? Illustrate. What is said of the different denominations of a compound number ? Is Viis a compound number : 3 rods, 2 pecks, and 6 pounds ? Why ? What is said of the first divisioa of the following tables ? What of the second ? 1 Shilling, " 1 Pound, " s. £ qr. = 4 = 48 — 960 EEDUCTION. 75 ENGLISH MONEY. 03« English Money is the Currency of Great Britain. TABLE. 4 Farthings (far. or qr.) make 1 Penny, marked d. 12 Pence '* 20 Shillings •* d. s. 1 £ 1 = 12 1 z= 20 = 240 Note. The numbers employed as multipliers and divisors in reducing a Compound Number are called a Scale ; thus, in reducing English Money, the Descending Scale is 20, 12, and 4 ; and the Ascending Scale is 4, 12, and 20. Ex. 1. In 7£ Is. 6d. 8qr. how many farthings ? Multiply the 7 by 20 to change the pounds to shillings ; to the product, 140, add the Is. givenjn the example, and the result is 141s. ; then multiply the 141 by 12 to change the shillings to pence ; to the product, 1692, add the 6d. in the example, and the result is 1 6 9 8d. ; so proceed till the example is solved. OPERATION. £ 8. d. qr. 7 1 6 3 20 14 1s. 1 2 1 6 9 8 d. 4 6 7 9 5 qr., Ans. 93. What is English Money ? Repeat the Table. What are the multi- pliers and divisors used in reducing a compound number called ? What is the descending scale in English Money ? What the ascending scale ? Explain Example 1, Explain Ex. 2. 76 REDUCTION. Ex. 2. Change 6795 qr. to pence, shillings, and pounds. OPERATION. First divide by 4 to change the 4 ) 6 7 9 5 qr. farthings to pence, giving 1698d. and 3qrs. ; then divide the 1698 12)1698 d. -|- Sqr. by 12 to change the pence to shillings, giving 141s. and 6d. ; 2 ) 1 4 1 s. + 6d. then divide the 141 by 20 to change the shillings to pounds, 7 £ 4" Is* giving 7£ Is., and thus obtain 7£ Is. 6d. 3qr., Ans. These examples are questions in Reduction. Hence, 04:. EEDUCiiiON consists in changing a number of one de- nomination to a number of another denomination, without changing its value. The process in Ex. 1 is called Reduction Descending^ because higher denominations are changed to lower. Hence, 95, Eeduction Descending consists in changing a num- ber from a higher to a lower denomination, and may be per- fopmed by the following EuLE. Multiply the highest denomination given, by the num- ber it takes of the next lower denomination to make one of this higher f and to the product add the number of the lower denom- ination ; multiply this sum by the number it takes of the next lower denomination to make one of this ; add as before, and so proceed till the number is brought to the denomination re- q 'lired. 06, The process in Ex. 2 is called Reduction Ascending, because lower denominations are changed to higher. Hence, 94. What is Reduction ? What is the process in Ex. 1 called ? Why ? 95. What is lleduction Descending ? Rule for performing it ? 96, What is the process in Ex. 2 called ? Why ? REDUCTION. 77 Eeduction Ascending consists in changing a number from a lower to a higher denomination, and may be performed by tte following EuLB. Divide the given number by the number it takes of that denomination to make one of the next higher ; divide the quotient by the number it takes of that denomination to niake one of the next higher , and so proceed till the number is brought to the denomination required. The last quotient, to- gether with the several remainders (Art. 69, Kote), will be the answer. 97. The processes in Eeduction Ascending and Eeduction Descending prove each other, as will be seen in Examples 3 and 4. In the same manner let the pupil prove all the examples in Eeduction, and the answers, for this purpose, will be omit- ted in the book. Ex. 3. How many farthings Ex. 4. Eeduce 15542 qr. to in IQ£ 3s. 9d. 2qr. ? pence, shillings, etc. OPERATION 16£ 3s. 20 323 s. 12 9d. 2qr. OPERATION. 4 ) 1 5 5 4 2 qr. 1 2 ) 3 8 8 5 d. + 2qr. 2 ) 3 2 3 s. + 9d. 3 8 8 5 d. 4 1 6 £ + 3s. Ans. 16£ 3s. 9d. 2qr. 15 5 4 2 qr., Ans. Note 1. In solving Ex. 3, the several numbers of the lower de- nominations are added mentally j and only the results are written ; .tlius, 20 times 16 are 320, and the 3s. added give 323s. Then mul- 96. What is Reduction Ascending f Rule for performing it ? 97. How are processes in Reduction proved ? In solving Ex. 3, how are the numbers of the lower denominations added ? 78 REDUCTION. tiplying the 323 by 12, and adding the 9d., we have 3886d. Finally, multiplying the 3885 by 4, and adding the 2qr., we have 15542qr. which is the Ans. Note 2. In solving Ex. 4, and other examples in Reduction As- cending, if any divisor is so large that the work is not easily done by Short Division, the numbers may be taken upon the slate and the work done by Long Division. 5. Reduce 27£ 16s. lid. Iqr. to farthings. 6. Ecduce 17375 qrs. to pence, shillings, and pounds. 7. Eeduce 54£ 9s. 3qr. to farthing.^. 8. Ecduce 25£ 3d, to farthings. 9. Eeduce 12497qr. to higher denominations. 10. Ecduce 23445 qr. to higher denominations. 11. A bookseller received from London fifty Oxford Bibles. The lot cost him 6£ 5s., how much was that apiece? TROY WEIGHT. 08. Troy Weight is used in weighing gold, silver, and precious stones. TABLE. 24 Grains (gr.) make 20 Pennyweights ** 12 Ounces " 1 Pennywei ght, dwt. 1 Ounce, oz. 1 Pound, lb. dwt gr- 1 = 24 20 = 480 240 =: 6760 oz. lb. 1 =: 1 = 12 = Scale. — Descending, 12, 20, 24; ascending, 24, 20, 12. 97. In Ex. 4, how is the work done? 98. For what is Troy "Weight used ? Repeat the table. What is the descending scale ? Ascending ? Ex- plain Ex. 1. REDUCTIOX 79 Ex. 1. How many grains Ex. 2. Reduce 4 5 9 5 4 gr. in 71b. lloz. 14dwt. 18gr. ? to pounds, ounces, etc. OPERATIOX. OPERATION. 7 lb. lloz. Udwt. 18gr. 2 4 ) 4 5 9 5 4 gr. 1 2 2 0)19 14dwt. + 18gr. 12) 9 5oz. + 14dwt. 71b. + lloz. Ans. 71b. lloz. 14dwt. 18gr. 4 5 9 5 4 gr., Ans, Note 1. In reducmg the pennyweights to grains in Ex. 1, we first multiply the 1914 by 4 and add the 18gr., giving 7674 ; then multiply the 1914 by the 2 tens, giving 3828 tens ; and finally add the results, giving 45954gr., Ans. 3. Reduce 61b. 4oz. ISdwt. 28gr. to grains. 4. Reduce 181b. lloz. 6dwt. 19gr. to grains. 5. Reduce 53649gr. to pennyweights, ounces, etc. 6. Reduce 63594gr. to higher denominations. 7. Reduce 151b. 6dwt. to grains. 8. How many spoons, each weighing 2oz. 3dwt. 18gr., can 9 5 oz. 20 1 9 1 4 dwt. 24 7 6 74 3 82 8 be made from lib. loz. 2dwt. 12gr. of silver? Ans. 6. 9. A jeweller made 6oz. 7dwt. 12gr. of gold into rings, which weighed 3dwt. 13gr. each; how many rings did he make? Note 2. In performing Exs. 8 and 9, and similar examples, both of the given quantities must first be reduced to the lowest denomin- ation mentioned in either. 80 REDUCTION. APOTHECAKIES' WEIGHT. 99. Apothecaries' Weight is used in mixing or com- pounding medicines ; but medicines are bought and sold by Avoirdupois Weight, TABLE. 20 Grains (gr.) make 1 Scruple , so. or B 3 Scruples <{ 1 Dram, dr. or 5 8 Drams tt 1 Ounce, oz. or § 12 Ounces tt 1 Pound, so. lb. or lb gr. dr. 1 z=z 20 oz. 1 = 3 =■ 60 lb. 1 — 8 — 24 = 480 I — \2 = 96 = 288 — 5760 Scale. Descending, 12, 8, 3, 20; Ascending, 20, 3, 8, 12. Note. The pound, ouDce, and grain, in Apothecaries' and Troy Weight are equal, but the ounce is differently subdivided. 1. Beduce 21b3i53 19 18gr. to grains. 2. Eeduce 13298gr. to scruples, drams, etc. 3. In 51b. 6oz. 5 dr. 2sc. 14gr. bow many grains ? 4. In 3 lb 5 § 33 9 24gr. how many grains ? 5. In 2543 7gr. bow many sciuples, drams, etc. ? 6. Eeduce 3764gr. to higher denominations. 7. What quantity of medicine will an apothecary use in preparing 365 prescriptions of 12 grains each? Ans. 9oz. Idr. 99. For what is Apothecaries' Weight used ? Eepeat the table. Descend- ing scale ? Ascending ? What denominations of Apothecaries' Weight are like those of Troy Weight ? What of tlie ounce ? REDUCTION. 81 AVOIRDUPOIS WEIGHT. 100. Avoirdupois Weight is used in weighing the coarser articles of merchandise, such as hay, cotton, tea, sugar, copi)er, iron, etc. TABLE. 16 Drams (dr.) make 1 Ounce, oz. 16 Ounces ti 1 Pound, lb. 25 Pounds <( 1 Quarter, qr. 4 Quarters ti 1 Hundredweight, cwt. 20 Hundredweight" 1 Ton, t. oz. dr. lb. 1 = 16 qr. 1 = 16 = 256 cwt. 1 25 =z 400 = 6400 t 1=4 zm 100 = 1600 = 25600 1 = 20 = 80 = 2000 = 32000 =: 512000 Scale. Descending, 20, 4, 25, 16, 16 ; Ascending, 16, 16, 25, 4, 20. Note 1. It was the custom formerly to consider 281b. a quarter, 1121b. a hundred weight, and 22401b. a ton ; but now the usual prac- tice is in accordance with the table. These different tons are distinguished as the long or gross ton = 22401b. and the short or net ton = 20001b. The gross ton is still used in the wholesale coal trade ; also in esti- mating goods at the U. S. custom-houses, etc. Note 2. A pound in Avoirdupois Weight is equal to 7000 grains in Apothecaries, and Troy Weight. 100. For what is Avoirdupois Weight used? Table? Scale? How many pounds now make a ton ? How many formerly ? What are the different tons called ? For what is the long ton now used ? One pound Avoirdupois equals how many grains Troy ? 82 REDUCTION, Ex. 1. Eeduce 2t. 6cwt. Iqr. 231b. to pounds. OPERATION. 2t. 6cwt. Iqr. 231b. 20 4 6 cwt. 4 1 8 5 qr. 25 925 370 2 31b. Ex. 2. In 46481b. how many tons, etc. ? OPERATION. 2 5 ) 4 6 4 8 lb. 4 ) 1 8 qr. + 231b. 2 0) 4 6 cwt. + Iqr 2t. -f 6cwt. Ans. 2t. 6cwt. Iqr. 231b. 4 6 4 8 lb., Ans. Note 3. Instead of mentally adding the numbers of the lower denominations, as in preceding examples and as is done with the 6cwt. and Iqr. in Ex. 1, the pupil may, if he chooses, write the numbers under the partial products, and then add, as is done with the 231b. in this Example. 3. Eeduce 6t. 7cwt. 3qr. 211b. looz. 7dr. to drams. 4. Eeduce 4t. 2qr. 15oz. to drams. 5. Eeduce 147683dr. to higher denominations. 6. Eeduce 1860861 dr. to ounces, pounds, quarters, etc. 7. If a cow eats 161b. of hay in 1 day, how many tons will she eat in 365 days? 8. In 7t. 16cwt. 3qr. 51b. net weight, how mz.iiy gross toils'^ CLOTH MEASUEE. 101. Cloth Measure is used in measuring cloths, rib- bons, braids, etc. 101. Explain Ex. 1. Explaia Ex. 2. 101. For what is Cloth MeaRTire used t Tabic ? Scale » 21 Inclies (in.) 4 Nails 4 Quarters REDUCTION. TABLE. make TJirlVia^ 1 Nail. ^ 1 Quarter 1 Yard, O?'^ na. qr. yd. yd. qr. 1 = - 4 = na. 1 4 in. 9 16 =z 36 Scale. Descending, 4, 4, 21 ; Ascending, 2J, 4, 4. 1. In 6yd. 2qr. 3na. how many nails? 2. In 107 nails how many quarters, etc. ? 3. Reduce 18yd. Iqr. 2na. to nails. 4. Reduce 47yd. 3qr. Ina, to nails. 5. Reduce 783 nails to quarters, etc. 6. Reduce 549 nails to higher denominations. 7. If 2yd. Iqr. of ribbon are used in trimming 1 bonnet, how many yards will be used in trimming 5 bonnets ? 8. If 2yd. Iqr. 3na. of cloth are used in making 1 coat, how many yards will be used in making 16 coats? Aiis. 19. 9. How many dresses can be made from 117yd. 2qr. of silk, if each dress requires 14yd. 2qr. 3na. Ans. 8. 10. What cost 18yd. 3qr. of velvet, at $2 per quarter? LONG MEASURE. 103. Long Measure is used in measuring distances; as, for example, the length of a line, or the length, breadth, height, or depth of any object. lOa For what is Long Measure used ? Table ? Scale ? 04 REDUCTION. TABLE. 12 Inches (in.) make 1 Foot, ft 3 Feet 1 Yard, yd. 5iYardsor le^Feet '' 1 Rod, rd. 40 Kods 1 Furlong, fur. 8 Furlongs " 1 Mile, m. 69^ Statute miles, nearly «' 1 Degree OB iCirc. of the Earth, V 360 Degrees 1 Circumference, circ. ft. in. yd. 1 = 12 rd. 1 = 3 — 36 fur. 1 r= H = m = 198 m. 1 r= 40 = 220 =z 660 = 7920 1 =: 8 = 320 1760 = 5280 = 63360 Scale. Descending, 360, 69j, 8, 40, 5J, 3, 12 ing, 12, 3, 5i, 40, 8, 69^, 360. Ascend- Note 1. The earth not being an exact sphere, the distance round it in difterent directions is not exactly the same. By the most exact measurements made, a degree is a little less than 69^ miles. Note 2. Besides the numbers given in the table, there are vari- ous other measures of length ; as, 3 barleycorns make 1 inch, 4 inches 1 hand, 9 inches 1 span, 3 feet 1 space, 6 feet 1 fathom, 3 geo- graphic miles 1 league, 60 geographic miles 1 degree, etc. 1. How many rods in 5m. 6 fur. 37rd. ? 2. Reduce 1877 rods to higher denominations. 3. Reduce 3659 rods to higher denominations. 4. In 301 furlongs how many miles? 5. In 5yd. 1ft. 9 in. how many inches ? 6. In 197 inches how many feet, etc. ? 7. The distance through the earth is about 7912 miles; how many rods is it ? REDUCTION. 85 8. The distance round the earth is about 8000000 rods J how many miles is it ? Ans. 25000. 9. The distance from the earth to the moon is about 240000 miles; how many rods is it ? 10. The distance from the earth to the sun is about 30400000000 rods ; how many miles is it? CHAIN MEASURE. 103. Chai?^ Measure is used by engineers and surveyors in measuring roads, canals, boundaries of fields, etc. TABLE. 7-1^0-0 Inches (in) make 1 Link, li. 25 Links ii 1 Rod, Perch »or Pole, rd. 4 Rods It 1 Chain, ch. 10 Chains « 1 Furlong, fur. 8 Furlongs • (( 1 Mile, li. rd. 1 m, in. ch. 1 =: 25 z= 198 fur. 1 z= 4 = 100 r= 792 m. 1 = 10 40 = 1000 — 7920 1 : = S = 80 izr J 320 = 8000 = 63360 Scale. Descending, 8, 10 , 4, 25, 7^%^j; Ascending 7x9/^, 25,4. 10, 8. Note. To measure roads, etc., engineers often use a chain 100 feet long. 1. Reduce 3m. 4fur. 8ch. 2rd. 20li. to links. 2. Reduce 28870 links to higher denominations. 3. Reduce 5 m. 7fur. 3ch. to links. 103 A degree upon the earth, how long ? What other measures of length ? 103 For what is Chain Measure used ? Table ? Scale ? Note ? SQ REDUCTION. 4. Eeduce 4m. Sch. 221i. to links. 6. Eeduce 35G47 links to higher denominations. 6. The distance from Boston to Andover is about 1 84000 links ; how many miles is it ? 7. The distance round a field is 5 far. 7ch. 3rd. ; what will it cost to fence this field at $ 2 per rod ? SQUARE MEASURE. 104:* A Surface is that which has length and breadth but no thickness. 10^. A four-sided figure having all its comers or angles, equal to each other, as A B C D, Fig. 1, is called a Rectangle^ A Fig. 1. B . A Fm. 2. B 1 2 3 4 5 6 7 8 9 10 11 12 .13 14 15 B D 106. A Rectangle whose sides are all equal to each other, as A B C D, Fig. 2, is called a Square. The small checks in Fig. 1 are squares. 107. The manner of finding the area or measure of any rectangle, as, for example, Fig. 1, may be understood by the following explanation: — Let A B represent (on a reduced scale) a line five feet long ; then, evidently, if we pass from A to e, a distance of 1 foot, and draw the line e/ the figure A B/e will contain 5 square feet, that is 5 X 1 square feet. So, in 104: What is a Surface? 105. A Rectangle? 107. How is the area of a Rectangle found ? 106. A Square? REDUCTION. 87 like manner, A B A^ will contain 10, or 5 X 2 square feet, and A B C D will contain 15, or 5 X 3 square feet. Hence we multiply together the numhtrs representing the length and breadth of a rectangle to find its area. 108. Reversing the process in Art. 107, r The area of a rectangle divided by its length will give its breadth, and the area divided by the breadth will give the length ; thus, in Fig. 1, 15 -j- 5 zzz 3, the breadth, and 15 -^ 3 r= 5, the length, 100. Square Measure is used for measuring surfaces. TABLE. 144 Square laches (sq. in.) make 9 Square Feet " 1 Square Foot, sq. ft 1 Square Yard, sq. yd. 30^ Square Yards or | 272i Square Fe^t j << 1 Square Rod, eq. rd. 40 Square Eods << 1 Rood. r. 4 Roods ti 1 Acre, a. 640 Acres n 1 Square Mile, sq. m. (a) Also in Chain Measure, 10000 Square Links or") 16 Square Rods ) make 1 Square Chair I, sq. ch. 10 Square Chains <« 1 Acre, sq. ft. a. sq. in. sq. yd. 1 = 144 sq. rd. 1 = 9 = 1296 r. 1 = 30i = 272^ = 39204 a. 1 = 40 = 1210: = 10890 = 1568160 sq. m. 1 = 4 = 160 = 4840: = 43560 = 6272640 1 = 640 = 2560 = 102400 = 3097G00 : = 27878400 = 4014489600 108. How is the breadth of a rectangle found when the area and length are known ? How the length when the area and breadth are known ? 109. For what is Square Measure used ? Table ? Scale ? Table in Chaui Measure? Note 1? Note 2? 8S REDUCTION. Scale. Descending, 640, 4, 40, 30^, 9, 144 ; Ascending, 144, 9, 30i 40, 4,640. "" Note 1. In measuring land, surveyors use a 4-rod chain com- posed of 100 links. Sometimes the half-chain of 50 links is used. Note 2. The 272^ before feet in the table is not a part of the scale. 1. Reduce 3sq. m. 325a. 2r. 37sq. rd. to square rods. 2. In 359317 square rods how many square miles, acres, roods, and rods ? 3. Reduce 30sq. yd. Isq. ft. 127sq. in. to square inches. 4. Reduce 39151 sq. in. to higher denominations. 5. How many square feet in Fig. 1 ? 6. How many feet round Fig. 1 ? 7. Suppose each side of Fig. 2 to be 7 rods, what is the distance round it ? How many square rods does it contain ? 8. How many square rods in a rectangular field that is 17 rods wide and 35 rods long? How many acres ? How many rods round this field ? 9. A board containing 45 square feet is 15 feet long ; how wide is it? 10. A flower-garden containing 288 square feet is 12 feet wide ; how long is it ? Ans. 24. 1 1. How many square yards of carpeting will be required to carpet a room that is 18 feet long and 15 feet wide ? 12. At $2 per yard for carpeting that is a yard wide, what will be the cost of carpeting a room that is 5 yards square ? CUBIC OR SOLID MEASURE. 110. A Solid or Body is any thing which has length, breadth, and thickness. 110. What is a Solid or Body ? REDUCTION. 89 111. A solid or "body bounded by 6 rectangular faces, as Fig. 3, is called a Rectangular Prism. D Fig 3. G 1 i = i « i 1 ...„ 1 1 i . ' ~ - — .... E 113. A rectangular prism 3 feet bounded by six square faces, as Fig. 4, is called a Cube. ^ Fig. 4. \ ^ ^ Length. ' 113. To find the volume or solid contents of a rectangu- lar prism, as Fig. 3, first find the area of the top face, AB C Dy as in Art. 107; then going from Ay B, and C down- ward 1 foot to a, h, and c, and passing a plane through a, h, and c, we shall cut ofi" 15 solid feet, that is 5 X 3 X 1 solid 111. A Rectangular rrism ? of a rectangular prism found ? 