L B /533 T3 I UC-NRLF ,.cM^ Digitized by tlie Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/exercisesinarithOOtaterich EXERCISES IN ARITHMETIC FOR ELEMENTARY SCHOOLS. AFTER THE METHOD OF PESTALOZZI. UNDER THE SANCTION OF THE GOmilTTEE OF COUNCIL ON EDUCATION, LONDON: BY JOHN W, PARKER, WEST STRAND. 1844 HARRISON AND CO., i'RrNTERS, ST, MARTIN'S LANE. CONTENTS. i BOOK I. WHOLE NUMBERS. Page Introduction - ----«. 5 First Exercise. — Addition and Subtraction ... - 7 Questions on the First Exercise, page 11. Second Exercise. — Multiplication and Division - - - 12 Questions on the Second Exercise, 17. Questions on Money, Weights, and Measures, 18. Third Exercise. — Division when there is a Remainder - - 19 Questions on the Third Exercise, 21. Questions on Money, &c.,22. Miscellaneous Questions on Shillings and Pounds, 23. Fourth Exercise. — Division, when the Remainder is expressed as a part of the Divisor - - - - - 23 Questions on the Fourth Exercise, 26. Fifth Exercise. — Ratios - - - - « 28 Case I. When a number is required which is a given part of a given number - - - - - 29 Questions on Case I., 33. Questions on Money, Weights, and Measures, 35, Case II. Where two numbers are given, to find their ratio 38 Questions on Case II., 42. Questions on Money, Weights, and Measures, 44. Case III. When a given number is a given part of a number required ------ 48 Questions on Case III., 49. Questions on Money, Weights, and Measures, 50. Miscellaneous Questions on the Fifth Exercise, 52. Sixth Exercise. — Proportion - - - - - 56 Case I. When the second term is divisible by the first - 57 Questions on Case I., 59. Questions on Money, Weights, and Measures, 60. Case II. When the first term is divisible by the second - 61 Questions on Case II., 62. Questions on Money, Weights, and Measures, 63. Case III. When the first and second terms have any ratio 64 Questions on Case III., 67- Questions on Money, Weights, and Measures, 68. Miscellaneous Questions on the Cases of the Sixth Exercise, 69. BOOK II. FRACTIONAL NUMBERS. Introduction - - - - ' . - - 71 First Exercise. — Formation and Nature of Fractions - 74 Questions on the First Exercise, 77- Second Exercise. — Reduction of Fractions - - - 79 Questions on the Second Exercise, 81. Third Exercise. — Addition and S abstract Ion of Fractions having a Common Denominator - - - g3 Questions on Third Exercise, 85. Fourth Exercise. — Multiplication and Division of Fractions by a whole Number - - - - - 37 Questions on the Fourth Exercise, 90. 10733? ^' IV CONTENTS. Page Fifth Exercise, — Reduction of Fractions to a Common Denominator 91 Questions on the Fifth Exercise, 97. Addition and Subtraction of Fractions having different Denominators - - - - - 100 Sixth Exercise. — Multiplication of Fractions - - - 103 Case I. Multiplication of whole numbers by a fraction - 103 Questions on Case I., 106. Money, Weights, and Measures, 107. Miscellaneous Questions on Case I., 108. Case II, To multiply a fraction by a fraction, or the fraction of a fraction, when the numerator of each fraction is 1 108 Questions on Case 11. , 109. Case III. When the numerators are whole numbers - 111 Questions on Case III., 114. Money, Weights, and Measures, 115. Qa&Q 111,^ continued ----- 115 Questions on Case III., continued, 118. Money, Weights, and Measures, 119. Questions on the Addition and Subtraction of Com- pound Fractions, 120. Miscellaneous Questions, 121. Seventh Exercise, — Multiplication by Mixed Numbers, and Divi- sion of Fractions - - - - - 121 I. On the Board of Simple Fractions - - - 122 Questions on the Seventh Exercise, 123. Miscellaneous Ques- tions, 124. II. On the Board of Compound Fractions - - 124 Questions on the Seventh Exercise, 127. Eighth Exercise, — Ratios of Fractional Numbers - - 128 Case I. When two fractions are given, to find their ratio 129 Questions on Case I., 135. Money, Weights^ and Measures, 137. Case I., continued - - - •■ - 139 Questions on Case I., continued, 140. Miscellaneous Questions, 141. Case II. When a given Fraction