113. A Cube? 113. How is the volume 90 REDUCTION. feet. So, if a plane be passed through d, e, and /, it will cut off 30, or 5 X 3 X 2 solid feet, etc.; that is, The continued product of the numbers representing the length, breadth, and height will give the volume or solid contents of a rectangular prism ; thus, in Fig. 3, 5 X 3 X 4 = 60, (solid feet,) the volume or contents. 114:. So, reversing the process in Art. 113 The volume divided by the area of the top face will give the height of the prism ; the volume divided by the area of one end will give the length ; and the volume divided by the area of one side will give the breadth or width; thus, in Fig. 3, 60 -^- 15 = 4, the depth; 60 -^ 12 = 5, the length ; and 60 -^ 20 = 3, the breadth. V1.5. Solid or Cubic Measure is used in measuring things which have length, breadth, and thickness. TABLE. 1728 Cubic Inches (c. in.) make 1 Cubic Foot, cu. ft 27 Cubic Feet " 1 Cubic Yard, c. yd. 16 Cubic Feet «' 1 Cord Foot, c. ft 8 Cord Feet or) 128 Cubic Feet j cu. ft c in. c. yd. 1 = 1728 1 = 27 = 46656 Scale. Descending, 27, 1728 ; Ascending, 1728, 27. Note 1. The numbers after 27, in this table, do not belong to the scale. 114. How the depth, when the volume and area of the top face are known ? How the length, when the volume and area of one end are known ? How the breadth, when the volume and area of one side are known ? 115. For what is Solid Measure used ? Table? Scale? Notel? 1 Cord, . c. REDUCTION. 91 Fig. 6. Note 2. A pile of wood, Fig. 5, that is 8 feet long, 4 feet wide, and 4 feet high, measures a cord, and one foot in length of such a pile is a cord foot. Note 3. A Perch of building stone or masonry contains 24^ cubic feet. A pile 16i feet long, Ih feet wide, and 1 foot high measures a perch. Note 4. Transportation companies often estimate freight, es- pecially of light articles, by the space occupied, rather than by the actual weight. In this estimate, from 25 or 30 to 150 or 175 cubic feet are called a ton. This is called arbitrary weight, and it varies with different transportation companies. 1. How many cubic inches in 33c. yd. 24cu. ft. 1635c. in. ? 2. Reduce 1582755c. in. to higher denominations. 3. Reduce 15c. yd. 18cu. ft. 1727c. in. to cubic inches. 4. In 5 c. 6c. feet, 9 cubic feet, 125 c. in. how niary cubic inches ? 5. If 40 cu. ft, make one ton, how many tons, cubic feet, etc., in 347859 cubic inches? 6. How many cubic feet are there in Fig. 3 ? How many square feet in the top face of Fig. 3 ? How many in the front side ? How many in the right-hand end ? How many in the whole surface of Fig. 3. 7. How many cubic feet in a cubical box whose edges are 2 feet in length? How many cubic inches? How many square feet in its surface ? 8. How many cords of wood in a pile that is 32 ft. long, 4 ft. wide, and 6 ft. high? How many cord feet? Cubic feet? Cubic inches? 9. A rectangular block of marble which contains 88 cubic feet, is 1 1 feet long and 4 feet wide ; how thick is it ? 115. Note 2? Notes? Note 4? 92 REDUCTION. 10. A grain-bin which holds 30 cuhic feet of grain is 3 feet deep and 2 feet wide ; how long is it? Ans., 5 feet 11. My cistern is 18 feet long, 15 feet wide, and 10 feet deep. By a pipe 6 cubic feet of water enter every minute ; in how many minutes will the cistern be filled ? LIQUID MEASURE. 116. Liquid Measure is used in measuring all liquids. TABLE. 4 Gills (gi ) make 1 Pint, pt. 2 Pints ti 1 Quart, qt. 4 Quarts tt pt. 1 Gallon, gal. qt. 1 = 4 gal. 1 = 2 = 8 1 = 4 =: 8 = 32 SCALE. Descending, 4, 2, 4 ; Ascend ing, 4, 2, 4. Note 1. The United States Standard Unit of Liquid Measure is the old English wine gallon, which contains 231 cubic inches. Note 2. It has been customary to measure milk, and also beer, ale and other malt liquors, by beer measure, the gallon containing 282 cubic inches, but this custom is fast going out of use. Note 3. Casks of various capacities, from 50 to 150 or more gallons, are indiscriminately called hogsheads, pipes, butts, tuns, etc. Those containing from 30 to 40 gallons are called barrels. 1. Reduce 9gal. 3qt. Ipt. 2gi. to gills. 2. Eeduce 318 gills to pints, quarts, etc. 3. Eeduce 12gal. Ipt. to gills. 4. Reduce 573 gills to higher denominations. 116. For what is Liquid Measure used ? Table ? Scale ? Note 1 ? REDUCTION. 93 5. How many bottles, each containing 3qt. Ipt. 2gi., can be filled from a cask which contains 46gal. 3qt Ipt. ? 6. How many gallons of molasses in 2i jugs, each contain- ing Igal. 2qt. Ipt. ? DRY MEASURE. 117. Dry Measure is used in measuring grain, fruit, potatoes, salt, charcoal, etc, TABLE. 2 pts. (pt.) 8 Quarts 4 Pecks make 1 I 1 Quart, . qt Peck, pk. Bushel, bush. bush. pk. 1 = qt. 1 8 pt. — 2 = 16 1 — 4 — 32 z= 64 Scale. Descending, 4, 8, 2 ; Ascending, 2, 8, 4. Note. The bushel measure is 18^ inches in diameter and 8 inches deep, and contains a little less than 2150^ solid inches, or nearly 9 J ■vrine gallons. Consequently 4 quarts or half a peck of oats should measure nearly 38 cubic inches more than a gallon of wine ; and a quart of berries, or any other article measured by Dry Measure, should contain nearly dh cubic inches more than a quart of wine or any other liquid. 1. Reduce 3bush. 2pk. 7qt. Ipt. to pints. 2. Reduce 239 pints to quarts, pecks, etc. 3. How many pints in 25 bush. Ipk. 5qts. Ipt. ? 4. How many pints in IZbush. 3qt. ? 116. Note 2 ? Note 3 ? 117. For what is Dry Measure used ? Table ? Scale ? What are the dimensions of the bushel measure ? How many cubic inches does it contain ? How many wine gallons ? How much ought^a quart of berries to exceed a quart of milk ? 94 REDUCTTOX, 5. Reduce 759 pints to higher denominations. 6. Reduce 8573 pints to higher denominations. 7. What is the cost of 2bush. 3pk. of grass seed, at $2 a peck? TIME. 118. Time is used in measuring duration. The natural divisions of time are days, months, (moons), seasons, and years. The artificial divisions are seconds, minutes, hours, weeks, etc. TABLE. 60 Seconds (sec.) make 1 Minute, m. 60 Minutes *• 1 Hour, h. 2 1 Hours "1 Day, d. 7 Days "1 AVeek, wk. 4 Weeks " 1 Lunar Month, 1. m. 13 Months, 1 Day, and 6 Hours ** 1 Julian Year, J. yr, 12 Calendar Months (=365 or 366 Days), 1 Civil Year, c. yr. 100 Years make 1 Century, C. 1. m. • yr. 1 1= 13^4 wk. 1 = 4 1 z= r= 28 = h. 1 z= 24=: m. 1 — 60 = 1440 = sec. 60 3600 86400 168 = 10080 = 604800 672 =: 40320 = 2419200 T\2 :52^\ =3651 —8766 =525960 =31557600 Scale. Descending, 4, 7, 24, 60, 60 ; Ascending, 60, 60, 24, 7, 4. 118.«For what is Time used? What are its natural divisions ? Artificial divisions ? Table ? REDUCTION. 95 Note 1. The names of the seasons and of the calendar months and the number of days in the several months, are as follows : — Seasons. Months. No. of Days. rrri--.-_ ( 1st. January 31 • } 2d. February 28, in leap year 29 ( 3d. March 31 Spring. ) 4th. April 30 ( 5th. May 31 ( 6th. June 30 Summer. ] 7th. July 31 < 8th. August 31 t 9th. September 30 Autumn. ] loth. October 31 < nth. November 30 Winter. 12th. December 31 Note 2. The number of days in each month may be easily re- membered by committing the following lines : — Thirty days hath September, April, June, and November j All the rest have thirty-one, Save the second month alone, Which has just eiglit and a score Till leap year gives it one more. Note 3. A solar year, that is, a year by the sun, is very nearly 365 days, 5 hours, 48 minutes and 50 seconds. 1. How many seconds in 18]a. 27m. 30sec. ? 2. Reduce 12850 seconds to higher denominations. 3. Reduce 4d. 22li. 57m. 54sec. to seconds. 4. Reduce 9wk. 15li. 19sec. to seconds. 5. Reduce 452897 seconds to higher denominations. 6. In 7 centuries how many calendar months ? 7. Reduce 10800 calendar months to centuries. CIRCULAR MEASURE. 110, Circular Measure is used in surveying, naviga- tion, geography, astronomy, etc., for measuring angles, deter- mining latitude, longitude, etc. 118. Scale ? What are the names of the calendar months ? How many. days in each ? In what season is each ? The number of each from the beginning of the year ? Length of a solar year ? y(5 REDUCTION TABLE. 60 Seconds (60'0 make 1 Minute, 1' 60 Minutes (< 1 Degree, 1° 30 Degrees it 1 Sign, s. 12 Signs, or 360° <( 1 Circumference, circ. V = 60" s. lo =: 60 = 3600 circ. 1 zn 30 = 1800 =z 108000 I = 12 =r 360 z= 21600 =z 1296000 Scale. Descending, 12, 30, 60, 60 ; Ascending, 60, 60, 30, 12. Note. A curved line is a jSgure bounded by a curved line, all parts of the curve being equally distant from the center of the circle. The Circumference is the curve which bounds the circle. An Arc is any portion of the circumference, as A B or B D. An arc equal to a quarter of the circum- ference, or 90°, is called a quadrant. A Radius is a line drawn from the center to the circumference, as C A or C B. A Diameter is a line drawn through the center and limited by the curve, as A D. 1. In 15° 38' 29" how many seconds ? 2. Eeduce 78695" to degrees, etc. 3. Keduce 2° 2T 39" to seconds. 4. In 5s. 17° how many minutes? 5. Keduce 276892" to signs, etc. 6. Keduce 17s. 21° 28' 3" to seconds. 119. For what is Circular Measure used? Table? Scale? What is a Circle? Circumference? Arc? Quadrant? Radius? Diameter? REDUCTION. 97 Miscellaneous Examples in Eeduction. 1. In 7£ 15s. 6d. 3qr. how many farthings? 2. Keduce 67219qr. to pounds sterling, etc. 3. Eeduce lOoz. 17dwt. 15gr. to grains. 4. Change 27&19gr. to pounds, etc. 5. In 7oz. 5dr. 2sc. 12gr. of opium, how many grains? 6. Reduce 17 tons 16cwt. 3qr. to quarters. 7. Change 627243oz. to tons, etc. 8. In 7yd. 3qr. 2na. lin. how many inches ? 9. Keduce 742 inches to yards, etc. 10. Change 5fur. 13rd. 7ft. lOin. to inches. 11. Eeduce 273894 inches to miles, etc. 1 2. In 27 fathoms, how many inches ? 13. John Smith's horse is 15 hands high ; how many inches high is he ? 14. In 7m. 3fur. 7ch. 2rd. how many links ? 15. Eeduce 3a. 2r. 27sq. rd. 127sq. ft. 126sq. in. to square inches. IG. How many cu. in. in 17 cords? 17. Reduce 76493c. in. to cords, etc. 18. IIow many gills in 27gal. 3qt. Ipt. 3gi. ? 19. Eeduce 643gi. to gallons, etc. 20. Change 46bu. 3pk. 6qt. Ipt to pints. 21. In 874qt how many bushels ? 22. Eeduce 17h. 56m. 433ec. to seconds. 23. Eeduce 178cwi 2qr. 101b. to ounces. 24. Eeduce 10yd. 2na. to nails. 25. Eeduce 726890 inches to miles. 98 DEFINITIONS AND GENERAL PRINCIPLES. DEFINITIONS AND GENERAL PRINCIPLES. 130. All numbers are even or odd. An Even Number is a number that is divisible by 2 ; as 2, 4, 8, 12. An Odd Number is a number that is not divisible by 2 ; as 1, 3, 5, 11, 19. ISl. AH numbers aro prime or composite. A Prime Number is a number that is divisible by no whole number except itself and one ; as 1, 2, 3, 5, 7, 11, 19. A Composite Number is a number that is divisible by other numbers besides itself and 1 ; thus, 6 is composite, because it is divisible by 2 and by 3 ; 12 is composite, because it is divisible by 2, 3, 4, and 6 ; 25 is composite, because it is divisible by 5 and 5. Factoring Numbers. 133. The Factors of a number are those numbers whose continued product is the number ; thus, 3 and 7 are the factors of 21 ; 3 and 6, or 3, 3, and 2 are the factors of 18; etc. The prime factors of a number are those prime numbers whose continued product is the number ; thus, the prime fac- tors of 12 are 2, 2, and 3 ; the prime factors of 36 are 2, 2, 3, and 3 ; etc. NoTK. Since 1, as a factor is useless, it is not here enumerated. 130. What is an Even Number? An Odd Number? 131. A Prime Number? What is the only even prime number? What is a Composite Number ? 133. What are the Factors of a number ? Wliat are the prime factors of a number ? DEFINITIONS AND GENERAL PKINCIPLES. 99 TABLE OF PRIME NUMBERS FROM 1 TO 997. 1 |41 101 167 239 313 397 467 569 643 733 823 911 2 43 103 173 241 317 401 479 571 647 739 827 919 3 47 107 179 251 331 409 487 577 653 743 829 929 5 53 109 181 257 337 419 491 587 659 751 839 937 ' 7 59 113 191 263 347 421 499 593 661 7Ci7 853 941 11 61 127 193 269 349 431 503 599 673 761 857 947 13 67 131 197 271 353 433 509 601 677 769 859 953 17 71 137 199 277 359 439 521 607 683 773 863 967 19 73 139 211 281 367 443 523 613 691 787 877 971 23 79 149 223 283 373 449 541 617 701 797 881 977 29 83 151 227 293 379 457 547 619 709 809 883 983 31 89 157 229 307 383 461 557 631 719 811 887 991 37 97 163 233 311 389 463 563 641 727 821 907 997 1^3. To resolve or separate a number into its prime factors we have the following : HuLE. Divide the given number hy -any prime number greater than one^ that will divide it ; divide the quotient by any prime number greater than one that will divide it, and so on till the quotient is prime. The several divisors and last quotient will be the prime factors sought. 1. What are the prime factors of 5768 ? OPERATION. 2)5768 2)2884 2) 1442 7) 721 10 3 Ans. 2, 2, 2, 7, 1 3. 123. Rule for resolving a number into its prime factors ' TOO DEFINITIONS AND GENERAL PRINCIPLES. 2. Eesolve 680 into its prime factors. Ans. 2, 2, 2, 5, and 17. 3. Eesolve 846 into its prime factors. Ans. 2, 3, 3, 47. 4. What are the prime factors of 200 ? Ans. 2, 2, 2, 5, 5. 5. Eesolve 984 into its prime factors. Ans. 2, 2, 2, 3, 41. Greatest Common Divisor. 134. A Common Divisor of two or more numbers is any number that will divide each of them without remainder ; thus 3 is a common divisor of 12, 18, and 30. 13^. The Greatest Common Divisor of two or more numbers is the greatest number that will divide each of them without remainder ; thus, 6 is the greatest common divisor of 12, 18, and 30. Note. A divisor of a number is often called a measure of the number, also an aliquot part of the number. ISO. To find the greatest common divisor vre have the follovring : EuLE 1. Divide the greater of two numbers by the less, andf if there be a remainder, divide the divisor by the remain- der, and continue dividing the last divisor by the last remainder until nothing remains ; the last divisor is the greatest common divisor of the two numbers. Ur, EuLE 2. If more than two numbers are g'ven^ find the greatest divisor of two of them, then of this divisor and a third number, and so on until all the numbers have b en taken ; tJie last divisor will be the divisor sought. 134. What is a common divisor ? 135. What is the greatest common divisor ? 136. Rule for finding the greatest common divisor of two numbers ? Second rule for finding greatest common divisor of more than two numbers ? I DEFINITIONS AND GENERAL PRINCIPLES. 101 1. What is the greatest com- 2. What is the greatest com- tnon divisor of 16 and 44 ? mon divisor of 8, 12, 28 ? OPERATION. OPERATION. '16)44(2 8)12(1 32 8 12)16(1 4)8(2 12 8 Ans., 4)12(3 Ans., 4 ) 2 8 (7 12 28 3. Find the greatest common divisor of 9, 12, 18, and 24. Ans. 3. 4. What is the greatest common divisor of 24, 40, 68 ? Ans. 4. 5. What is the greatest common divisor of 144, 17, and 1728? 6. What is the greatest common divisor of 72, 45, 999 ? 7. What is the greatest common divisor of 1825, 640, 60? Least Common Multiple. 137. A Multiple of a number is any number which is divisible by that number ; thus, 15 is a multiple of 5 and also of 3 ; 21 is a multiple of 7 and of 3. 138. A Common Multiple of two or more numbers, is any number which is divisible by each of the given numbers ; thus, 48 is a common multiple of 4, 6, and 8. 139. The Least Common Multiple of two or more num- bers, is the least number that is divisible by each of the given numbers ; thus, 24 is the least common multiple of 4, 6, and 8. VZ7. What is a Multiple of a number? Ii28. A Common Multiple of two or more numbers ? 1*9. The Least Common Multiple? 102 DEFINITIONS AND GENERAL PRINCIPLES. 130. To find the least common multiple of two or more numbers : Rule. Having set the given numbers in a line, divide hj any prime number that will divide two or more of them, and set the quotients and undivided numbers in a line beneath ; fro- ceed with this line as with the first, and so continue until no two of the numbers can be divided by any number greater than one ; the continued product of the divisors and numbers in the last line will be the multiple sought, Ex. 1.. What i3 the least common multiple of 6, 8, 12, 16, 18, 24? OPERATION. 2) 6, 8, 12, 1 6, 18, 2 4 2)3, 4, 6, 8, 9, 12 2)3, 2, 3, 4, 9, 6 3)3, 1, 3, 2, 9, 3 1, 1, 1, 2, 3, 1 2X2X2X3X2X3 = 144, Ans. 2. Find the least common multiple of 5, 10, 12, 15, 20, 24. 120, Ans. 3. rind the least common multiple of 7, 8, 12, 14, 16, 21. Ans. 336. 4. Find the least common multiple of 24, 72, 18, 48. 5. Find the least common multiple of 10, 15, 24, 18, 32. 6. Find the least common multiple of 21, 7, 36, 42, 84, Vo. 130. Rule for fiading the Least Common Multiple ? coMMOx rr.ACTioxs. 103 COMMON FEACTIONS. ISl. If a single thing (an apple, for instance,) is divided into two equal parts, one of these parts is called one half, (written i) ; if divided into three equal parts, one of these parts is called one third (J). €> « # Halves. Thirds. Fourths. And so, if we divide a unit or single thing into four, five, six, etc. equal parts, one of these parts is called one fourth (i), one fifth (I), one sixth (^), etc. ONE UNIT. 1 i 1 i 1 1 i i 1 1 i i 1 1 1- i ± -+- J_ -+- i -+- i -i- i _^ J_ -+- J_ — 1 H 13^. A Fraction is an expression representing one or more of the equal parts of a unit. 133. A Common or Vulgar Fraction is expressed by two numbers, one above and the other below a line ; thus ^ (one half), f (two fifths), &c (a) The number above the line is called the Numerator, and the number below the line is called the Denominator. (b) The Denominator shows into how many parts the unit is divided, and gives the name to the fraction. 133, VThat is a Fraction? 133, A Common Fraction? (a) Where do we write the numerator? Denominator? (b) What does the denominator show ? (c) What the numerator ? (d) What are both called ? 104 COMMON FRACTIONS. (o) The Numerator shows how many of those parts are taken or expressed by the fraction. (d) The numerator and denominator are called the terms of the fraction. Write the following fractions : three fourths, two thirds, seven eighths, nine tenths, seven elevenths, eight fifteenths. Bead the following fractions : ^, |, |, f , f , ^j, |f , ij. 134:. A Simple Fraction has but one numerator and one denominator ; as |, f, ^-. 135. A Compound Fraction is a fraction of a fraction ; as S of f , f of -^%. 136. A Proper Fraction is one whose numerator is less than the denominator ; as f , |, f . 137. An Improper Fraction is one whose numerator equals or exceeds the denominator ; as |, ^, f , ^-. 138. A Mixed Number is a whole number and a frac- tion united ; as, 7^, 5|, 27f. 139. The terms of a fraction sustain to each other the relation of dividend and divisor, the numerator answer- insT to the dividend and the denominator to the divisor. That is, a fraction may be regarded as an expression of division. Hence, The VALUE of a fraction is the quotient of the numerator divided by the denominator, asf = 9-i-3 = 3. It follows from this that the General Principles of Division (Arts. 85, 86, and 87) apply to all fractions. 134, What is a Simple Fraction ? 135. Compound? 136. Proper? 137. Improper ? 138. What is a mixed number ? 139. What relation do the terms of a fraction sustain to each other? Wliich term answers to the dividend ? Which to the divisor ? How may a fraction be regarded ? To what is the value of a fraction equivalent f What principles before stated apply to fractions ? COMMON FRACTIONS. 105 1 . Multiplying the numerator ^ if the denominator remains unaltered^ m,ultiplies the value of the fraction hy the same number, as ^"^ ^ z=z^. 2. Dividing the numerator, if the denominator remains unaltered, divides the value of the fraction hy the same number, asi^2 =1, In the above cases it will be seen that the size of the parts, (fourths,) remains the same, but the number of the parts is in- creased or diminished. 3. Multiplying the denominator, if the numerator remains unaltered, divides the value of the fraction by the same ninnber, «« I X 2 = f • 4. Dividing the denominator, if the numerator remxiins unaltered, multiplies the value of tJie fraction by the same num- ber, as f ^ 2 = f . In the last two cases it will be seen that the number of parts (numerators) remains the same, but the size of the parts (denominators) is increased or diminished. 5. If the numerator and denominator are both multiplied or divided by the same number the value of the fraction is not ;, , 2X2 4 2-^2 1 altered, as - .. c. z=z -or ~ ; « = — 4X2 y 4—2 2 Hence, the following general law in regard to Fractions may be stated. That any change in the numerator causes a like change in the t^alice of the fraction ; and any change in the denominator causes an opposite change in the value of the fraction, LTpon these principles all the following operations upon fractions depend. 139. Give the Ist principle and illustrate it. The 2d principle. The 3d principle. The 4th principle. The 5th principle. "What general law is given ? 106 COMMON FRACTIONS. Case 1. 14:0. To reduce a mixed number to an improper fraction. Ex. 1. In 7f how many fifths ? 17 3 ' In 2l unit there are five fifths ; and in g^ seven units there are seven times five fifths, or 35 fifths, which with the 3 3 3 . fifths in the example =z 38 fifths = ^^-. EuLB. Multiply/ the whole number by the denominator of the fraction ; to the 'product add the numerator ^ and under the sum write the denominator, 2. Eeduce 1 7f to an improper fraction. Ans. ^K 3. Eeduce 26 14- to an improper fraction. Ans. ^^, 4. Eeduce 43f to an improper fraction. Ans. ^i. 5. Eeduce 56 § to an improper fraction. Ans. ^^, 6. Eeduce 85^j to an improper fraction. Ans. -^f^ 7. In 19y^j how many fourteenths? 8. How many seventeenths in 38|f ? Ans. ^^ 9. Eeduce 49^^ to an improper fraction. Note. To reduce a whole number to a fraction having any given denominator, multiply the whole number by the proposed denomi- nator, and under the product write the denominator. Case 2. 141 • To reduce an improper fraction to a vrhole or mixed number. Ex. 1. How many units in ^- ? In one unit there are four fourths, and in seventeen fourths JLT. — 17 • 4 — 4-1- ^ • there are as many units as four is contained times in seventeen. 140. Explain the operation in Case 1. Rule for reducing a mixed number to an improper fraction ? 141. Bule for reducing an improper fraction to a whole or mixed number ? COMMON FRACTIONS. 107 Rule. Divide the numerator by the denominator ; if there is any remainder, place it over the divisor, and annex the fraction so formed to the quotient. 2. Reduce 4^^ to a mixed number. Ans, 2f . 3. Reduce ^ to a mixed number. Ans. 5f . 4. Reduce ^f to a mixed number. Ans. ^^. 5. Reduce \^ to a mixed number. 6. Reduce ^/- to a mixed number. Ans. 9^|. 7. Reduce ^^- to a mixed number. Ans. lO^f . 8. Reduce W- ^^ * mixed number. 9. Reduce ^^ to a whole number. Ans. 7. 10. Reduce ^^' to a whole number. Note. The denominator of a fraction being a divisor, it follows that whenever the denominator exactly measures the numerator, the quotient will be a whole number. (See Exs. 9 and 10.) Case 3. 14^. To reduce a fraction to its lowest terms. Ex. 1. Reduce f| to its lowest terms. 1st OPERATION ^^^^ ^- ^*^^' ^^^^ ^^^ ^y ""''y /«^' 2)2^ tor common to them, then divide these quo- \)i^~& tients hy any factor common to them, and ^^^rC^ 2 *^ proceed till the quotients are mutually 8 ) ^ = f Ans ^^^^ ^^^^^ j3^^ .^j^^ 2d OPERATION. Find the greatest common divisor, (Art. 125,) l|}f|=:| Ans. and by it divide both terms of the fraction. Rule 2. Divide each term hy their greatest common divisor. 2. Reduce -^| to its lowest terms. Ans. |. 3. Reduce |-f to its lowest terms. Ans. ^. 4. Reduce ^^f to its lowest terms. 14t2. Rule for reducing a fraction to its lowest terms ? Second rule for reducing a fraction to its lowest terms ? 108 COMMON FRACTIONS. 5. Reduce /^ to its lowest terms. Ans. |. 6. Reduce f |^f to its lowest terms. 7. Reduce ^-^-^ to its lowest terms. Ans. ^. 8. Reduce :|f f to its lowest terms. Ca^e 4. 143. To multiply a fraction by a whole number. Ex. 1. Multiply I by 4. i„4. ^^ .,.„^.. I* is just as evident that 4 times 7 Ist OPERATION. •' r w ^ 20. i eighths (t) are 28 eighths (%8-) as it is that 4 times 7 boys are 28 boys. 2d OPERATION. If we divide the denominator by 4 we 5^4==^ obtain the same result as before. In the first operation we increase the number of the parts four- fold, and in the second, we increase the size or value of the parts four- fold while the number of parts remains the same. Hence the following Rule 1 Multiply the numerator hy the whole number. Or, Rule 2 Divide the denominator by the whole number, 2. Multiply A ^y 3. Ans. ^. 3. Multiply -/^ by 8. Ans. f f or |. 4. Multiply i^ by 5. Ans. f f =: 2f |. 5. Multiply ^g by 14. Ans. || or f = 3. 6. Multiply jS^- by 3. Ans. f|- or ^. 7. Multiply ^W by 15. Ans. |||. 8. Multiply ^853^ by 12. 143. First rule for multiplying a fraction by a whole number? Second rule? COMMON FRACTIONS. 109 9. Multiply ^^j\ by 21. 10. Multiply fl by 117. Ans. ef^s. 11. Multiply II by 17. 12. Multiply ^Yf ^y 3^- ^ns. 4-V¥- (a) To multiply a mixed number by a whole number. 13. Multiply ^ by 8. 1st OPERATION. 2 J OPERATION. ^ — ^f- ^ X 8 = 3_2 ^ ^ ^ X 8 z=z -i|2 _ 3S| Ans. 4 X 8 =: 32. 32 + 6|z=38f. Hence the following KuLE. Reduce the mixed number to an improper fraction^ (^r#. 140,) and then multiply. Or^ multiply the fraction and whole number separately and add the products together. 14. Multiply 6f by 9. Ans. 60f 15. Multiply 7 rV ^1 26. Ans. 190. 16. Multiply 28^\ ^y 42. Ans. 1186^j,. 17 Multiply 46fx by 39. 18. Multiply 89-t-V by 68. 19. Multiply 246J-I by 142. 20. Multiply 392/^ by 257. 21. Multiply 1501- by 27. as multiplying a fraction by a whole number, e. g. yX4=:4Xf. Case 5. 14:4:. To divide a fraction by a whole number. 143. How is a mixed number mxiltiplied by a whole number? Another method ? 110 COMMON FKAOTIONS. Ex. 1. Divide ? by 3. Ans. f . One third of 6 apples is 2 apples ; it is 1st OPERATION, equally clear that one third of 6 sevenths f ^ 3 = f (f ) is 2 sevenths (f .) If I divide f by 1, the quotient will be 2d OPERATION, f. Now if I divide it by 3 instead of 1, I f X 3 = A obtain a quotient only one third as great, or ^ of f z= ^\. In this instance the number of parts remains the same, while the size of the parts is diminished. EuLE 1. Divide the numerator hy the whole number, Or^ EuLE 2. Multiply the denominator hy the whole number, (Art. 139, 2nd and 3rd.) 2. Divide Jf by 8. Ans. /-. 3. Divide J-f by 6. Ans. j^^. 4. Divide \\ by 5. Ans. -^J. 5. Divide -i-f by 12. Ans. y^^. 6. Divide |f by 13. Ans. J3. 7. Divide |f by 7. Ans. j^^-. 8. Divide f f by 14. 9. Divide ^%% by 35. 10. Divide JfJ by 42. Note. If the dividend be a mixed number first reduce it to an improper fraction, or divide the whole number and fraction sepa- rately and add the results. 11. Divide 261 by 6. 12. Divide 161 by 7. Ans. 2\%, 13. Divide 28| by 7. 144. First rule for dividing a fraction by a whole number ? Second rule ? A mixed number how divided by a whole number ? COMMON FRACTIONS. Ill 14. Divide 69i by 13. Ans. 5|. 15. Divide 21 1| by 12. Ans. 17f. Case 6. 14:5. To multiply a fraction by a fraction. Ex. 1. Multiply I by i Ans. fj-. We first multiply the fraction | by 7, (Art. 143, Eule 1.) and obtain ^^'. Now, as 7 is eight times the true multiplier I, the product is 8 times too large ; and we obtain the true prod- uct by dividing -\^~ by 8 (Art 141, Kule 2.) | X 7 — sj-, and -2J- -^ 8 =z |i. Hence, Kule. Multiply the numerators together for a new nume- rator jand the denominators for a new denominator. 2 6 2 2. Multiply izX^ -^^s. f . 7 p In the above example, we have the factor 3 in the numerator of the f , and also in the denominator of the f . These we reject in the operation, since this is equivalent to dividing both terms of the product by 3, which (Art. 139, 5th) does not alter the value of th j fraction, and obtain the answer in its lowest terms. This process of cancellation may be em- ployed advantageously in many cases, as the principle is the same, as when applied in division. (See Art. 88.) 17 5 17 8. Multiply ^* X g = 56, Ans. 7 4. Multiply ^\ by by ^%, Ans. i|. 5. ' Multiply If by Jf. Ans. -g-ff. 6. Multiply tf by ^f. Ans. ^\\, 7. Multiply f^ by ff. 145 . Rule for multiplying a fraction by a fraction ? 112 COMMON FRACTIONS. 8. Multiply |6-| by U^-. 9. Multiply 6 1 by ^y- Ans. Ih Note- 1. Eeduce the mixed numbers to improper fractions. 10. Multiply 5f by f . 11. Multiply 2^ by §. Ans. 9/,. 12. Multiply 58f by 5^. Ans. 298f. A compound fraction may be reduced to a simple one by the rule for multiplying one fraction by another. 13. f of f equals what ? Ans. t\ = ^ 14. I of -^^ of ^ equals what ? 15. /tj- of f f of II equals what ? Case 7. 140. To divide a fraction by a fraction. Ex. 1. Divide f by f . Ans. | = 1^ We first divide | by 2, and obtain OPERATION. f , Art. (144, Eule 2,) but the divisor |-^-f = l X | = f used is 3 times too great, and conse- quently the quotient f is only ^ of the required quotient, and hence must be multiplied by 3 to obtain the correct result. From the above we have the following Eule. Invert the divisor, and then proceed as in multipli- cation (Art. 145). 2. Divide ^ by |A. Ans. |. 3. Divide if by f. Ans. |f = l^\. 4. Divide |f by I, Ans. f ^. 5. Divide 4f by f . Ans. ^^ = 6-J-i. 6. Divide -^^^ by ^§. 7. Divide fax by f 146. Rule for dividing a fraction by a fraction ? COMMOJT FRACTIONS. 113 8. Divide I by if i. Anfl. |f|. 9. Divide ff by ^^. 10. Divide f by ^ of ^. Ans. 8. Note. If either of the quantities is a mixed number it must be re- duced to an improper fraction. Case 8. 147. To reduce fractions that have different denom- inators to equivalent fractions having a common denom- inator. Ex. 1. Eeduce | and f to equivalent fractions having a common denominator. We multiply both terms of each frac- tion by the denominator of the other 5 w ? ?J fraction ; this (Art 139, 5th) does not ^ 9 36 alter the value of either fraction, and of 5 4 20 necessity it makes the denominators alike 9 ^ 4 ^^ 36 as they are both the product of the two de- nominators, 4 and 9. EuLE. 3Iultiply both terms of each fraction by the con- tinued product of the denominators of all the other fractions. Ex. 2. Pieduce f , f , and f , to equivalent fractiofh having a common denominator. 3. Beduce f , f , f to equivalent fractions having a common denominator. Ans. ^i^, \^l, J-||. 4. Eeduce ^\, f , ^^ to equivalent fractions having a com- mon denominator. Ans. 4^a, ^^%, f f^. 5. Eeduce f , f, | to equivalent fractions having a common denominator. Ans. ||8, |o o, |^. 6. Eeduce -f^j f , -^-^ to equivalent fractions having a com- mon denominator. 147. Kule for reducing fractions to equivalent fractions having a common denominator ? 114 COMMON FRACTIONS. Though the above rule will give a common denominator, yet it will not always give the least common denominator. Ex. 7. Ecduce |, §, ■^^, i| to equivalent fractions having the least common denominator. OPERATION. 4 )4, 8, 12, 16 2 ) 1 , 2, 3, 4 1, 1, 3, 2 4 X 2 X 3 X 2 =r 48 = L. C. M. = Least Com. Denom. (Art. 130,Eule.) 48 -^ 4 = 12, and 12 X 3 zrr 3G = 1st numerator. 48 -^ 8 = 6, and 6 X 5 =r 30 = 2d numerator. 48 -^ 12 — 4, and 4 X 7 = 28 r= 3d numerator. 48 -f- 16 — 3, and 3 X 13 = 39 = 4th numerator. Ans., If, ff, ff, and f|, the several equivalent fractions. Explanation. We first find the least common multiple of the denominators 4, 8, 12, and 16, which is the least com- mon denominator. We then divide this denominator by each of the denominators of the given fractions, and multiply each quotient b^ its corresponding numerator. Hence we have the KuLE. Reduce all the fractions (if necessary) to their lowest terms. Find the least common midtiple of all the denominators for a common denominator. Dicide this multiple hy each of the given denominators, and multiply the several quotients hy their respective numerators for neio num- erators. IToTE. It will be seen that both the above rules are based upon the principle Art. 139, 6th. 147. Rule for obtaining the least common denominator of several fractions ? On What principle does this rule and the former one depend ? Explain. COMMON FRACTIONS. 115 8. Eeduce f , t> |» i^ *^ equivalent fractions having the least common denominator. Ans. ||, §^, f |, f f . 9. Eeduce f , ^, |f to equivalent fractions having the the least common denominator. 10. Eeduce f, f, ^^, J to equivalent fractions having the least common denominator. Case 9. 14:8, Numbers that are of the same kind can be added together. For example, 6 pen?-|- 7 pens -(-5 pens =: 18 pens. 2 hats -f- 5 hats z= 7 hats ; and for the same reason I + f + i =r f . f + f , + ^ == f , etc. But numbers which are not alike, cannot he added. We cannot say 6 knives -|- 4 pens =10 pens, or 10 knives. Neither can we say f bush. -(- f qt. =: I bush, or \ qt. Numbers must be of the same jiiND if we would add them together. Hence, To add fractions we have the following Eule. Reduce the fractions to equivalent fractions having a common denominator ; after which, write the sum of the numerators over the common denominator. Ex. 1. Add -j^^, y3^, and -J^ together. Ans. -^f. 2. Add I, |, and f together. Ans. f§f — Ifgf. 3. Add f , I, ^\ and f . Ans. f f f = 2^^. 4. f + t + ^+| = what? 5. tV + l + f + i = what? . Ans.|H = 2x||. ^- ^ + § + 1 + 1 = what? Ans. 5^2_i ~ ^tV- 7. f of f -ft of i==^liat? Ans. If = 1^1^. Note 1. Compound fractions must be first reduced to simple frac- tions. 8. t of T^y + f of I = what ? Ans. f | =: l^ 9. I of f -(-i|of f = what? 148. Can numbers not alike be added ? Rule for adding fractions ? 116 COMMON FRACTIONS. 10. f of I of I + f of j\ = what? 11. Add 41, Gl, and of together. 4+6+5=:15,| + ^ + fz:=2,y^, aiidl5 + 2TV^=rl7TV^, Ans. Note 2. Mixed numbers may be reduced to improper fractions ; or the whole numbers and the fractions may be added separately, as above, and tlien their sums united. 12. Add 4i and 7| together. Ans. llf^. 13. Add 5f , 7|, and f together. 14. 16| + 14| + 18f z= what ? Practical Questions i:i Addition of Fractions. 1. John bought a top for i of a dollar, a knife for | of a dollar, and a "ball for i of a dollar ; how much did they cost him? Ans. $U. 2. Jane bought a work-basket for | of a dollar, a pair of gloves for If dollars, and gave i dollar to some poor persons ; how much did she expend in all ? 3. A lady bought several remnants of cloth. One piece was I yd. long, another J yd., and a third ^ yd. ; how much cloth did she buy in all ? Ans. 2^ yds. 4. A coal dealer sold coal to three men, as follows : To one 2| tons, to another 5§ tons, to the third 6^ tons; how much did he sell to all ? 6. Three men bought a horse. One paid 51^ dollars, another 67f dollars, and the third paid 76|- dollars; how much did the horse cost? Case 10. 14:9. The remarks under Case 9 apply with equal force to questions in Subtraction. We cannot take 4 pens from G COMMON FRACTIONS. 117 knives ; nor can we take | of a gallon from § of a pound. Numbers must be of the same kind or the subtraction can- not be performed. Hence To subtract fractions we have the following EuLE. Reduce the fractions to equivalent fractions having a commcm denominator^ and tJien write the difference of the numerators over tJie common denominator. Ex. 1. From I take |. Ans. f. 2. From f take f. Ans. f. 3. From f take f . Ans. /^. 4. From f take y\. Ans. ff. 5. From ^^ of ^V take ^ of f . Ans. ^Vf- 6. From 4| take 2f . Ans. fj z= If ^. Note 1. Whenever mixed numbers occur in the question they must be reduced to improper fractions. 7. Subtract f from Ih Ans. f f — Iff. 8. 41 — 24= what? 9. 5y\ — 4f = what ? 10. Ql-—6\=z what ? Ans. 1 ^V- 11. I of iJ-— 2 of^^ — what? 12. f of 5| — f of S^zzzwhat? 100. Practical Questions in Subtraction of Fractions. 1. A boy having | of a qt. of nuts, gave away J of a qt ; what part of a quart had he left? Ans. ^|. 2. A merchant having a piece of cloth containing 12f yds. sold 7f yds. ; how much had he left ? Ans. 4f f yds. 3. A farmer had a field containing 23 1 acres. Of this, 6 1 acres were planted with corn, and the remainder bore grass ; how much grass land was there in the field ? Ans. 16ff acres. 149. What is the rule for subtraction of fractious ? 118 COMMON FRACTIONS. 4. Bought a cask of wine containing 37f gal.; 16 1 gallons having leaked out, what quantity remained in the cask ? Ans. 20^^! gal. 5. A boy while fisting for pickerel, lost part of his pole ; and on. measuring, he found that the part saved was | of the original length. What part was broken off ? 