is a given part of a Frac- tion required - - - - - 143 Questions on Case II., 147- Case II., continued - ----- 148 Questions on Case II., continued, 149. Miscellaneous Questions, 149. Addenda to the Eighth Exercise. — Division of Fractions - - 151 Ninth, Exercise. — Proportion of Fractional Numbers - - 153 Case I. When the second term of the proportion is divisible by the first - - - - - - 153 Questions on Case I., 156. Case II. When the first term of the proportion is divisible by the second - - - - - 15G Questions on Case II., 158. Case III. When the first and second terms have any ratio 159 Questions on Case III., 161. Miscellaneous Questions, 162. Ques- tions on Money, Weights, and Measures, 163. Tenth Exercise. —Sqaeire and Solid Measure - - 165 Questions on Square Measure, 168. Questions on Solid Measiu'c, 171. y" OF THE ^>^ 'university BOOK I. OF WHOLE NUMBERS. INTRODUCTION. The following exercises are based upon the simple facts, that number consists of distinct units, and that those units may be grouped in different ways. 123456789 10 1 2 8 4 5 G 7 8 10 1 The Board of Simple Units. 6 INTRODUCTION. Board of Simple Units, The foregoing figure represents the first board, upon which the relations of simple unity are to be demonstrated. The first line, or line of ones, contains ten marks, or units, placed in separate compartments, or squares. The second line, or line of twos, has two marks, or units, placed in each of the ten squares. The third line, or line of threes, has three marks or units in each square; and so on, concluding with the tenth line, or line of tens, which has ten marks, or units, placed in each square. Thus the whole board is com^ posed of ten lines and ten columns. The number of any line is shown by the numeral at the left hand side of the board, while the number of any column is shown by the numeral at the top of the board. All the exercises are to be recited by the pupils without the use of figures, and always in connexion with the appro- priate board. The questions are to be solved mentally, and those annexed to each exercise are to be given while the atten- tion of the pupil is directed to the particular line which they are intended to illustrate. In the solution of these questions the pupil must be encouraged to pay attention not only to the results, but also to the language of demonstration. No new step should be taken, until the preceding one has been sufii- ciently secured. The teacher will constantly direct the observation of the pupils to the numbers upon the boards, by means of pointers from three to four feet in length. The whole class, or a a portion of the class, may recite the exercises simultaneously, or each individual in the class may be called upon in rotation to repeat the successive steps. Commencing wath the origin of numbers, the exercises proceed, by easy gradations, from the simple to the more complex relations of numbers. The sixth and the latter part of the fifth exercise on unity, are perhaps exceptions to this rule. These exercises, therefore, may be deferred, until the first four exercises on fractions shall have been explained. FIRST EXERCISE. Addition and Suhtraction. The Board of Simple Units being placed in front of the class, the teacher commences this exercise by requiring the pupils to count, in the ordinary way, the marks upon the first line, and afterwards in succession the numbers in each of the different compartments of the other lines. This simple process must be repeated until the children thoroughly comprehend the construction of the board, and can name the number of marks contained in any square. The class may then proceed with the following tables, first reciting the column of Addition and then repeating it in connection with the column of Subtraction; the teacher being careful always to point out the marks corresponding with the numbers referred to. Thus, whilst the pupils say " 4 and 3 make 7," the teacher, with the left-hand rod, counts off four marks, and then, with the other rod, counts off three more. In going over the tables of this exercise for the first time, the marks are to be mentioned in connexion with the number, as for example, " 3 marks and 2 marks make 5 marks." In the following Table the first line of squares, or the first and second columns on the Board of Units, is to be referred to. Addition. By Ones. 1 and 1 are 2 2 and 1 are 3 3 and 1 are 4 And so on. By Twos. 1 and 2 are 3 2 and 2 are 4 3 and 2 are 5 And so on. By Threes. 