6. From a chest of tea weighing 62 1 lb. 39| lb. were sold. How many pounds remain unsold ? 1^1. Miscellaneous Examples in Fractions. 1. Reduce f | to its lowest terms. 2. Ptcduce 7 1 to an improper fraction. 3. Eeduce -\^- to a mixed number. 4. Multiply §^ by 9. 5. Multiply jW by 6. 6. Divide |f by 4. 7. Divide -V- by 10. 8. Divide 23^ by 7. 9. Multiply 20 by f . 10. Multiply 100 1 by I. 11. Multiply f^ by if. 12. Eeduce f of f of -^^ to a simple fraction. 13. Divide ^9- by f . 14. Divide 207 by |. 15. Eeduce |-, y^j, -^^ to equivalent fractions having a common denominator. 16. Addf and V-. 17. Add ^^^V and i. 18. Add 10^ and 6f. 19. Addi, §,|, i 20. Subtract! from |. 21. Subtract | from |§. COMMON FRACTIONS. 119 22. Subtract y\^ from 1. 23. Subtract 2i from 3i 24. Keduce |f, -j^y^g* 1^1 *^ equivalent fractions with a least common denominator. 15. Eeduce -//j of ^V^ to a simple fraction. Examples in Analysis. 1^9. We analyze an example when we solve it according to its own conditions without being guided by a^ particular rule. 1. If 1 pound of tea cost $1.20, what will f of a pound cost? Analysis. If 1 lb. cost $ 2.20, 3^ of a pound will cost ^ of $ 1.20, or 24 cts. ; and |^ of a pound will cost 4 times 24 cts., or 96 cts., which is the answer. Note. A period called the decimal point, which will be here- after more fully explained, is placed at the right hand of dollars, and the first two places at the right of the points always express cents. In the following examples, if no cents are named with the dollars, their places may be supplied with two ciphers. 2. If 1 cord of wood cost $ 9, what will f of a cord cost? Ans. $5.62J. 3. What will f of a ton of hay cost if 1 ton cost 21 dollars? Ans. $ 15. 4.. When oil is $2.25 per gall, what will f of a gall, cost? Ans. $1.50. 5. If 1 acre of land cost $ 140, what will -^j of an acre cost ? Ans. $ 89.09 y^j. 6. If I of a yd. of cloth cost $ 2.50, what will be the cost of 1 yd. ? Analysis. If f of a yd. cost $ 2.50, } will cost i of $2.50, or $.83^ and | or 1 yd. will cost 5 times $.83^, or $ 4.16f ., which is the answer. 7. Bought f of an iron foundry for $6783, what was the value of the whole foundry? Ans. $ 8478.75. 120 COMMOJ^ FRACTIONS. 8. "When we pay 8 62 for ^ of an acre of land what is the cost per acre ?. Ans. $ 70.85^. 9. If I of a gallon of molasses cost $.83, what is the cost per gallon ? 10. If 3 A. 2R. 30sq. rd. is | of a field, what is the entire area ? Ans. 6A. 2R. 22sq. rd. 11. A grocer sold from a cask ISgal. 3qt. Ipt. of oil, which was^ of what the cask contained ; how much remained in the cask? Ans. 21 gal. Octt. l^pt. Note. It is evident that if he had sold 3. of what the cask at first contained, f remained. 12. If 6 doz. eggs cost $ 1.68, what will 12| doz. cost? Analysis. If 6 doz. cost $ 1.68, 1 doz. will cost ^ of $ 1.68, or 2Scts. 12| doz. = \t doz. If 1 doz. cost 28cts. i doz. will cost i of 28cts. or 7cts., and -^^ will cost 5 1 times 7cts., or $3.57. Ans. 13. If 5 rods of wall can be built for $ 8.65 what will it cost to build 17| rods ? Ans. $ 30.05|. 14. If 16 bushels of corn cost $22.72, what will 47§ bushels cost ? Ans. $ 67.68f . 15. If a family consume 4 barrels of flour in 7 J months, how long would 9 J barrels last them ? Ans. 17^ months. 16. If i gal. wine cost $ 4.75, what will 6f gal. cost? Analysis. If J gallons of wine cost $4.75, | will cost ^ of $ 4.75 or $.67f , and f or 1 gallon will cost 8 times 67« cts. or $5.42f. 6 1 gallons = -\^ gallons. Now if 1 gallon cost $5.42f, -1 gallon will cost ^ of $ 5.42f, or $ 1.08f, and -\*- will cost 34 times $ 1.08^ or $ 37.33f. Ans. It. When f of a dollar will purchase 3 qts. of cherries, how many can you purchase for 2i dollars ? Ans. 9| qt. COMMON FRACTIONS. 121 18. If you can buy 4| tons of hay for $ 70, what will 12| tons cost? Ans. $ 188i 19. When 17^ cents are paid for If lbs. of nuts, how many pounds will 48| cents buy? Ans. 4| lbs. 20. If 6f bushels of wheat cost $15, how many bushels can you buy for 68| dollars ? 21. Sold 5 1 cords of wood to one man, and 12f cords to another ; how much did I sell to both ? 22. How many hours will it take a man to travel 65 f miles, if he travel 3| miles in an hour ? Ans. 19^^ h. 23. Bought a horse for $176 J, and a wagon for 8 67| ; how much more did the horse cost than the wagon ? Ans. $108j. 24. Paid 8 16 for some cloth, at the rate of 8 f per yd. ; how many yards were there ? Ans. 20 yards. 25. If 4 doz. oranges cost f of $ 4, what will 7 oranges cost? Ans. $ /o* 26. A man who owned |^ of a farm sold f of his share ; what part of the farm did he sell and what part did he still own ? Ans. sold y^^, and had left ^|. 27. If a man has 28f gal. of wine, and sells | of it, how many gallons has he left? Ans. 7f. 28. A man has 7 pieces of broadcloth, each piece contain- ing 26| yd. This he makes up into overcoats requiring 4f yards each ; bow many garments can he make and how much cloth will he have left ? Ans. 40 garments,and {^yd. left. 29. A field containing 157^sq. rd. is 9|rd. wide; how long is it? Ans. 16| rd. 30. If f of a ton of coal can be bought for $ 7, what part of a ton can be bought for one dollar ? Ans. 5^. 31. If |- of a ton of hay is worth $ 12|, what is the value of a ton? Ans. 8 15|. 122 COMMON FRACTIONS. 32. One man earns $ If per day, and another earns $ 2|- ; how much will they both earn in 5 days ? Ans. $19^. 33. A merchant buys flour at $ 9f per barrel and sells it for $ 12| ; how much will he make on 5 barrels? Ana. e 15|. 34. A tailor made 3 suits of clothes, each containing 3f yards of cloth, from a piece 35 yards long; how much was left? Ans. 24fyd. 35 What will 12| cords of wood cost at $ 8f p3r cord ? Ans. $109^^. 36. A farm containing 247 acres was sold for $ 21118J ; what was the price per acre ? Ans. $ 85^^. 37. Four partners purchase, goods to the amount of $ 1264f, and sell the same for $ 1586f. The profits being divided equally, what was each one's share ? Ans. $ 80xVu- 38. A tailor paid $ 12| for cloth, and $ 5| for making up the same into a coat, vest, and pants, and sold the same for $ 26i ; what were his profits? Ans. $ 7j\>-. 39. A merchant sold to a customer 5|yd. broadcloth, 6|yd. doeskin, and 4|yd. cassimere ; how many yards in all were there? Ans. 16:|f. 40. When flour is $ 15 per bbl. how many barrels can be bought for 46^ bushels of wheat at $ 3^ per bushel? Ans. 9|^ bbl. 41. A person owning ^ of a ship, sold y\ of his share for $ 6000, which was $ 950 more than it cost him ; what did he pay for his entire share ? 42. A gentleman has $ 9750 invested in United States Bonds, which is f of his fortune ; how much is he worth ? DECIMAL FRACTIONS. 123 DECIMAL FKACTIONS. 1^3. A Decimal Fraction is a fraction whose denomi- nator is 10, 100, 1000, or 1 with one or more ciphers annexed. 104. The denominator of a Common Fraction may be any number whatever. Every principle and every operation in Common Fractions is equally applicable to Decimals. \55, The denominator of a decimal fraction is not usually expressed, since it can be easily determined, it being 1 with as many ciphers annexed as there are figures in the given decimal. 1^6. A decimal fraction is distinguished from a whole number by a period, called the decimal point or scparatrix, placed before the decimal ; the first figure at the right of the point is tenths ; the second, hundredths ; the third, thousandths ; etc. ; thus, .6 =: -^^, ,06 = y§^, .006 :r= i^^is^ etc., the figures in the decimal decreasing in value from left to right, as in whole numbers. \SK • Since whole numbers and decimal fractions both decrease by the same law from left to right, they may be ex- pressed together in the same example, and numerated as in the following NUMERATION TABLE. (-1 cJ .S - ^ O TO- . CL rS ^ 2 ^ °Q ;2rt3 ..TO.SrQ-TSSE-lTj KhPRhWhh jT-'S 5 9 8. 7 2 vv: 124 DECIMAL FRACTIONS. 158. A whole number and decimal fraction written to- gether, as in the above table, form a mixed number. The integral part is numerated from the decimal point towards the left, and the fraction from the same point towards the right, each figure, both in the whole number and decimal, taking its name and value from its distance from the decimal point. Hence, 159. Moving the decimal point one place towards the right, multiplies the number by 10; moving the point two places multiplies the number by 100, etc. Also moving the point one place to the left, divides the number by 10 ; moving the point two places divides by 100, etc. 100. A decimal is read like a whole number, giving in addition, the name of the denomination of the right-hand figure to the entire number. Thus, .46 is read forty-six hun- dredths ; .073 is read seventy-three thousandths; .0068 is read sixty-eight ten thousandths; 42.045 is read forty-two, and forty-five thousandths, etc. 101. Since multiplying both terms of a fraction by the same number does not alter its value (Art. 139, 5th), annex- ing one or more ciphers to a decimal does not affect its value; thus, -fjs — j%o^ — f^OoV etc. ; i. e. .2 = .20 = .200, etc. lOS. Prefixing a cipher to a decimal, i. e. inserting a cipher between the separatrix and a decimal figure, diminishes 153. VThat is a Decimal Fraction ? 154. A Common Fraction, what is its denominator ? Are the principles of common fractions applicable to deci- mals? 155. la the denominator of a decimal usually expressed? 15G. How is a decimal fraction distinguished from a whole number ? What is the first fi^re at the right of the point ? Second ? Third ? 157. Read the Numera- tion Table. 158. What is a mixed number ? Which way is the integral part numerated ? ' Which way the decimal ? What determines the name and value of a figure ? 159. How does moving the decimal point to the right affect the value of a number? How moving it to the left 7 160. How is a decimal read? Illustrate. 161. How does annexing one or more ciphers to a deci- mal affect it ? Why ? 16^. Prefixing one or more ciphers to a decimal how affects it? Why? DECIMAL FRACTIOITS. 125 ^e value of that figure to i^^j its previous value ; for it removes the figure one place further from the decimal point (Art. 159) ; thus, .3 =z ^jj but .03 = only y§^, which is but yV of yV Notation and Numeration ov Decimal Fractions. 103. Let the pupil write in figures the following num- bers : 1. rifty-six hundredths. 2. Eighty-seven thousandths. 3. Two hundred sixteen ten-thousandths. Note. To avoid ambiguity in expressing a whole number and a decimal, we should use " and" only once, and that, between the whole number and decimal, as two hundred three, and six thou- sandths, (203.006) ; not, two hundred and three and six thousandths. By observing the above rule, much trouble will bo obviated, and numerous mistakes avoided. 4. Twenty-eight, and one thousand nine, ten- thousandths. 5. One hundred sixty-eight, and thirteen millionths. 6. Eight hundred forty, and forty-two hundred thou- sandths. 7. Seven, and seven hundredths. 8. Four hundred eighty-seven, and three hundred forty- four ten-thousandths. 9. Sixteen thousand, four hundred thirty nine, and ninety- two thousandths. 10. Fifty-one million two thousand eighty-five, and seven- teen hundredths. 11. Four thousand eight hundred twenty-eight, and nine hundred fifty-six hundred-thousandths. 12. Eighty-seven thousand three hundred forty-nine, ten- millionths. 163. How are operations in decimal fractions performed? What of the the decimal point ? Proof? 126 DECIMAL FRACTIONS. KoTE 2. Let the teacher give many examples, similar to the above, continuing the exercise till the pupil can write decimals with ease and accuracy. Numerate and read the following : 1. .7 11. 4587.9506 2. .03 12. .000001 3. 40.6 13. 17.75851 4. 601.75 14. 7.805 5. 77.7 15. 84591.57 6. 9005.847 16. 42.2222 7. .0001 17. 10000.001 8. 1000.1 18. 45671.3501 9. 45678.951 19. 1000.001 0. .37558 20. 1846561.07 As all the operations in decimal fractions are performed precisely as the same operations in whole numbers, no explana- tions are necessary, except to determine the true place of the decimal point in the several results. The methods of proof, also, are the same as in whole numbers. Case 1. 164:. To add decimal fractions ; Rule. Place tenths under tenths^ hundredths under hun- dredths^ etc. ; then add as in whole numbers, and place the point in the sum directly under the points in the numbers added. Ex. 1. 2. 4.3 7 4 6 9.037 6 5.0 8 2 609 3.0 08406 9.0 9 5 6.9 005 4 6 3.0 8 4 790 0.0 56209 Sum, 5 4 1.6 224 Sum, 149 6 9.00 1 6 65 Proof, 5 41.6 2 24 Proof, 1 4 9 6 9.0 1 6 6 5 164. Bole for addition of decimals ? DECIMAL FRACTIONS. 127 3. Add 469.05309, 27.039, 8056.00963. Ans. 8552.10172. 4. Add 904.0602,6095.8095, 600.06, and 29076.004069. Ans. 36675.933769. 5. Add 2307.085063, 65.0047, 3S0.30027, and 70580. 060309. Ans. 73332.450342, 6. Add 4.063, 85.605, 74608.37, 63.704. 7. Add two hundred forty- three, and sixty-five thousandths ; seventy-one, and eighty-four ten-thousandths; two thousand, and two thousandths ; six hundred, and six hundredths. Ans. 2914.1354. 8. Add six hundred fifty-eight, and seven hundred two ten- thousandths ; ninety- seven, and ninety-seven hundredths; two thousand sixty-five, and eight thousand three hundred six hundred-thousandths. 9. Add sixty-eight millionths ; two hundred forty-seven ten-thousandths ; nine hundred seventy-two hundred-thou- sandths. Ans. .034488. 10. Add three thousand nine hundred sixty- two, and 'seven hundred thousandths ; five hundred seventy-three, and ninety- three ten-thousandths; eight thousand forty- four, and seven hundredths. Case 2. 16^. To subtract a less decimal from a greater : EuLE. Place the less number under the greater, tenths under tenths, etc. ; then sithtract as in whole numbers, and place the point in the remainder ^ directly under the points in the minuend and subtrahend. 165. Rule for Rubtraction of decimals? When the number of decimal places in the subtrahend exceeds the number of decimal places in the minuend what is done? 12^ DECIMAL FRACTIONS. Ex. 1. 2. 3. 5.9 2 3 5 3.08 76 4 9 6 3.0 3 2 2.8 6 7 47.19 84 8 7 4.0 8 5 7 6 9 Eem. 3.0 5 6 5.8 8 92 408 8.9 17 43 1 Proof, 6.9 2 3 6 3.087 6 Note. "Whenever there are more decimal figures in the subtra- hend than in the minuend, as in Ex. 3, we may supply the deficiency by annexing ciphers, or supposing them annexed, to the minuend. 4. From 68.0473 take 39.0027. Ans. 29.0446. 6. From 234.0023 take 97.013005. G. From 608.01004 take 290.020635. Ans. 317.989405. 7. From 5901.632 take 807.000956. Ans. 5094.631044. 8. From 20.006 take 7.020407. 9. From one hundred eighty-three, and twenty-four thou- sandths, take seventy-six, and three thousand seven hundred ninety-eight ten-millionths. Ans. 107.0236202. 10. From five thousand six hundred nine, and one hundred thirty-two hundred-thousandths, take nine hundred eighty-five, and four hundred ninety-six ten-thousandths. Case 3. 160. To multiply one decimal by another : EuLE. Multi'ply as in whole numbers, and point off as many figures for decimals in ike product as there are decimal places in both factors counted together. 166. Rule for multiplication of decimals ? Reason of the rule for point- ing the product ? Suppose there are not figures enough in the product for observing the direction for pointing off? DECIMAL FKxVCTIONS. 129 Ex. 1. Multiply .37 by .28. OPEEATION. Multiplicand, Multiplier, .3.7 .2 8 296 74 PROOF. fo^TrXy^a^iVVoV Product, .10 3 6 Note. 1. The reason of the rule for pointing the product will be obvious if we change the decimals to the form of common fractions and then perform the multiplication ; hence, the proof above. Thus we have, .24 X .16 = ^^^ X t'o% = tM^ = -0384. And again, .00G9 X .09 = y^6_9^^ X toxt =tQ-=z.(j25 ; and hence we have the following EuLE. Annex one or more ciphers to the numerator and di- vide the result by the denominator, continuing the operation until there is no remainder, or as far as is desirable. Point off as many decimal places in the quotient as there are ciphers annexed to the numerator, 2. Eeduce | to a decimal. I X 100 -= iao _ -J5 . and 7^ -^ 100 = .75 Ans. 3. Reduce f^ to a decimal. Ans. .5625. 4. Reduce |^f to a decimal. Ans. 1.171875. 5. Reduce f | to a decimal. 6. Reduce -^^ to a decimal. Ans. .4166 -f-. 7. Reduce ^, f, f, f, |, ^^ to decimals. Case 6. 109. To reduce a decimal to a common fraction : Ex. 1. Reduce .75 to a common fraction. .75=:^^^^, and this, reduced to its lowest terms z= | Ans. Hence, Rule. Write the denominator to the decimal, omitting the decimal point, and then reduce the common fraction to its lowest terms, 2. Change .625 to a common fraction. Ans. f . 168. Rule for reducing a common fraction to a decimal ? 169. Rule for reducing a decimal to a common fraction ? 134 DECIMAL, ru ACTIONS. 3. Change .375 to a common fraction and to its lowest terms. Ans. |. 4. Keduce .0625 to a common fraction. Ans. ^^^. 5. What common fraction is equivalent to .4375? Ans. ^^. 6. Eeduce .68 to a common fraction. 7. Change .875 to a common fraction. 8. Change .0075 to a common fraction. Miscellaneous Examples in Decimals. 1. What is the sum of one- tenth, one hundredth, and for- ty-seven thousandths? Ans. .157. 2. What is the difference between seven hundredths and eight thousandths? Ans. .062. 3. .065 — .0098 z= what? 4. Multiply eighty-four hundredths by forty-seven ten- thousandths. Ans. .003948. 5. Divide two by four-tenths. 6. Divide eighteen thousandths by six millionths. Ans. 3000. 7. From seven- tenths take four millionths. Ans. .699996. 8. Paid $ 480 for a piece of land at $ 62.50 per acre ; how many acres were there ? Ans. 7.68. 9. What cost 8 acres of land at $ 68.75 per acre? 10. What cost 6.75 lb. of coffee at $ .24 per lb. ? 11. How many casks each holding 37.5 gallons can be filled "with 168'?. 5 gallons of wine? 12. Bought 6.5 tons hay at $ 16.875 per ton; what was the entire cost? Ans. $ 109.68.75. 13. What common fraction is equal to the sum of .625 and .0625 ? Ans. ^. DECIMAL FRACTIONS. 135 14. When $ 18.5625 is paid for 148.5 yds. of cloth, what is the cost per yard ? 15. What will 17 pairs of boots cost at $ 10.875 per pair? 16. Change .68 to a common fraction. Ans. ^^^ ^^il- 17. Change 5.25 to a common fraction. Ans. if^^-V-^Si- 18. How many pairs of shoes at $1.25 can be purchased for $ 45 ? 1^. How many pounds of sugar at 8 .18 per pound can you buy for $6.12? Ans. 34. 20. What will 12 bales of cotton cost, each bale weighing 5.25 cwt. at $ 46.50 per cwt.? 21. If .625 of a ton of coal cost $5. 75, what will one ton cost? Ans. $9.20. 22. What cost .875 of a ton of coal at 8 12 per ton ? xVns. $ 10.50. 23. What will 8.75 cords of wood cost at $ 10 per cord? 24. If a boat will sail 7.5 miles in 1 hour, how far will she sail in 9.75 hours? 25. Divide one hundred by one hundredth. 2Q. Multiply one thousandth by one thousandth. 27. If 9564.75 rods of wall can be built in 87.75 days, how many rods can be built in one day ? Ans. 109. 28. I have a room 15.50 feet wide, 16.75 feet long; how many square feet does the floor contain? Ans. 259.625. 29. How many square yards of carpeting would it take to carpet the above room ? 30. How much would the above carpet cost at $ 1.75 per yard ? 31. A load of hay weighs 1675.25 lb. how much will it cost at 8 2.50 per cwt.? Ans. $ 41.88. 32. A ship carries 725 bales of cotton, each bale weighing 400 pounds; how much will the freight amount to at $.0125 per pound ? 136 UliTITED STATES MONEY. UNITED STATES MONEY. ITO. United States Money, sometimes called Federal Money t is the currency of the United States. TABLE. 10 Mills (m.) make 1 Cent, Marked c. 10 Cents " 1 Dime, '' d. 10 Dimes " 1 Dollar, " % 10 Dollars " 1 Eagle, '' e. Cents. Mills. Dimes. 1 = 10 Dollars. 1 — 10 = 100 Eagle. I — \0 = 100 = 1000 1 := 10 = 100 = 1000 = 10000 Note. The terms eagle and dime are seldom or never used in com- putation ; eagles and dollars being read collectively and called dollars, and dimes and cents being called cents ; thus, 3 eagles and 5 dollars are called $ 35, and 4 dimes and 3 cents are called 43 cents. When mills are written with dollars and cents they are set in the third place at the right of the period; thus, thirty-five dollars, forty-three cents, and seven mills, expressed in figures, is ^ 35.437. 171. A coin is a piece of gold, silver, or other metal, stamped by authority of the General Government, to be used as money. 173. The coins authorized by our Government, and stamped at the U. S. Mint, are the following: 170. What is United States Money ? Repeat the Table. Are the terms eagle and dime much used ? How are eagles and dollars read ? Dimes and cents ? What place do mills occupy ? Illustrate. 171. What is a coin ? UNITED STATES MONEY. 137 Gold Silver. Double Eagle. S 20.00 Dollar, $1.00 Eagle, 10.00 Half Dollar, .50 Half Eagle, 5.00 Quarter Dollar, .25 Quarter Eagle, 2.50 Dime, .10 Three Dollar Piece. 3.00 Half Dime, .05 One Dollar, 1.00 Three Cent Piece, .03 Also of copper, bronze, and nickel, we have the One Cent, Two Cent, Three Cent, and Five Cent Pieces. Note 1. The mill is not coined. Note 2. Our Government, by enactment of Congress, may recall any of these coins, or issue new ones of different values and com- posed of different metals, at any time. Note 3. The greater part of the currency in general use in this country, consists of bank hills and notes of the General Government^ which are much more convenient for most purposes than gold and silver. Operations in United States Money are performed precisely like those in Decimal Fractions, the dollar being considered the UNIT. Therefore no special rules are needed. Practical Examples. 1. Paid $ 12.50 for a barrel of flour, % 2.375 for a box of sugar, $ 17.875 for a tub of butter, and $ 5.25 for a cheese ; what did I pay for all ? Ans. $ 38. Having set dollars under dollars, o o 7 r cents under cents, etc., add as in Art. - '^ „ ^ 164, and set the sum below, remember- 5 2 5 i'^o ^^^^ ^^® point, or period, should TT—— — — - be placed directly under the points in $3 8,0 0, Ans. ^, ^ , ,/, ^ the numbers added. 17ii. What gold coins are authorized by our Government ? What silver coins ? What other coins ? What is said of the mill ? What of changing the coins in use ? What of paper money ? How are operations in U. S, Money performed ? 138 UNITED STATES MONEY. 2. Bought a coat for $ 21.75, a vest for $ 5.35, a pair of pantaloons for S 8.40, a hat for $ 5.25, a pair of boots for $ 7.50, and various other articles for $12.75 ; what must I pay for all?- 3. A farmer paid $ 125.50 for a pair of oxen, $ 52 for a cow, $350.75 for a horse, and $45.25 for a harness; how much did all costf Ans. $ 573.50. 4. A merchant in returning from the city found he had expended $ 1050.375 for dry goods, $ 850.75 for groceries and $ 250.875 for hardware ; what was the amount of his purchases ? 5. A broker has $ 19505.00 in one bank, $4550.50 in another bank, and $ 6750.37 in another ; how much money has he in the three banks ? , Ans. $ 30805.87. 6. The property of a gentleman is divided as follows ; he has $5750 in bank stock, $3100.50 in notes at interest, a manufactory worth $ 19587, two farms, one worth $ 5780 and the other twice as much, and $ 6850.75 due him on accounts; how much is he worth ? 7. A man who owed $87.37, paid $16.52; how much did he still owe ? OPERATION. Having set the less number under the 8 7.3 7 greater, dollars under dollars, etc., subtract as 1 ^-5 2 in Xvi. 165, remembering to place the point in $ 7 0.8 5, Ans. the remainder under the points in the minuend and subtrahend. 8. Paid $ 175.625 for a pair of oxen, and $ 132.375 for a horse ; how much more did I pay for the oxen than for the horse? Ans. $43.25. 9. A gentleman purchased a city residence for $ 19570, which was $8957.75 more than his country place cost him; what did his country place cost him ? 10. A drover paid out for cattle $ 5767.50 ; he received for the same lot, besides expenses of taking them to market, $ 6530 ; how much were his profits? Ans. $ 762.50. UNITED STATES MONEY. 139 11. A merchant went to the city to buy goods, with $ 3575.50 in cash ; he bought to the amount of $ 5050 ; how much did he buy on credit? Ans. ^ 1474.50. 173. To find the cost of any number of things when the price of one thing is given. 12. Bought 6 cows at $35,375 each; what did I pay for the lot? OPERATION. Six cows will evidently cost 6 times as $ 3 5.3 T 5 much as one cow. All similar examples are p solved in like manner. Hence, the fol- S 2 1 2.2 5 0, Ans. lowing EuLE. Multiply the price of one by the number. 13. "What is the cost of 7 barrels of flour at $8.50 per barrel ? 14. Bought IG yards of silk, at $1.75 per yard; what was the cost of the piece? Ans. $ 23.00. 15. Bought 33 sheep, at $8.25 per head ; what was the cost of the flock ? Ans. $ 272.25. 16. What are 85 pounds of butter worth, at 3T cents per pound? Ans. $31.45. 17. "What are 625 cords of wood worth at $ 8.75 per cord? Ans. $ 5468.75 18. What is a cargo of coal of 2070 tons worth at $ 11.25 per ton? Ans. $23287.50. 19. What are 1625 bushels of wheat worth at $3.75 per bushel? What would be the freight on the above at $.625 per bushel ? 20. Supposing the above wheat to make 425 barrels of flour, how much would it be worth at $ 17.50 per barrel? Ans. $7437.50. 173. How can you find the cost of any number of things when you know the price of one? 140 UNITED STATES MONEY. 174L. To find the price of an article when the cost of a given number of articles is known. 21. Paid $1129.50 for 9 horses; what was the average price per horse ? OPERATION. One horse is worth one ninth as g\ 2 129. 50 much as 9 horses. To obtain one- . ninth of any number we divide the * number by 9. Hence, the Rule. Divide the cost hy the number. 22. Paid $19.61 for 53 pounds of butter ; what was the price per pound ? Ans. ^7 cents. 23. If I pay $315.75 for 25 barrels of flour, what is the price per barrel ? Ans. $ 12.63. 24. If a mechanic earns $ 65.24 in 28 days, what is his daily •wages? 25. Paid $74.75 for 13 weeks' board; what was the price per week? ' Ans. $5.75. 26. Seventy-seven boys paid $2233 for 1 year's tuition; what did each boy pay ? Ans. $ 29. 27. Bought a farm containing 87 acres for $4763.25 ; what was the price per acre ? 28. If 8 barrels of flour cost $75, what is the price per barrel? Ans. $9.3T5. OPERATION. When the division is incomplete 8 ) $ 7 5.0 and there are no cents and mills in $ 9.3 7 5, Ans. the dividend, ciphers may be annexed to the dividend and the division continued. 1T4. How can you find the price of one article when you know the cost of a given number of articles ? Explain Ex, 28. f v^ <^ UNITED STATES MONEY. ^ w ST * niSfc ^"^ \ Cj ^ ^^ PROOF. • Ke versing the above p^^t^MpFiO^ $ 9.3 7 5 the proof, will cause the ccnl 8 mills to disappear and bring back the $ 7 5.0 original dividend. 29. If 21: men earn $ 63 in a day, what will 1 man earn in the same time? Ans. $2.G25. 30. Paid S 6300 for 36 horses ; what was the price of each ? 31. If 5 barrels of flour are worth $47,267, what is 1 barrel worth? Ans. $9,453+. 32. Paid $ 34.88 for 9 yards of cloth ; what was the price per yard? Ans. $3,875+. 33. If a cargo of wood is worth $19275, and the number of cords is 2850, how much is the price per cord? Ans. $ 6.76+. 34. If $ 1000 will buy 850 bushels of com, what is the price per bushel ? Ans. $1.17+. 35. If a merchant's bill for flour was $18500 in one month, and he purchased 1500 barrels, what did it cost him, on an average, per barrel? 175, To find the quantity when the cost of the quantity and the price of one are given. 36. At $8 a ton, how many tons of coal can I buy for $240? OPERATION. $8')S240 '^^ many times as $ 8 is contained in $ 24 so many tons I can buy. Hence, the 3 0, Ans. BuLE. Divide the cost hy the price of one. 175. How do you find the quantity when the total cost and the price of one are given ? 142 UNITED STATES MONEY. 37. At 16 cents a pound, how many pounds of sugar can I buy for $ 19.96 ? Ans. 124|. 38. How many yards of cloth, at $ 2.56 per yard, can I buy for $642.56? Ans. 251. 39. How many sheep, at $ 7.75 a head, can be bought for $193.75? Ans. 25. 40. A farmer paid $ 3562.50 for land, at $ 37.375 per acre ; how many acres did he buy ? 41. A merchant paid $ 4498.83 for a lot of broadcloth ; the average price per yard was $ 3.33 ; how many yards were there ? Ans. 1351. 42. How many books at $1.75 each, can be bought for $2625? 43. An agent has $ 925 with which to purchase flour ; at $ 12.25 per barrel how many whole barrels can he buy ? How much money will he then have left? 176. To find the cost or value of any number of articles when the price of one is an exact or aliquot part of a dollar. Table of Aliquot or Exact Parts op a Dollar. 50 cents := ^ of a dollar, 20 cents = ^ of a dollar, 331 cents := ^ of a dollar, 16f cents = ^ of a dollar, 25 cents = :J- of a dollar, 12^ cents = | of a dollar. 44. What cost 45 yards of calico, at 33^ cents per yard ? 331 cents is ^ of a dollar ; hence, 45 yards will cost $45-^3 = $15, Ans. 45. What cost 84 pounds of butter, at 50 cents a pound ? 46. What cost 48 pounds of honey, at 25 cents a pound ? 47. What cost 32 bushels of corn, at 87^ cents per bushel? 176. What is an aliquot part ? Repeat the table of aliquot or exact parts of a dollar. UNITED STATES MONEY. 143 OPERATION. $32 = cost of 32 bush., at S 1. 16= cost of 32 bush., at 50 c, ot J- of $ 1. 8 :=cost of 32 bush., at 2 5 c, or ^ of 50c. 4 = cost of 32 bush., at 12^ c, or ^ of 25c. Ans. f2S = cost of 32 bush., at sTi c. That is, the cost at $ 1 is evidently as many dollars as there are bushels ; the cost at 50c., is half as much as at $ 1 ; the cost at 25c., half as much as at 50c. ; and the cost at 12^c., half as much as at 25c. Then the cost at 50c., at 25c., and at 12Jc., added, gives the cost at 87ic. 48. What is the value of 736 yards of gingham, at 37^ cents a yard ? Ans. $ 276. 49. What shall I pay for 1832 bushels of oats, at 62i cents per bushel ? This process is usually called Praciice,{oT which we have the following KuLE. Take such cdiQUot pqrts of the number of articles as the pHce is an aliquot part of %\, 50. What cost 24 barrels of apples, at % 3.75 per barrel? OPERATION. $2 4 = cost at S 1. _3 $7 2 = costat$3. 12 = cost at .5 or J- of $ 1. 6 = cost at .2 5 or I of 50c. Ans. $90 = cost at $ 3.7 5. 51. What are 348 barrels of flour worth, at % 9.87^- per barrel? Ans. $3436.50. 52. What are 165 thousand of brick worth, at $ 11.75 per thousand ? 176. How do you find the cost of any number of articles when the price is an aliquot part of a dollar ? What is this process called ? 144 UNITED STATES MONEY. 53. Wliat are 84 cases of mercliandise worth, reckoning each case at $ 475.3 7^? 54. What would be the cost of 336 yards of carpeting at $2.G2J per yard? Ans. $ 882. 65. How much would 1250 cords of wood cost at $ 8.75 per cord ? Ans. 8 10937.50. 177, To exchange or barter goods. 6Q. How many pounds of sugar, at 20 cents a pound, shall I give for 50 bushels of com, at 80 cents a bushel ? OPERATION. 4 This example is best solved $ X 5 by cancelling as in the margin. (\1\ ^^ ^^^' "^^^ ■''■*' ^^^ ^^^^ ^^ analyzed as fol- ^ ^ lows ; 50 bushels at 80 cents are worth 50 times 80 c. = 4000g., and 20c. in 4000c., 200 times, the number of pounds of sugar required, Ans. 200. 57. How many cords of wood, at $ 8 per cord, will pay for 6 tons of hay, at $20 per ton? Ans. 15. 58. How many tons of coal, at % 12.50 per ton, will pay for 16 yards of cloth, at % 6.25 por yard? Ans. 8. • 59. How much flour at $ 10.50 per barrel can be obtained for 150 bushels of potatoes at 75c. per bushel ? 60. How many entire yards of broadcloth at $ 5.75 per yard, can be bought for 3^ cords of wood at $ 6.75 per cord, and what money will remain duo ? Ans. 4 yards, and 62^c. due. 61. How many bushels of wheat at $ 3.50 per bushel would purchase 50 bushels of corn at $1.50. per bushel? Ans. 2\f. 62. It requires 16 thousand shingles to cover the roof of a certain house, and they cost $4.75 per thousand; how many days work at % 2.50 per day would it require to pay for them ? 177. Explain the operation in Ex. 56 by cancellation. How else mfey this example be solved ? UNITED STATES MONEY. 145 63. How many yards of cloth, at $ 3.50 per yard, can be had for 3^ tons of hay, at $19.50 per ton ? BILLS. 178. A Bill of Goods is a written statement of articles sold, giving the price of each article and the cost of the whole. An Account is a written statement of the items of debt and credit between two persons or companies. The person or company who owes is the Debtor, and the one to whom something is due is the Creditor. When a bill is paid it is usually receipted or signed by the (Creditor or by his authorized agent. Receipts for an amount of $20 dollars or upwards, according to the laws of Congress, now require a revenue stamp to be affixed. Find the cost of the several articles, and the amount or footing of each of the following bills. (1.) Mr. John Low, 2 5 lb. Sugar, 4 2 lb. Butter, 1 5 yd. Cloth, Boi Bought of at n (( ved Payment, tton, Sept. 6, 186T. David Flint, 16 c. 2 5 c. $3.3 3i 1 ! Stamp. 1 Recei 1 $64.5 David Flint. 178. What is a Bill of Goods ? What an Account ? Who is Debtor ? Who Creditor? When should a bill be signed or receipted? By whom? What is said of affixing a revenue stamp ? 146 unit:ed states money. (2.) New York, Oct. 15, 1867. Messrs. Smith & Co., Bought of Abel Adams, 2 4 gal. Molasses^ at 8 7 1^ c. 3 2 gal. Syrup, " $ 1.1 2 1 4 8/5. Coffee, " ^7^c. 16 lb. Tea, ** 6 2ic. Stamp. 8 8 5, Received Payment, Abel Adams, By L. Snow. (3.) iVew' Orleans, Dec. 19, 1865. 3/r. James Fitoh, 1865. To Henry Day & Co., Dr. June 4. To 12 Day's Algebras, at S7^c. Aug. 9. '* 4: Beams Paper, " $2.75 16. " 2 4.S'/a^es, " 3 7ic. JVbv. 11. *' 3 Webster's Diction- aries, at $ 8.7 5 Stamp, $5 6.7 5 Received Payment, John Smith, For Henry Day & Co» (4.) Norwich, July 5, 1867. Mr. R. B. Allen, Bought of James Robinson & Co. 1 2 pairs Men's Calf Boots, at $ 4.7 5. 12 " " Thick " ♦* 3.7 5. 18 " Boys' " '* " 2.1 2 J. Stamp.! Received Payment, I James Robinson & Co. UNITED STATES MONEY. 147 (5.) Messrs. John P. Jones & Co., 1866. Baltimore, Dec, 15, 1866. To E. C. Johnson «& Co., Dr. Mar. 4. T'o 1 tons Ice, at Jpr. 8. "25 hbl. Flour, June 8. "10 bush. Com, " " ♦< 5 bush. Wheat, " 1866. Cr, May 14. By Cash, June 8. " ^Merchandise, Sept. 6. " 5 cords Oak Wood, at Dec, 1. " 2 tons May, ** Stamp. $12.37^ 9.2 5 8 7c. 81.75 $5 2 9.5 $2 2 5.5 115.7 5 9.7 5 15.4 5 $420.90 Balance due E. G. J. ^ Co. $ 1 8.6 Received Payment, E. C. Johnson &Co. Miscellaneous Examples in U. S. Money. 1. If 4 cords of wood cost $ 34.50, what is the price per cord? Ans. $8.62i 2. What shall I pay for 7 tons of hay, at $ 16.75 per ton? Ans. $117.25. 3. My farm cost $ 3476.50 and my house cost $ 2347.75 ; how much more did I pay for the farm than for the house ? 4. When beef costs 12 J cents per pound, what shall I pay for 1936 pounds? ' Ans. $242. 5. Bought 4 lb. tea at 75c., 6 yd. sheeting at 33|c., and 5yd. broad cloth at $ 3.25 ; what was the cost of all ? Ans. $21.25. 6. If 8 yards of cloth cost $ 12, what will 12 yards cost? 7. If 16 barrels of flour cost $ 144, what will 12 barrels cost? Ans. $108. 148 UNITED STATES MONEY. 8. If 4 tons of coal cost $ 35, what will 64 tons cost? 9. If 4 tons of hay cost $ 62.50, what will 48 tons cost? 10. Paid $ 4050 for 75 acres of land ; at what price per acre shall I sell it to gain $ 225 ? Ans. $ 57. 11. Bought 22 pounds of sugar at 9 cents, 4 pounds of cof- fee at 66 cents, 3 pounds of tea at 75 cents, 5 gallons of mo- lasses at 48 cents, and 5 barrels of flour at $ 9.75, and gave the merchant 6 ten-dollar bills, how much change shall he return tome? Ans. $2.38. 12. Bought 6 pounds of butter at 30 cents, 10 pounds of cheese at 18 cents, 24 pounds of rice at 6 cents, 7 pounds of raisins at 25 cents, 2 bushels of potatoes at 75 cents, 1 bushel of beans at $ 1.50, and 10 yards of sheeting, and gave 2 ten- dollar bills to the merchant, who returned $ 7.91 ; what was the price per yard of the sheeting ? Ans. 23 cents. 13. A merchant bought 8 ''boxes of tea, containing 60 pounds each, for $ 312 ; but it being damaged he sold it at a loss of $ 72; at what price per pound did he sell it? How much did he lose on each pound ? 