1 and 3 are 4 2 and 3 are 5 3 and 3 are 6 4 and 3 are 7 And so on. Subtraction. By Ones. 1 from 2, and 1 remains 1 from 3, and 2 remain 1 from 4, and 3 remain And so on. By Twos. 2 from 3, and 1 remains 2 from 4, and 2 remain 2 from 5, and 3 remain And so on. By Threes, 3 from 4, and 1 remains 3 from 5, and 2 remain 3 from 6, and 3 remain 3 from 7, and 4 remain And so on. FIRST EXERCISE. Addition. By Fours. 1 and 4 are 5 2 and 4 are 6 3 and 4 are 7 4 and 4 are 8 And so on. Subtraction. By Fours. 4 from 5, and 1 remains 4 from 6, and 2 remain 4 from 7, and 3 remain 4 from 8, and 4 remain And so on. By Fives. 1 and 5 are 6 2 and 5 are 7 3 and 5 are 8 And so on. By Fives. 5 from 6, and 1 remains 5 from 7, and 2 remain 5 from 8, and 3 remain And so on. In like manner the tables may be extended to the addition and subtraction by sixes, sevens, &c. When the pupils have been sufficiently practised in these tables, they may then be repeated in an inverted form ; for example, inverting the form of the last line, we shall have. 5 and 1 are 6 5 and 2 are 7 5 and 3 are 8 And so on. 1 from 6, and 5 remain 2 from 7, and 5 remain 3 from 8, and 5 remain And so on. These tables are so constructed that every successive addi- tion is obtained from the precedmg sum by increasing that sum by unity ; so that the pupil remembering, for example, that 3 and 5 make 8, readily concludes that 3 and 6, or 6 and 3, make 9. Also in the repetition of the same numbers, if the pupil know that 3 fives are 15, he may infer that 4 fives are 15 and 5 more, that is, 20. Addition. Of Twos. 2 and 2 are 4 4 and 2 are 6 6 and 2 are 8 8 and 2 are 10 And so on. Subtraction. Second Line. Of Twos. 2 from 4, and 2 remain 2 from 6, and 4 remain 2 from 8, and 6 remain 2 from 10, and 8 remain And so on. ADDITION AND SUBTRACTION. Addition, Of Threes. 3 and 3 are 6 6 and 3 are 9 9 and 3 are 12 12 and 3 are 15 And so on. Subtraction. Third Line. Of Threes. 3 from 6, and 3 remain 3 from 9, and 6 remain 3 from 12, and 9 remain 3 from 15, and 12 remain And so on. Fourth Line. Of Fours. Of Fours. 4 and 4 are 8 4 from 8, and 4 remain 8 and 4 are 12 4 from 12, and 8 remain 12 and 4 are 16 4 from 16, and 12 remain And so on. And so on. The other lines are to be gone over in a similar manner. The teacher now proceeds to show the connection between multiplication and addition. Repetition of Ones. First Line. 1 and 1 are twice 1, or 2 1 and 1 and 1 are 3 times 1, or 3 1 and 1 and 1 and 1 are four times 1, or 4 1 and 1 and 1 and 1 and 1 are 5 times i, or 5 And so on. Repetition of Twos. Second Line. 2 and 2 are twice 2, or 4 2 and 2 and 2 are 3 times 2, or 6 2 and 2 and 2 and 2 are 4 times 2, or 8 2 and 2 and 2 and 2 and 2 are 5 times 2, or 10 And so on. Repetition of Threes. Third Line. 3 and 3 are twice 3, or 6 3 and 3 and 3 are 3 times 3, or 9 3 and 3 and 3 and 3 are 4 times 3, or ,12 3 and 3 and 3 and 3 and 3 are 5 times 3, or 15 And so on. A 3 10 FIRST EXERCISE. Repetition of Fours. Fourth Line, 4 and 4 are twice 4, or 8 4 and 4 and 4, are 3 times 4, or 12 4 and 4 and 4 and 4 are 4 times 4, or 16 4 and 4 and 4 and 4 and 4 are 5 times 4, or 20 And so on. The same process may be performed upon any of the other lines. As a further exercise in addition, by adding the units con- tained in any of the columns, we have, 1 and 2 are 3 2 and 3 are 5 3 and 4 are 7 4 and 5 are 9 And so on. 3 less by 2 are 1 5 less by 3 are 2 7 less by 4 are 3 9 less by 5 are 4 And so on. 1 and 10 are 2 and 9 are 3 and 8 are 4 and 7 are 5 and 6 are And soon. Again, let it be required to show the different ways in which any number is made up by the addition of two numbers. For example ; 1 and 8 are 9 2 and 7 are 9 3 and 6 are 9 4 and 5 are 9 And so on. In like manner we have for subtraction : 10 less by 1 are 9 1 1 less by 2 are 9 12 less by 3 are 9 13 less by 4 are 9 And so on. Here, it will be observed, that in order that the sum of two numbers may remain the same, we must increase the one number by the same quantity that we diminish the other, and in like manner, in order that the difference may remain the same, we must increase or diminish the two numbers by the same quantity. ADDITION A>5D SUBTRACTION. 