14. A family, consisting of father, mother, and 2 children, desires to board by the sea during the summer, and can afford to pay $ 126 ; how many weeks can they remain, if the board of each parent costs $5.50, and of each child $3.50 per week? Ans. 7. 15. A laborer bought a bushel of potatoes for 75 cents, 6 pounds of sugar at 15 cents, a barrel of flour for $ 8.75, and 12 pounds of meat at 10 ceirts; he paid $5.35 in cash, and the balance in work at $ 1.25 per day ; how many days did he work? Ans, 5. 16. A merchant found that for one year his whole profits were $8750; of this he paid $1250 for store rent, $2750 for clerk hire, and $ 1850 for other expenses; how much clear profit remained ? UNITED STATES MONET. 149 17. A bookseller went to the city and bought a bill of books" as follows : 12 readers at $ 1.25, 18 spellers at $.37^, 10 geo- graphies at $ 1.75, 8 primary geographies at $.62^. In pay- ment he gave a hundred dollar bill, how much should he receive back? 18. It took 25 yards of carpeting at 8 1.87 J per yard for my sitting room, 50 yards of matting at $.65 per yard for my chambers, and 39 yards of oil-cloth at $ 1.25 per yard for my kitchen and halls; what was the amount of my bill? Ans. $128.12^. 19. A ship carried to London from New Orleans 4500 bales of cotton, each bale weighing 475 pounds, at a freight of 2^ cents per pound, and other merchandise upon which the freight was $ 8595 ; what was the whole amount of the ship's freight? 20. A drover bought stock as follows : 7 horses, at an average price of $ 225 ; 50 sheep at $ 3.50; and one pair of oxen for $200 ; what did the whole cost him? 21. A gentleman found that his household expenses for one month were as follows : provisions $175.50, groceries $150.50, house rent $83.33 ; what was the amount? What would be the amount for one year ? 22. A farmer sold the produce of his farm as follows: 150 bushels of potatoes at $.55, 175 bushels of com at $1.25, and 50 bushels of wheat at $4.50 per bushel; what was the amount he received ? 23. A builder took a contract to build a house ; he paid for brick and stone work, with materials, $ 3500 ; for carpenter work, with materials, $2575.50; for painting $675, and for other work $1550; he received $10,000; how much were his profits ? 150 COMPOUND NUMBERS, COMPOUND NUMBEES. ADDITION. 179. A Compound Number is composed of two or more denominations (Art. 92) "whicli do not usually increase deci- mally from right to left ; consequently, in adding the diflferent denominations, we do not carry one for ten, hut for the numher it takes of the particular denomination added, to make a unit of the next higher denomination ; thus, in adding Sterling or English money, we carry 1 for 4, 12, and 20, because 4qr. make Id., 12d. make Is., and 20s. make 1£. Ex. 1. Add together 5£ 10s. 7d. 3qr., 6£ 18s. lid. 2qr., 9£ 13s. 5d. Iqr., 17£ 16s. 9d. 3qr. We first arrange the numbers as in the margin. Then add the right-hand column as in simple numbers, and find the amount to be 9qr. := 2d. and 3 9 19 10 1 Iqr. We write the Iqr. under the column of farthings, and add the 2d. to the column of pence ; the amount of which we find to be 34d. = 2s. and lOd. We set the lOd. under the column of pence, and add the 2s. to the column of shillings, and find the amount to be 593. =:z2£ and 19s. We write the 19s. in the column of shillings, and add the 2£ to the column of pounds; the amount of which we find to be 39£, and the whole amount £ 8. d. qr. 3 9 19 10 1 Ans. 179, What is a compound number ? How do they increase ? What is said of carrying ? OPERATION. £ s. d. qr. 10 7 3 6 18 11 2 9 13 5 1 1 7 16 9 3 ADDITION. 151 180. The principle of this process is precisely the same as in addition of simple numbers. Hence, To add compound numbers, EuLE. Write the numbers so that each denomination shall occupy a separate column, the lowest denomination at the rights and the others towards the left in the order of their values. Add the numbers in the lowest denomination, divide the amount by the number it takes of this denomination to make one of the next higher, set the remainder under the column, and carry the quo- tient to the next column. So proceed until all the columns are added. Proof. The same as in Addition of Simple Numbers 2. 3. £ s. d. gal. qt. pt, 27 17 e 5 3 19 15 10 4 2 14 6 1 1 7 3 28 19 9 4 Sum, 91 22 1 Proof, 91 22 1 NoTK. In writing, and also in adding the numbers of a single DENOMINATION, the Tules of simple addition must be observed ; thu» in writing the pounds in Ex. 2, set imits under units, tens under tens. 4. 5. lbs. oz. dwt. grs. A. R. rd. 17 10 19 23 7 3 27 13 7 13 19 2 2 31 7 11 17 21 6 3 28 27 10 15 20 9 3 39 Sum, 6 7 5 7 1 1 27 2 5 Proof, 6 7 5 7 11 27 2 6 152 COMPOUND NUMBERS. 6. 7. bush. pk. qt. pt. t. cwt. qr. lb. oz. 71 3 7 1 7 ■ ■19 3 20 13 19 2 5 1 5 1 4 2 16 14 3 3 3 17 3 23 2 13 2 4 1 4 1 6 1 19 8 8. 9. yd. qr. na. in. ^rd. yd. , ft. in. 5 3 3 2 7 4 2 10 7 2 3 U 1 5 2 11 9 3 2 2 6 3 1 ■ 7 7 3 3 2 4 4 2 9 21 2 il I or 21 2 2 7 Note. A fraction occurring in the amount may sometimes be reduced to whole numbers of lower denominations ; thus, in Ex. 9 ; we reduce the iyd. to lower denominations = 1ft. 6in., this we add to the ft. and in. in the example, and have 21rd. 2yd. 2ft. Tin. 10. A trader bought 4 hhd. of sugar: the first weighed lOcwt. 3qr.. 171b.; the second 13 cwt. Iqr. 191b.; the third 12cwt. 3qr. 18lb.;and the fourth llcwt. 3qr. 271b.; what did the whole weigh? Ans. 2t. 9cwt. Iqr. 61b. 11. I have my winter's wood in four piles; in one are 4c. 5 c. ft. 12 cu. ft; in another 2 c. 7 c. ft. 9 cu. ft.; in another 1 c. 6 c. ft. 13 cu. ft. and in the fourth 3 c. 5 c. ft. 11 cu. ft.; how much wood have I in all? Ans. 13 c. 1 c. ft. 13 cu. ft. 12. A vintner has wine in 3 casks; in the first, 68gal. 3qt. Ipt. 3gi.; in the second, 79gal. 2qt. Ipt. Igi.; in the third, 94gal. 3qt. Ipt. 3gi.; how much has he in the three casks? 180. Rule for addition of compound numbers? Principle? Proof? Numbere of a single denomination, how written and added ? SUBTRACTION. 153 SUBTKACTION. 181. The principle is like that of subtraction of simple numbers. Hence, To subtract compound numbers, EuLE. 1 . Write the less quantity under the greater, arrang- ing the denominations as in addition. 2. Beginning at the right, take each denomination of the subtrahend from the number above it, and set the remxiinder beneath. 3. If any number of the subtrahend is greater than the number above it, add to the upper number as many as it takes of that denomination to make one of the next higher, and take the number in the subtrahend from the sum ; set down thQ re- mainder, and considering the number in the next denomination in the minuend one less, or that in the subtrahend one GREATER, proceed as before. Proof. As in subtraction of simple numbers, Ex. 1. From 12£. 9s. 6d. 3qr. take 8£. 7s. 9d. 2qr. We take 2qr. from 3qr. and have Iqr. remaining, which we write under the qr. in the sub- trahend. We see that we cannot take 9d. from 6d., we therefore borrow one from the 9 shillings, and reduce it to pence, which with the 6d. in the examples ISd, We now say 9d. from 18d. leave 9d. which we write in its proper place, under the pence in the example. Now, as one 181. Rule for subtraction of compound numbers ? Principle ? Proof? OPERATION. £ s. d. qr. 12 9 6 3 8 7 9 2 Sum, 4 19 1 Proof, 12 9 6 3 154 COMPOUND NUMBERS. of the shillings has been borrowed, we say, 7d. from 8d., or what is practically the same, 8d. from 9d. leave Id , and so proceed through the example, 2. From S£ 5s. 7d. Iqr., take 3£ 12s. 4d. 3qr. Ans. 4£ 13s. 2d. 2far. 3. 4. t. cwt. qr. lb. lb. oz. dr. sc. grs. Min., 19 12 1 20 13 5 3 1 10 Sub., 13 17 3 22 7 9 12 17 Kern., 5 14 5. 1 23 5 8 11 6. 13 yd. qr. na. gal. qt. pt. 9 1 2 29 1 1 3 3 3 13 3 1 7. 8. lb. oz. dwt. gr- mi. fur. rd. yd. ft. 19 6 1 2 10 8 2 21 2 1 12 10 1 7 21 3 7 33 3 2 4 2 2 7 3^ 2 4 2 2 7 4 6 in. Note. A fraction occurring in the answer, may, when reduced, contain a denomination higher or lower than any in the given exam- ple ; as in Ex. 8, the iyd. = 1ft. Gin. The 1ft. added to the 2ft. in the first remainder = 3ft. = 1yd. Add this to the 3yds. and we have 4:mi. 2fur. 27rd. 4yd. Oft. Gin. Explain Ex. 1. SUBTRACTION. 155 10. deg. mi. fur. rd. yd. ft. in. '643321114 3 62 5 37 1 2 7 11. If I cut loyd. oqr. 2na. from a piece of clotli con- taining 31yd. 2qr., bow much will remain? Ans. 15yd. 2qr. 2na. 12. A grocer had a box of sugar containing 15cwt. Iqr. 131bs. After taking out 9cwt. 3c[r. 211bs., bow much remained in the box? Ans. 5cwt. Iqr. 171bs. 13. An invoice of broadcloth, which cost 187£ 17s. 6d., was sold for 257£ 9s. 3d.; what was the gain? Ans. 69£ lis. 9d. 14. What is the difference in the longitude of two places, one 63° 30' 15" east, and the other 23° 45' 30" east? 18S. To find the time between two dates. Ex. 1. What is the difference of time between June, 11, 1856, and Oct. 4, 1859? OPERATION. In subtracting an Min., 18 5 9 10 4 earlier from a later Sub., 1856 6 11 date, we call 30 Kem., 3y^ 3 mo. 2 3 Ans. days a month. We write first, the number of the year, month, and day of the latest date, and under it, the number of the year, month, and day of the earliest date, and subtract as in Art. 181, and the re- mainder will be the difference of time between the two dates. 2. Find the time from Sept 23, 1862, to May 13, 1866. Ans. 3y. 7mo. 20d. 3. Find the time from Aug. 17, 1858, to June 11, 1863. Ans. 4y. 9 mo. 24d. 156 COMPOUND NUMBERS. 4. Find the time from Feb. 8, 1856, to Aug. 1, 1860. 5. Find the time from March 7, 1857, to Nov. 20, 1865. MULTIPLICATION. 183. In the multiplication of both simple and compound numbers, the multiplier is always a simple abstract number. The product is of the same kind as the multiplicand ; for re- peating a number does not change its nature. The principle is the same as in multiplication of simple numbers. Hence, To multiply a compound by a simple number vre have the following Rule. Multiply the lowest denomination in the multiplicand, divide the product hy the number it takes of that denoinination to make one of the next higher, set down the remainder, add the quotient to the product of the next denomination, and so proceed till all the denominations are multiplied. Peoof. 3Iultiplication and Division of Compound Numhers prove each other, , 2o[r. by 9. We first say, 9 times 2qr. = 1 8qr. = 4d. and 2qr. ; write the 2qr. under the farthings, and then say 9 times 3d. = 27d. and the 4d. added give 31d. := 2 s. and 7d., and so proceed. Note. As raultiplication and division prove each other, it is profit- able to teach the reverse operations simultaneously. 183. How is the time between the two dates found ? 183. What kind of a number is the multiplier in all cases? What the product? The rule for multiplying a compound number ? Proof? Explain Ex. 1. Ex. 1 . Multiply 7£ 63. 3d, OPERATION. £ s. 7 6 d. qr. 3 2 Multiplicand. 9 Multiplier- 65 16 7 2 Product. 2 Multiply By • rd. yd. 9 4 ft. 2 in. 7 8 Product, 7 8 5 2 8 4. lbs. oz. dwt. grs. 6 7 13 17 5 33 2 8 13 6. yd. qr. na. 7 2 3 in. 2 8 8. gal. qt. pt. 9 2 1 gi- 3 . 3 MULTIPLICATION. 157 3. gal. qts. pt. gi. 9 3 13 7 69 3 1 5. lb. oz. dr. so. grs. 3 9 6 2 14 6 . 7. o / It 17 30 45 5 9. bush. pk. qt. 9 3 6 pt. 1 11 10. How much vinegar in 6 casks, each holding 37gal. Iqt. Ipt. 3gi. each? Ans. 224gal. 3qt. Opt. 2gi. 11. What will be the weight of 3 loads of coal, if one load weighs 1 ton, Scwt. 3qr. 2 7 lbs. (long ton) ? 12. What is the entire produce of a field of 8 acres, if one acre produce 38bu. 3pk. 6qts. ? Ans. 311bu. Ipk. 13. If the moon's daily motion through the heavens is 33° 10' 35", how much of her orbit does she traverse in 17 days ? 14. If a horse travel 42mi. 3fur. 37rd. in one day, how far will he travel in 16 days? 1,58 COMPOUND NUMBERS. DIVISION. 184:. Here, as in the three preceding sections, the prin- ciple is the same as in the corresponding operation in simple numbers. Hence, To divide a compound number we have the following BuLE. Divide the highest denomination of the dividend, and set down the quotient ; if there is a remaindery reduce it to the next lower denomination ; to the result add the given nmn- her of that denomination^ and divide as hefore, setting down the quotient and reducing ihe remainder j and so proceed till all the denominations are divided. Proof. Division is proved hy' multiplication. Ex. 1. Divide 27£ 15s. 6d. 3qr. by 8. OPERATION. • "VVe first divide 27£ by 8 and have £ s. d. qr. a quotient of 3, and 3£ remaining. 8)27 15 6 3 3£ reduced to shillings, with the 15 3 9 5 1| shillings in the example, give 75 shil- lings, which divided by 8, give 9 and a remainder of 3s. This we reduce to pence, and add the 6d. in the example, and have 42d., which we divide as before, etc. 2. Divides? tons, 15cwt. Iqr. 211b. by 12. Ans. 3 tons, 2cwt. 3qr. 20ilb. 3. Divide 3 5y. 3mo. 17da. 13h. by 3. 4. Divide 76a. 2r. 25rd. by 5. Ans. 15a. Ir. 13rd. 5. If 5 loads of wood contain 9c. 7c. ft. lOcu. ft., what are the contents of 1 load ? 6. How far will a man travel in one day, if he travel 1 71mi. Ifur. 29rd. in 7 days? Ans. 24mi. 3fur. 27rd. 7. If it take 250yd. 3qr. 2na. of carpeting to carpet nine 184. Rule for dividing a compound number ? Principle? Proof? DIVISION". 159 rooms, how many yards will it take to carpet one of the floors, they being of equal siz* ? Ans. 27yd. 3qr. 2na: 8. If 4doz. spoons weigh 6lbs. lOoz. 16dwt., what will one dozen weigh? Ans. lib. 8oz. 14dwt. 9. A farmer put his wheat, consisting of 359bu. 3pk. 2qt., into 12 boxes of equal size ; how much did each box contain? Ans. 29bu. 3pk. 7qt. Ipt. 10. If24hhd. of sugar weigh 4t. 14cwt. 3qr. 51b., what is the weight of Ihhd. ? Ans. 3cwt. 3qr. 201b. 11. A farmer divided his farm consisting of 446a. 3r. 30 rd., equally among his 8 children; what was the share of each? Miscellaneous Examples. 1. A blacksmith bought 5cwt. 2qr. 211b. of iron at one time, It. 1 Icwt. 181b. at another ; how much did he buy in all? Ans. It. 16cwt. 3qr. 141b. 2. How many pounds of iron did the above blacksmith buy, and what did it cost him at 5 cents per pound ? 3. A farmer raised in one field 302bu. 2pk. 7qt. of oats, in another 290bu. 3pk. 4 qt. ; how much more did he raise in one than the other? Ans. llbu. 3 pk. 3 qt 4. I have a piece of land containing 50a. ; if I sell 25a. 3r. 25rd. of it, how much shall I have left? 5. If a ship sail 2° 2' 30" in one day, how far will she sail in a week? Ans. 14° 17' 30". 6. How much wood in 5 loads, each containing Ic. 3c. fb. 16cu. ft. ? 7. What would be the crop of hay on 10a. if the product of 1a. was 3t. 10 cwt. 2 qr. ? Ans. 35 t. 5 cwt 8. Divide 2t 7 cwt. 2qr. 10 lb. by 7. 160 PERCENTAGE. 9. If 1 cubic yard of stone weigh 2 tons, 7cwt. 251K, what is the weight of 1 cubic foot? • Ans. 1751b. PERCENTAGE. 185. The term Per Cent means by the hundred ; thus, by jive per cent of a ton of coal, we mean five one hun- dredths of it ; i. e. five parts out of every one hundred parts ; 6 per cent of a sum of money, is six one-hundredths of the sum, i. e. $6 out of every $100. Note. Instead of the WGrds per cent it is quite customary in writing to use the sign %\ thus, 6 per cent is written 6 ^; 4i per cent 4:i %. 186. The Rate per cent is the number far each hun- dred; thus, 6 Gjo is ToTT» or .06, i. e. 6 parts for each hundred 18T. The Percentage is the sum computed on the given number; thus, the percentage on % 200 at 6 per cent is $ 12. Note. The pupil should be cautioned not to confound j»cr cent and 'percentage. The distinction should clearly be borne in mind. 188. The Base of percentage is the number on which the percentage is computed ; thus, we say the percentage on % 500, at 8 per cent is % 40. Here $ 500 is the base, 8 is the PER CENT and $ 40 is the percentage ; also, 10 per cent, of 2000 lb. (a ton) of coal is 2001b. ; here 2000 lb. is the hasey 10 is the j»er cent and 2001b. is the 'percentage. 185. Meaning of per cent ? 186. Kate per cent ? Illustrate. 187. Percentage? Illustrate. 188. Base of percentage ? Explain the three last mentioned terms by an example. PERCENTAGE. 161 189. The rate per cent being a certain number of hun- dredths, may be expressed either decimally, or by a common fraction, as in the following TABLE. Decimals. Common Fractions. 1 per cent is .01 = t^^* 2 per cent .02 =: 3^, 5 per cent .05 = ^^, 6i per cent .0625 — yV 8i per cent .08^ =: yV- 10 per cent .10 =: yV 12i per cent .125 — \. 16f per cent .16| = f 181 per cent .1875 — yV 20 percent .20 = |. 25 per cent .25 = \. 33i per cent .33^ = \. 60 per cent .50 = i. etc. etc. Note. When the per cent is expressed by a decimal of more than 2 places, the figures after the second decimal place must be regarded as parts of 1 per cent ; thus, (in the seventh line of the fore- going table,) .125 is 12i^ or 12^ per cent. Ex. 1. Write the decimal for 6 per cent. Ans. .06. 2. Write the decimal for 4 per cent; 12 per cent; 8 per cent; 15 per cent; 25 percent; 16J per cent. 3. Write the common fraction for 5 per cent; 10 per cent; 12 J per cent ; 6i per cent; 33 J per cent. 189. In what ways may the rate be expressed ? If expressed decimally by more than two figures, what are the figures after the second decimal place ? 162 PERCENTAGE. Note. Too much pains can not be taken to make the pupil thorough in exercises like those in the last two examples. Case 1. 190. To find the percentage, the base and rate per cent being given. Ex. 1. John Dow had $ 360, hut lost 5 per cent of it, how many dollars did he lose ? $ 3 6 Since 5 per cent is .05 = ^i^, we find the loss, 'Q ^ (percentage), by multiplying $360 by .05 or by $ 1 8.0 s^j. Hence, the KuLE. 3Iultiplt/ the base hy the rate 'per cent expressed decimally or as a common fraction, and the product will he the percentage. 2. What is 20 per cent of $ IGO. ? Ans. $ 32. 3. The base is 560 and the rate per cent 40 ; what is the percentage ? 560 X .40 = 224, Ans. Or, 560 X f = 224, Ans. 4r. What is 16| per cent of 180 barrels of flour? Ans. 30 bbl. 5. What is 5 % of $ 200 ? Ans. % 10. 6. What is 8i % of 240 tons of coal? Ans. 20 tons. 7. In a certain school there are 720 pupils, 33| per cent are more than 12 years of age; how many are over 12 years old ? Ans. 240. 190. Rule for finding the percentage when the base and rate are given ? PERCENTAGE. 163 8. A flour merchant bought 1200 bbls. of flour, but 8 per cent of it was injured by rain ; how much was injured? 9. A pupil had a lesson of 40 words, but failed on 10 per cent of them ; on how many words did he fail ? 10. A city containing 35000 inhabitants, had 15 per cent of the number in school children ; how many school children were there ? An s. 5250. 11. A merchant fails in business, owing $ 12G00, and can pay but 35 per cent of his debts ; how much will his credi- tors lose ? 100 — 35 =65 12600 X. 6 5 =$8 19 0, Ans. 12. Bought 600 boxes of oranges, but on opening them, I find 8 % of them spoiled ; how many were lost? Ans. 48. 13. A gentleman sold his house for $6500. $600 he re- ceived in cash and took a note for the balance ; how much cash did he receive ? 14. I have $ 1580 on deposit in the bank. If I draw out 12 J per cent of it, what per cent will remain ? What amount of money will remain? Last Ans. $1382.50. 15. In an orchard consisting of 1200 trees, 30 % bear apples, 45 % bear pears, and the remainder bear peaches ; how many bear peaches? Ans. 300 trees. Case 2. 191 • To find the rate per cent when the base and percentage are given. Ex. 1. What per cent of 8 24 is $ 6. ? S 6 is i of $ 24 and I re^ 2^i =: J = .2 5, Ans. duced to a decimal = .25 L e. $ 6 = 25 per cent of $ 24. Hence tho 164 PERCENTAGE. Rule. Mahe the 'percentage the numerator of a common fraction and the base the denominator^ and then reduce this fraction to a decimal, 2. "What per cent of $ 20 is $ 5 ? Ans. .25. 3. What per cent of $ 400 is ^ 50 ? Ans. .12^-. 4. A man having $ 6000, paid % 1200 for a piece of land ; what per cent of his money did he expend? Ans. .20. 5. My salary is $1800 per annum, and my expenses $ 1600 ; what per cent of my income do I spend? What per cent save ? 6. Bought a cask of vinegar containing 84 gallons; 21 gallons have leaked out ; what per cent have I lost ? 7. Purchased a horse for $ 160, and sold him for $ 128, what per cent did I lose ? Case 3. 193. To find the base when the percentage and the rate are given. Ex. 1. $ 12 is 4 per cent of what sum? Ans. | 300. If $ 12 is 4 per cent, 1 per cent will be | of $ 12 which is I 3, and if $ 3 is 1 per cent, 100 per cent will he 100 times $ 3 =: S 300. The same result is obtained by multiplying by 100 first, and then dividing by 4; thus, 1200-^4 = 300. Hence, EuLE. Multiply the percentage by 100, and divide the prod- uct by the rate, and the quotient will he the base. 2. $ 12.60 is 6 % of what sum? Ans. $ 210. 3. $ 15 is 8 % of what sum? Ans. $ 187.50. ' 191. Rule for finding the rate per cent when the base and percentage are given ? 19'^. Kule for finding the base when the percentage and rate are given ? INTEREST. 165 4. $36.30 is 3 % of what sum? 5. $ 75 is 6 % of what sum ? Ans. $ 1250. 6. $ 12 is 4 % of what sum ? Ans. $ 300. 7. A gentleman purchased a farm for $ 6900, which was 20 per cent of his entire property. What was he worth ? 8. James Marvin cures fish for Thomas Tarlton, receiv- ing in pay 12^ per cent of the quantity cured. His share this season is 46801bs ; what quantity did he cure? 9. A merchant saves $3000 annually, which is 16| per cent of his entire receipts; what arc his receipts? Ans. $18000. 10. A farmer sold 56 sheep, which was 12^ per cent of his whole flock. How many sheep had ho in all ? Ans. 448. INTEEEST. 103* Interest is money paidybr the use of money. The Principal is the sum for which interest is paid. The Amount is the sum of the principal and interest. 194:. An example in interest is only a question in per- centage. HhQ principal is the base of percentage, the interest is the percentage, and the interest on $1 for a year is the rate written decimally. 100. The rate is usually Jixed hy law, and a higher rate than the law allows is called usury. In New England and most of the United States the legal or lawful rate is 6 per cent ; in New York, 7 per cent. Note. In Massachusetts a higher rate than six per cent may legally be agreed upon. In this book 6 per cent is understood when no per cent is mentioned. 193. What is Interest ? What is the Principal ? What is the Amount f 194. What is said of an example in Interest ? Explain how the latter is like one in percentage. 195. What is the Rate ? The legal Rate ? 166 PERCENTAGE. 100. To find the interest on any sum at 6 per cent for a given time. Ex. 1 . What is the interest of $ 240, for one year 6 months ? OPERATION. ^240. Principah Interest for 2 mo. Interest for 1 mo. Months z= ly. 6 mo. 2) 2.4 0, 1.2 0, 1 8 960 120 ANALYSIS. Any sum of money at 6 % will gain jwo of it- self in 2 months. AVe find this hy removing the decimal point two places to the left, that is, by di- ^21.60, Interest for 18 mo. viding by one hundred. We then divide this interest by 2 which gives the interest for one month. This, we multiply by the number expressing the given time in months, and have the interest of $ 240 for ly. 6mo. = $21.60. Ex. 2. What is the interest of $ 244.40 for 1 year 4 mo. 12 days? OPERATION. $2 44.40, Principal. 2 ) 2 .4 4 4 =r Interest for 2 mo. 1.2 22 = Interest for 1 mo. 1 G.4 mo. = ly. 4mo. 12d. ANALYSIS. In this example we proceed as in Ex. 1, till we come to the days. 12 days = -J^ of a month, which we reduce to a decimal = .4 and annex to the 16 months, and $2 0. 0408 = Int. for IG^V ^^- multiply as before. Hence to find the interest on any sum for any time at 6 per cent we have the following Rule. Bemove the decimal point in the principal^ two places to the left ; divide this result hy two and multiply this 196. How do you find the interest on any sum for any time at 6 per cent ? Expiaiu the stepe. 4888 T332 1 222 INTEREST. 167 quotient hy the time in months and tenths of a month {the days divided hy ^ =z tenths of a month), and the product will be the interest at 6 per cent for the given time, Ex. 3. What is the interest of $ 46 for 1 month? $46. -^ 100 = 46, and .46 -^ 2 zrr $ .23, Ans. 4. What is the interest of $ 246.58 for 1 month? $ 246.58 -^ 100 = 2.4658, -^ 2 = $ 1.2329. 5. What is the interest of S 4, for 1 month ? $ 4 -^- 100 =: .04, -^2 = .02, Ans. 6. What is the interest of $ 1, for 1 month? $ 1. _i- 100 = .01, -^2 — .005, Ans. 7. What is the interest of $ 56.^98, for 1 month? 8. What is the interest of $ 864.25, for 1 month? Ans. $4.32125. 9. What is the interest of $ 69.42, for 1 month ? 10. What is the interest of S 2468.20, for 1 month? 11. What is the interest of $59,278, for 1 month? 12. By what will you multiply the interest for one month to find the interest for 1 year, 4 months ? ly. 4mo. =: 16mo. Ans. 16. 13. By what will you multiply to find the interest for 2 years, 7 months? 14. By what will you multiply to find the interest for 15 days? 15d. = -^^mo. =^mo. =.5mo. Ans. .5. 15. By what will you multiply to find the interest for 21 days? 16. By what will you multiply to find the interest for 22 days? 22d. = ffmo. = .7jmo. Ans. .Zj. 17. By what will you multiply to find the interest for 9 months, 13 days? 9rao. 13d. = 9|§mo. :;:=: 9.4jmo. Ans. 9.4J. 168 PERCENTAGE. 18. By what will you multiply to find the interest for ly. 7mo. 19d.? 19. By what will you multiply to find the interest for 2y. 9mo. 20d. ? 20. What is the interest of S 164.30, for 18 days? 164.30 -^ 100 = 1.6430, -^ 2 z= .8215, X .Q~$A929, 21. What is the interest of $ 58.64, for 2y. 3mo. 21d. ? $ 58.64 -^ by 100 = .5864, -^ 2 = .2932, X 27.7 = $8.12+. 22. Find the interest on $ 619.28, for 3y. 4mo. 26 days. Ans. $132.66. 23. Find the interest on $ 384.92, for 2y. 5 mo. 27 days. 24. Find the interest on $ 87.25, for ly. Imo. 10 days. Ans. $5,816. 25. What is the amount of $ 142.80, for ly. 6mo. 24 days? Ans. $156,223. Note. The amount is the sum of the principal and interest added together. 26. What is the amount of $ 234.60, for 6y. 7mo. 8 days? Ans. $ 327.57+. 27. What is the amount of $ 104.20, for 6mo. 19 days? Ans. $ 107,655+. 28. What will $ 380.50 amount to in ly. 5rao. 10 days? 29. Find the interest of $ 60, for 60 days. Ans. $ .60. 30. Find the interest of $ 30, for 90 days. 31. Find the interest of $240.60, for 2y. llmo. 28d. Ans. 8 43.227. 32. Find the amount of $ 350, for 3y. 7mo. lOd. 33. Find the interest on $ 3, for 7 days. Ans. .0035. 34. Find the interest on $ .80 for 10 days. Ans. .0013i. 35. Find the interest on $2.42, for 25 days. INTEREST. 169 36. Find the amount of $ 5, for 12 days. Ans. $ 5.01. 37. rind the amount of $ 75.60, for 8mo. 29d. Ans. $78,989. 38. Find the amount of $ 3000, for 2y. 8mo. 39. Find the amount of $ 230, for 7y. 6mo. Ans. $333.50. 40. Find the interest of $ 394.27, for 8y. 23d. 41. Find the amount of $ 6000, for 9y. 7mo. 23d. 197, In all the previous examples, the interest has been computed on the basis of 6 per cent. In Ex. 42, we first cast the interest at 6 per cent, as before, and find it to be $1 7.001 ; this we divide by 6, which gives the interest at 1 per cent; and lastly, we multiply this interest by 7 and have the interest at 7 per cent = $ 19.83+. 42. What is the interest of $ 1 64, for ly. 8mo. 22d. at 7 percent? Ans. $19.83+. 43. \Yhat is the interest of $270.60, for 3y. llmo. 19d. at 8 per cent? Ans. $ 85.93. 44. What is the interest of $ 492.75, for 2y. 7mo. at 7i per cent ? 45. Find the interest on $ 75.87, for 5y. 3mo. at 9 percent. Ans, $ 35.848. 46. Find the interest on $ 894.20, for 3y. 6mo. 15d. at 8 per cent. Ans. S 253.35.+ 47. Find the amount of $382.85, for 4y. lOmo. at 9 per cent. 48. Find the amount of $ 69.47, for 3y. 8mo. at 6^ per cent. Ans. $ 86.02. 197. How is interest found for any other rate than 6 per cent ? Esplaia the steps. How is the difference of time between two dates found ? 170 PERCENTAGE. 49. Find the interest of $609.42, for 8y, 7mo. 6d. at 4 percent. Ans. $209.64. 50. Find the interest on $ 493.85, for 2y. 3mo. at 3^ per cent. Ans. $38.89. 51. Find the interest on $6000, for 87. 7mo. 17d. at SJ per cent. 52. What is the interest of $ 64.82 from June 24, 1856, to >Oct. 9, 1859? Ans. $ 12.80. Note. First find the difference of time = 3y. 3rn. 15d. 53. What is the interest of $ 85.93 from Jan. 6, 1850, to June 1, 1854? 54. What is the interest of $ 942.87 from Aug 13, 1861, to Nov. 7, 1864? Ans. $182.91. 55. What is the interest of $ 293.80 from Feb. 19, 1860, to Sept. 4, 1863 ? Ans. $ 62.43. bQ. What will $ 843.92 amount to, from Aug. 28, 1862, to Jan. 1, 1866? Ans. $1013.12. 57. What is the interest of $ 59.75 from Dec. 29, 1858, to June 7, 1861, at 7t per cent ? Ans. $ 10.92. 58. What will be the amount of $ 642.90 from July 4, 1862, to the day of Gen. Lee's surrender? 59. Find the interest of $ 8942, from the fall of Fort Sumpter to the evacuation of Eichmond ? PEOFIT AND LOSS. 198. Profit and Loss, are commercial terms, used to indicate the gain or loss in buying and selling goods, and in business transactions generally. 198. "What is Profit and Loss ? PROFIT AND LOSS. 171 Case 1. 199, To find the absolute gain or loss on a quan- tity of goods sold at retail, the purchase price of the whole quantity being given. Ex. 1. Bought 2001b. of coffee for $ 50 and sold it af S .40 per lb.; how much did I gain on the whole ? § 200. X .40 == $ 80.; 80 — 50. z= 80. Ans. | 30. Hence the EuLE. Find the whole sum received for the goods, and the difference between this and the purchase price will he the gain or loss. 2. -Bought 32 yards of cloth for % 48 and sold it at $ 2.20 per yard; what was my whole gain? Ans. S 22.40. 3 Bought 4cwt. 3qr. 161b. sugar for $61,375, and sold it at $ .20 per lb. How much did I make on the whole ? 4. Bought 960 oranges for % 19.20 and sold them for 5 cents apiece ; what was my entire gain ? 5. Bought 15 doz. pencils at $ 1.20 per doz. and sold them at $.15 each; what did I make ? Case 2. 300. To find the per cent of gain or loss when the cost and selling price are given. Ex. 1. Bought 12 yards of cloth for $ 18. and sold it at $ 2.50 per yd. What was my whole gain, and what my gain per cent? 199. Rule for finding absolute gain or loss ? aOO. Kule for finding the per cent of gain or loss on the cost price ? 172 PERCENTAGE. $ 2.5 Selling price. "We first find the whole gain to bo , L? y^'^^^- $ 12 which is if — § of the cost. 3 0.0 = Sum rec'd. This we reduce to a decimal, and IQ =Cost. have .66|; i. e., the gain $12, is $12 = Whole gain. 66 f per cent of the cost $ 18. Hence, J| = I =3 .66|. the following. EuLE. Having found the total gain or loss hij Case 1, make a common fraction hy writing the gain or loss for the numerator and the cost of the article for the denominator, and then reduce this fraction to a decimal. 2. Bought a gross of steel pens for $ 1.20, and sold them at 1 cent apiece. What was my gain per cent ? Ans. 20%. 3. I purchased 1 bushel of cherries for $ 3, and sold them at 15 cents per quart. What was my gain per cent? 4. Sold a horse for $ 100 which cost mo $ 150. What was my loss per cent ? / . Case 3. 301. To find the selling price, the cost and gain or loss per cent being given. Ex. 1. Bought cloth at $ 6 per yard and wish to sell it at an advance of 20%. What is my selling price? $ 6.0 I shall sell what cost me $ 1. for $ 1.20 1-2 and what cost me $ 6 for six times $ 1.20 $7.2 0, Ans. r= 7.20. Ex. 2. Bought goods for $ 300, but they being damaged I am willing to sacrifice 15 % of their value ; for what shall I sell them? '^01. Ilule to find the selling price, the cost and gain or loss per cent bein^j given ? PROFIT AND LOSS. 173 $ 3 0. If I lose 15%, I shall sell what •8 5 cost me S 1. for S. 85, and shall sell 15 what cost me $ 300. for 300 times 2400 $.85=8 255. $ 2 5 5.0 Hence, we have the following Rule. Multiply the cost by 1 minus the loss or plus the GAIN PER CENT, and the product will be the selling price. 3. For what must I sell sugar that cost me 15 cents per pound, to gain 30% ? Ans. $ .19^. 4. Bought a pair of skates for $ 4.00 ; for how much must I sell them to gain 10% ? Case 4. dO^. To find the first cost of an article, the selling price and gain or loss per cent being given. Ex. 1. Sold wine at $6 per gal. and by so doing I made 20 per cent on the cost ; what was the cost ? That which cost $ 1. sold for S 1.20, 12.^ zzr I therefore, the cost was j^^o z= |. of the $6.Xf^=S5.=: cost, selling price ; henee, the cost was f of $6. = $5. Hence v^^e have the following Eule. Make a fraction hy writing 100 /or tlw numerator and 100 minus the loss or plus the gain per cent for a denomi- nator, then multiply tlie selling price by this fraction. 2. Flour selling at S 15 perbbl., yields a profit of 25%; what is the cost? Ans. $ 12, 3. Sold 8 yards of cassimere at $3 per yd., and made 20% by the sale; what was the cost? Ans. $2.50 per yard. 4. Bought a quantity of wheat, but it being damaged, I sell it at % 1.50 per bushel, and by so doing lose 25 per cent on the cost ; what was the cost ? Ans. 8 2.00. 202. Rule for findinjf the cost, the selling price and gain or loss being given ? 174 MISCELLANEOUS. MISCELLANEOUS EXAMPLES. 1. Add 120 X 2 to 972-^36. Ans. 267. 2. The difference between two numbers is 16, and the larger is 92 ; what is the smaller ? 3. What is the difference between 2446352 4-694701 and 2146705 — 8392-1:1? 4. What is the difference between 246001 X 1641 and 245897 X 321 ? Ans. 324754704. 5. Multiply 12948 by 287, subtract 58672, and divide the remainder by 218. 6. How many cords in a pile of wood 86 ft. long, 4 ft. wide, and 9 ft. high? Ans. 24^^^. 7. Divide | of f by | of J-^. Ans. h 8. What number multiplied by 8| gives 205 ? 9. What number divided by 19f gives 36 ? *Ans. 708. 10. At S10§ per ton, what will be the cost of | of a ton of coal? 11. If If yards of cloth are required for 1 coat, how many coats may be made from 2 2f yards? Ans. 13. 12. What is the difference between eight hundred thousand and eight hundred-thousandths? Ans. 799999.99992. 13. If 18 gentlemen have $646.70 each, what sum have they all? Ans. $11640.60. 14. Bought 19 bbls. of flour for $261.25; what was the price per bbl.? 15. If the crop of hay on 1 acre is 1 ton, 18 cwt. 3 qr. 15 lbs. what will be the crop on 10 acres? Ans. 19 t. 9 cwt. 16. The population of a certain city is 29460; what will it be a year hence if it gains 5 % ? MISCELLANEOUS . 175 17. If a farmer raises 2650 bushels of wheat one year and 2968 the next ; what per cent did his crop increase ? Ans. 1 2 per cent. .18. A man with a salary of $1600, spends $1200 ; what per cent of his salary did he save ? 19. What is the interest of $1.00 for 8 yrs 8 mos. and 6 days? Ans. $0,521. 20. What is the interest of $500.00 for 60 days, at 7y3^%? 21. Find the interest of $460 for 2 yrs. 4 mos. and 1 8 days. Ans. $65.78. 22. What is the interest of $84.75 for 10 yrs. 6 mos. 24 days, at 9% ? Ans. $80,597. 23. The credit side of an account is composed of the fol- lowing items: $1500.75, $655.30, $175,875; what is the whole amount ? 24. The debit side of the above account has the following items: $576.37, $1025.50, and $1850.00; what is the whole amount ? 25. On which side of the above account is the balance, and how much is it? Ans. debit; $1119.945. 26. How many yards of cloth at $2.25 per yard should be received for 8 cords of wood at $6.20 per cord ? Ans. 22/^ yds. 27. In 1 week, 1 day, 16 hours, and 40 min. how many seconds? Ans. 751200. 28. Resolve 1820 into its prime factors. 29. Add f , |, }, and f Ans. 2|f. 30. Bought a house-lot 120 ft. long and 90 ft. wide, at 10 cts. per square foot. The cost of the lot was 12% of the cost of the house ; what was the cost of both ? Ans. $10080.00. 31. If 12 bbls. flour cost $135.00, what will 50 bbls. cost? 32. Divide four thousand by eight thousandths. 176 MISCELLANEOUS. 33. If yS^ of a vessel cost ^3690, what will f cost ? Ans. $6396. 34. What is the interest of $1245.60 from Jan. 20, 1866, to May 2, 1867? 35. The population of the New England States, in round numbers* is as follows: Maine 628000, New Hampshire 326000, Vermont 315000, Massachusetts 1231000, Rhode Island 175000, Connecticut 460000 ; what is the entire population of New England ? 36. Bought corn at $2.00 a bag, and sold it for $1.60 ; how much did I lose per cent ? Ans. 20%. 37. If 1 ton of coal costs $7.75, how many tons can be bought for $147.25? 38. Bought a pile of wood 40 ft. long, 12 ft wide, and 15 ft. high, at $6.00 a cord ; how many cords were there, and what was the expense ? Ans. 56 J cords, and $337.50. 39. Eind the greatest common divisor of 1504 and 3478. 40. Eind the least common multiple of 6, 12, 18, 36, and 54. 41. If it cost $25.00 for 1000 miles travel, what is that a mile ? Ans. 2| cents. 42. Bought a house for $1800.00, which was f of what I paid for my farm ; what was the cost of the farm ? Ans. $4200.00, 43. A manufacturer sends 34 cases of shoes to a com- mission merchant. These are sold at $140 a case. The commission is 2i per cent. What is the whole amount of the commission received, and how much is due the manu- facturer ? 44. Multiply 25893 by .000402. Ans. 10.408986. - 45. If I give .35 of a cord of wood for 1 day's work, how much should I give for 64.50 days' work? How much would it be worth at $5.00 a cord? MISCELLANEOUS. 177 46. What is the interest of $800.50 from June 19, 1860, to Nov. 4, 1865 ? If the above interest is payable in gold, at a premium of 40%, what would be the current value? 47. Sold a horse and carriage and lost $50 by the trans- action. This was 10% of the cost What was the cost and selling price? Ans. $500.00, cost; $450.00, selling price. 48. If $200 gain $24 in 2 years, what will $50 gain in the same time ? 49. Bought 15 shares of bank stock for $1650, and sold them at an advance of $5.50 a share. What per cent did I gain? Ans. 5%. 50. Bought 160 acres of land for $15 an acre. Sold 10 house-lots at $50 each, a quantity of lumber and wood for $1725, and the remainder of the land for $1550. What did I gain ? THE METEIC SYSTEM or WEiaHTS AND MEASURES. d03. In the Metric System the Scales are all decimal as in United States Money. It is so named from the Meter, which is one ten-millionth of a quadrant, or one forty-millionth of the circumference of the earth measured over the poles. Long Measure. S04:. The principal unit of length is the Meter ^ which is 39.31 inches long. i403 . What is said of the scales of the Metric System ? Why is this System 80 called f iJ04. Whatis the principal unit of Long Measure? What is its length? 178 THE METRIC SYSTEM. TABLE. 10 Millimeters (™'") make 1 Centimeter. 10 Centimeters 1 Decimeter. 10 Decimeters 1 METEE n. 10 Meters 1 Dekameter. 10 Dekameters 1 Hectometer. 10 Hectometers 1 Kilometer C^'"). Centiin. Miu. Dccim. 1= 10 Meters. 1 = 10= 100 Dekam. 1 = 10 = 100= 1,000 Hectom. 1 = 1 Z= 100 = 1,000= 10,000 K„. 1= 10= 100 = 1,000 = 10,000= 100,000 1 =10=100 = 1,000 = 10,000 = 100,000=1,000,000 Note 1. About twenty-five (more exactly 25.4) millimeters make one inch. The meter is about three feet, three inches, and three-eighths of an inch, which may be remembered as the rule of the three threes. Note 2. The kilometer is the common unit for road measure' and is about two hundred rods, or five-eighths of a mile. Five meters make about one rod. The accompanying scale exhibits one decimeter divided into ten centimeters, each centimeter being divided into ten milli- meters. With it is a four-inch scale divided into eighths of an inch. Give the Table of Long Measure. What is the Common Unit of Road Meas- ure ? Its length ? Draw a section of the accompanying scale, and explain it. THE :.IETRIC SYSTEM. 179 These measures, as well as all the other metric measures and weights, are written like whole numbers and deci- mals. Thus, 3 kilometers, 8 hectome- ters, 7 meters, and 5 decimeters, are written 3807. 5°^. Large distances, as in road measure, are given as kilome- ters and decimals. Thus, 47.34'"'" stands for 4 myriameters, 7 kilometers, 3 hectometers, and 4 dekameters. Small distances are usually expressed in millimeters, or in centimeters. The names of the several larger units of length are formed from the word Meter, by prefixing Myria for 10,000, Kilo for 1000, Hecto for 100, and Deka for 10. The smaller units are denoted by Deci for yV* G^nti for y^^, and Milli for -t^jVu- I^ ^^ same way, as will be seen hereafter, are formed the names of weights and of measures of surface and capacity. Note 1. The first series of prefixes is from the Greek, the second from the Latin language. Note 2. The terms Dime, Cent, and Mill, in United States money, for the tenth, hundredth, and thou- sandth parts of a dollar, are analogous to the terms Decimeter, Centimeter, and Millimeter. Note 3. The Metric System is used in France and many other countries, and is legalized in the United States and Great Britain. How are these weights and measures written ? Illustrate. What prefixes indicate the larger denominations ? What the smaller ? o Q CO M^ o E H H ::; 2 o, — I H - H —I f - ~ 4 - p « — E w o * 1 180 THE METRIC SYSTEM. ^Oo. To reduce ti larger denomination in the Met- ric Syitem to a smaller, or a smaller to a larger : Multiply or divide hy 10, 100, 1000, ^c, as the case may require. {Art, 159.) Ex. 1. Keduce 64 meters to millimeters. !■" = 1000™™ 64" — 1000™" X 64= 64000™™. Ans. 2. Ecduce 8500 millimeters to meters. Ads. 8.5™. 3. Keduce 95000 meters to kilometers. Ans. 95^™. 906. Metric measuies and weights are added, subtracted, multiplied and divided like whole numbers and decimals. Ex. 1. Add 4.5™, 26.25™ and 9450™™. Ans. 40.2™. 2. From 978™ take 392.64™. Ans. 585.36™. iJ. Multiply 736.45™ by 7. , 4. Divide 1840.86™ by 63. Ans. 29.22™. f Square Measure. d07. The principal units of square measure are the Are and the Square Meter. The Are is a square whose side is 10 meters, and therefore contains 100 square meters. TABLE. 100 Sq. Centimeters make I Sq. Decimeter. 100 Sq. Decimeters " 1 Centare, or sq. meter. 100 Centares, or sq. meters '* 1 Are (*"■). 100 Ares '* 1 Hectare C»). 100 Hectares '* 1 Sq. Kilometer. Sq. Decim. Sq. Centini. Sq. Meters, \ IQO or Centurus. — Ares. 1 _ 100 — 10,000 1 — 100 - 10,000 — 1,000,000 8q. 1 _ 100 — 10,000 =: 1,000,000 — 100,000,000 1 = 100 - 10,000 — 1,000,000 — 100,000,000 — 10,000,000,000 ;i05. How is Reduction performed? ^06. How are these measures added, subtracted &c. ? 307. What are the principal units of square measure? Give the Table. THE METRIC SYSTEM. 181 Note 1. The hectare, which is a common unit for land measure, is a square whose side is a hundred meters ; hence it is equal to 10,000 square meters. It is 2.471 acres. Notes 2. Since the scale in square measure is 100 (two dimen- sions, 10 X 10), there will be two figures for each denomination. Thus, 25 hectares, 7 ares, 17 centares, and 20 square decimeters, would be written 2507.172 ares, or 250717.2 square meters. Ex. 1. Reduce 15 hectares to sq. meters. Ans. 150,000. 2. Reduce 456000 sq. decimeters to ares. Ans. 45.6. 3. Reduce 78 kilometers to ares. 4. Reduce 9624 ares to hectares. Ans. 96.24. 5. In a field 300 meters long and 78 meters wide, how many ares ? Ans. 234. 6. How many hectares in a field 275 meters long and 500 meters wide? Ans. 13.75. Cubic Measure. 208. The principal unit of cubic measure is the Cubic Meter or Stere, It is 1.308 cubic yards. TABLE. 1000 Cub. Centimeters make 1 Cub. Decimeter, or Z*Ver. 1000 Cub. Decimeters " 1 Cub. Meter, or Stere C*). The tenth part of the Stere is the Decistere, and ten Steres make a Dekastcre. Note. Since the scale is a thousand (three dimensions, 10 x 10 X 10), three figures will be required for each denomination. 207. The common unit for land measure ? Equal to how may acres ? How many figures required for each denomination ? Why ? 308. What is the principal unit of cubic measure ? Equal to how many cubic yards ? Give the Table. How many figures required for each denomina- tion? Why? 182 THE METRIC SYSTEM. Ex. 1. Reduce 12 cubic meters to cubic decimeters. Ans. 12000. Reduce 41,930,000 cubic centimeters to cubic meters or 2. stores. 3. 4. 5. 6. Ans. 41.93^*. Ans. 290. Reduce 29 steres to decisteres. Reduce 495 steres to dekasteres. Reduce 5230 decisteres to steres. Ans. 523. In a pile of wood 10 meters long, 1 meter wide and 3 meters high, how many steres ? Ans. 30. 7. How many cubic meters in a box 3 meters long, 1.25"" wide and 1.2™ deep? Dry and Liquid Measure. d09. The principal unit, both for Dry and for Liquid Measure, is the Cubic Decimeter, or Liter, It is a little larger than a wine quart. TABLE. 10 Milliliters, or cub. centimeters, mako 1 Centiliter (°^). 10 Centiliters " 1 Deciliter. 10 Deciliters " 1 Liter Q, °'-<=»b<^«">"iter. 10 Liters " 1 Dekaliter. 10 Dekaliters " 1 Hectoliter (^^). 10 Hectoliters " 1 Kiloliter, o^cub. meter. Centiliters. Milliliters. Deciliters. 1 ^= 10 1= 10 = 100 Delcaliters. 1== 10= 100 = 1,000 1= 10= 100= 1,000 = 10,000 Cnb. Meter. 1= 10= 100= 1,000= 10,000 = 100,000 1 - - 10 = 100 = 1,000 = 10,000 = 100,000 = 1,000,000 309. What is the principal unit of Dry and Liquid Measure ? How does it compare with a wine quart ? Give the Table. THE METRIC SYSTEM. 183 Note 1. These measures are usually written as liters and decimal parts; or as hectoliters and decimal parts. Thus, 2 kiloliters, 7 hectoliters, 7 liters, and 5 deciliters are written 27.075^, or 2707.51. NoTB 2. The calculation of the contents of a bin or cistern, is very simple in the metric system. The product of the length, breadth and thickness in decimeters, gives the contents in liters. Ex. 1. Eeduce 24000 milliliters to liters. Ans, 24. 2. Eeduce 3560 liters to hectoliters. Ans. 35.6. 3. Eeduce .917 liters to centiliters. 4. Kow many liters in a tank, 4.5™ long, 3™ wide, and 2.5"" deep ? 45 X 30 X 25 = 33750. Ans. 5. Eeduce 61250 liters to cubic meters. Ans. 61.25. 6. What is the value of a hectoliter of molasses at 45 cts. a liter? Ans. $45.00. Weight. dlO. The principal units of weight are the Gram and the Kilogram. The Kilogram is the weight of a liter of water, and is a little more than 2-^ pounds avoirdupois. TABLE. 10 Milligrams ('"^) make 1 Centigram. 10 Centigrams " 1 Decigram. 10 Decigrams " 1 Gram (^). 10 Grams *♦ 1 Dekagram. 10 Dekagrams •* 1 Hectogram. 10 Hectograms " 1 Kilogram, or Kilo, (^^), 10 Kilograms ** 1 Myriagram. 10 Myriagrams " 1 Quintal. 10 Quintals " 1 TmineauQ), 309. How are these measures usually written? 310. What are the principal units of weight? What is a kilogram? Equal to how many pounds avoirdupois ? Give the Table. 184 THE METRIC SYSTEM, OentJgr. MilHgr. Deoigr. 1= 10 Oiuni. 1= 10= 100 Dektgr. 1= 10= 100= 1000 Heotogr. 1= 10= 100= 1000= 10,000 Kilogr. 1= 10= 100= 1000= 10,000= 100,000 Myriagt. 1= 10= 100= 1000= 10,000= 100,000= 1,000,000 Quta. 1= 10= 100= 1000= 10,000= 100,000= 1,000,000— 10,000,000 ,. 1— 10— 100— 1000= 10,000= 100,000= 1,000,000= 10,000,000= 100,000,000 1=10=100=1000=10,000=100,000=1,000,000=10,000,000=100,000,000=1,000,000,000 Note. A cubic centimeter of water weighs a gram, and a cubic meter of water weighs a tonneau. The kilogram is often called kilo, for brevity. Ex. 1. Eeduce 64.73 kilos to grams. Ans. 64730. 2. Eeduce 7490 kilos to tonneaus. Ans. 7.49. 3. Eeduce 28500 milligrams to grams. Ans. 28.6. 4. "What is the weight of 36 liters of water? Ans. 36=^^. 5. If 2^^^' of sugar cost 80 cts. what will be the expense of le'^*^- at the same rate? Ans. $6.40 6. If o5e tonneau of coal cost $10, what will 7^ tonneaus cost? The names of the metric weights and measures are formed accord- ing to a simple law, as will be seen by inspection of the following scheme : Lengths. Surfaces. Capacities. Weights. Eatios. Myria - meter. Myria - gram. Kilo - meter. Kilo - liter. Kilo - gram. Hecto - meter. Hect - are. Hecto - liter. Hecto - gram. Deka -meter. Meter. Are. Deci - meter. Centi - meter. Cent - are. Milli -meter. Deka - liter. Deka - gram. Liter. Gram. Deci - liter. Deci - gram. Centi - liter. Centi - gram. Milli - liter. Milli - gram. 10000 1000 100 10 1 1 TTnTTT THE METRIC SYSTEM. 185 311. The following equivalents of the metric measures and weights have been established by Congress for use in all legal proceedings : Measures op Length. METRIC r)ENO> VAL nXATIONS AND UES. EQUIVALENTS IN DENOMINATIONS IN USE. Myriameter . . 10,000 meters . . 6.2137 miles. Kilometer . . . 1,000 meters . . 0.62137 mile, or 3,280 feet and 10 in. Hectometer . . 100 meters . . 328 feet and 1 inch. Dekameter . . 10 meters . . 393.7 inches. Meter .... 1 meter . . . 39.37 inches. Decimeter . . . ytj. of a meter 3.937 inches. Centimeter . . ^^^ of a meter 0.3937 inch. Millimeter . . . TT)Vo-o^»™^*«'' 0.0394 inch. Measukes op Surface. METRIC DENOMINATIONS AND VALUES. Hectare Are . , Centare 10,000 eq. meters 100 sq. meters 1 eq. meter EQUIVALENTS IN DENOMINATIONS IN USE. 2.471 acres. 119.6 square yards. 1550 square inches. Measures op Capacity. METRIC DENOMINATIONS AND VALUES. Names. Kiloliter or stere Hectoliter Dekaliter . Liter . . Deciliter . Centiliter. Milliliter . No. of Liters. 1000 100 10 1 1 1 Tot> Cubic Measure. 1 cubic meter ^j^ of a cu. meter 10 cu. decimeters 1 cu. decimeter -Ig-ofacu. decim. 10 cu. centimeters 1 cu. centimeter EQUIVALENTS IN DENOMINA- TIONS IN USE. Dry Measure. Liquid or Wine Measure. 1 .308 cu. yards 204.17 gallons. 2bu.&3..35pks. 26.417 gallons. 9.08 quarts . . 2.6417 gallons. 0.908 quart . . 1.0567 quarts. 6.1022 cu. inches 0.845 giU. 0.6102 cubic inch 0.338 fluid ounce. 0.061 cubic inch 0.27 fluid drachm. 186 THE METRIC SYSTEM. Weights. METRIC DENOMINATIONS AND VALUES. Names. Millier or tonneau Quintal , . . Myriaffram . . Kilogram or kilo Hectogram . . Dekagram . . Gram .... Decigram . . Centigram . . Milligram . . No. of Grams. 1000000 100000 10000 1000 100 10 1 tV Weight of what quanti ty of water at maximum density. 1 cubic meter .... 1 hectoliter 10 liters 1 liter 1 deciliter 10 cubic centimeters . . 1 cubic centimeter . . Y*^ of a cubic centimeter 10 cubic millimeters . . 1 cubic millimeter . . EQUIVALENTS IN DE- NOMINATIONS IN USE. Avoirdupois Weight. 2204.6 pounds. 220.40 pounds. 22.046 pounds. 2.2046 pounds. 3.5274 ounces. 0.3527 ounce. 15.432 grains. 1.5432 grains. 0.1543 grain. 0.0154 grain. 212, To reduce metric weights or measures to those in customary use. Ex. 1. Eeduce 2.5 kilos to pounds avoirdupois. Ans. 55.115 lbs. Since one kilo is 2.2046 pounds, 25 kilos will be 2.2046 X 25 1= 55.115 lbs. Hence, we have the Etjle. Multiply the number of metric units hy the corres- 'ponding numlter in the table. 2. Reduce 30 meters to inches. Ans. 1181.1. 3. Eeduce 18 liters to wine quarts. Ans. 19.0206. 4. Eeduce 50 hectares to acres. Ans. 123.55. '■ZVif. How are metric weights and measures reduced to those in custom- .ary use? THE METRIC SYSTEM. 187 313. To reduce customary weights and measures to those of the metric system. Ex. 1. Eeduce 195 inches to meters. Ans. 4.953. Since one meter is 39.37 inches, the numher of meters in 195 inches is the numher of times that 195 contains 39.37, that is, 4.953 -|- meters. Hence, we have the KuLE. Divide the numher of the customary denomination by the corresponding numher in the table. 2. Eeduce 85 gallons to hectoliters. Ans. 3.217. 3. Eeduce 28 ounces avoirdupois to grams. Ans. 7.938. 4. Eeduce 674 square yards to ares. Ans. 5.635. Miscellaneous Examples. 1. Add 65". and 8000-™. Ans. 73-. 2. Add 58.29^ and 136^ Ans. 6965^ 3. Add 4». 60''^. and 3620^. Ans. 4063.62'^^. 4. Erom4'^^. take 371^. Ans. 3629^. 5. From 5^^ take 45^ Ans. 4558'. 6. Multiply 24.5"^™. hy 160. Ans. 3.92°», 7. Multiply 42.35^. by 40. Ans. 1694^. 8. Divide 9^ hy 15^=^ Ans. 60. 9. Divide 43.46'^'". hy 106. Ans. 410™. 10. Divide 126^. hy 42^. Ans. 3. 11. Eeduce 68.49^^*. to ares. Ans. 6849". 12. Eeduce lOt. to pounds avoirdupois. Ans. 220461h. 13. In 100 cords of wood, how many { sterea ? Ans. 362.4**. 14. What cost 15m. of cloth at $3.00 a meter ? Ans. $45.00. 313 How are the customary measures reduced to those of the Metric System? 188 THE METRIC SYSTEM. 15. How many kilograms of sugar at 50 cts. a kilogram can be bought for $17.50 ? 16. How many kilometers from Boston to Albany, the dis- tance being 200 miles?" Ans. 321.86''"-. 17. In 4 cubic meters of water, how many gallons? Ans. 1056.68. 18. What is the weight in pounds avoirdupois of 2 cubic meters of water? Ans. 4409.2. 19. If a hectoliter of corn costs $2.70, what is the price of a dekaliter? Ans. $27. 20. How many steres will a pile of wood contain that is 20™ long, 2™ wide and 3°» high? Ans. 120**. 21. Bought 500 hectares of land at $75.00 a hectare and sold it at $100.00 a hectare. What was the whole gain, and gain per cent? Ans. $12500, and 33 J percent. 22. A meter of cloth costs $5.00 ; how many yds, can be bought for $205.00 ? Ans. 44.83 23. How many hectoliters in 94 gallons? Ans. 3.557. 24. What is the cost of a quintal of coffee at 60 cents a kilogram ? Ans. $60.00. 25. Mount Washington is 6226 feet above the level of the sea ; what is its height in meters ? 26. The difference in latitude, between New Orleans and Alton Illinois, is 9 degrees, (one tenth of a quadrant. Art. 203, ) what is the distance in kilometers between the places ? Ans. lOOO'^™. VB 17363 ^^ 1 EATON'S ABITHMETICS, THIS SERIES PRESI ViS A FULL AND PRACTICAL EXPOSITION OF THE METRIC SYSTEM OF WEIGHTS AND MEASURES, AND THE LATENT AND AIOST IMPROVED METHODS OF TRACHING ARITHMETIC. The Primary Arithmetic. Beautifully illustrated, and made very attractive for beginners. It teaches Arithmetic by the use of object lessons. The Intellectnal Aritlimetic. on the plan of Warrent Colburn's First Lessoks, bur j more gradual in the devtlopnieni'»f principles. It contains a lar;^o number of practical exercises, and brief and business-like forms of { analysis. ' The Elements of Arithmetic. A short cour^e of Written Arithmetic, with a large immber oi ■ Mciii 'Exercises. This work is designed for those pupils who have j i«j. 'V ... to^pursue a thorough course in Written Arithmetic. ! The Common School Arithmetic. This work presents business-like examjdcs, clear and accurate rules ard definitions, and is full Enough for all ordinary purpose^ It is especially designed for Common and Grammar Schools. The Hi^h 8chooi Arithmetic. A thorough and exhaustive treatise for High Schools and Academies. The Grammar ISchool Arithmetic. This work unites the < oimon School and High School Arith- metics, and iurnishes an exc- dent treatise when only one Written Arithmetic is needed. These books have the unfjualijied recommendation of the most i prominent educators. I TAGGARD & THOMPSON, Publishers. I 29 CornhiU, Boston.