11 Questions on the First Exercise. The following questions will show to the teacher some of the various ways in which questions may be put. In order that the results of the foregoing exercise may be thoroughly im- pressed upon the memory of the pupil, the teacher will find it necessary to extend these questions. The words, First Line, Second Line, &c., intimate that the teacher must point out on what line of squares on the board the pupils are to seek for an answer to the question proposed. First Line. i. How many times must one be repeated to make ten ?.... Ans, Ten times. ii. One and one and one make what number ?....Ans, Three. iii. One from three leaves how many ?....Ans. Two. Second Line. i. Three twos make what number ?....Ans. Six, because two and two make four, and four and two make six. ii. Two and six make what nmnhev ?....Ans, Eight, be- cause six and one make seven, and seven and one make eight. iii. A man receives 2 shillings a day as his wages ; ho\v many shillings will he receive for 6 days' work ?....Ans. Twelve. iv. If John has 2 marbles in one pocket, and 3 in the other ; how^ many has he altogether ?....A7is. Five. V. Give the addition of tw^os as far as twenty. Third Line. 1. Count the squares of glass in the window by threes. ii. Show in how many ways 5 marbles may be made up. iii. Give the addition of threes as far as twenty-one. iv. If 1 article cost three-pence, how much will 5 articles cost ?....^w^. Fifteen pence. Fourth Line. i. Give the addition of fours as far as twenty. ii. Repeat 4 six times. iii. If I have 4 shillings in my pocket, and afterwards pay 3 shillings away ; how many shillings have I left?.... -4??^. One. 12 SECOND EXERCISE. Fifth Line. i. John had 5 nuts, and gave 2 of them away; how many had he left?. ...4/1^. Three. ii. What must I take from five-pence to leave three- pence ?..../4w*. Two-pence. iii. Count the marks on the fifth line by fives. SECOND EXERCISE. Multiplication and Division. Addition is an operation by which we collect two, or more numbers into one whole. Subtraction is the reverse of Addition. The sign of Addition is +, or plus: that of Subtraction •— , or minus. When equal numbers are to be added, we proceed by a more compendious method, called Multiplication, as for instance 4 and 4 and 4, or the number 4 repeated 3 times, equals 12. This operation is called the multiplication of 4 by 3, and it is symboHcally expressed by 3 X 4 = 12. The numbers 4 and 3 are called the factors of 1 2 ; the number 4 receives the name of the multiplicand, 3 that of the multiplier, and 12 that of the product. To multiply a number therefore we have to add it as many times as there are units in the multiplier. The first column of the following table shews the manner in which " the Multiplication Table" is formed, and is a recapi- tulation of some of the results obtained in the first exercise. Thus on the second line, each successive product is obtained by adding 2 to the preceding product ; and on the third line, each successive product is obtained by adding 3 to the preceding one, and so on. In the second column the operation is reversed, that is, the units contained in the product are resolved into factors, and then one of these factors is said to be contained in the product a certain number of times. In the third column, the pupil is led to consider the relation of one of these factors to the whole product. By these progressive steps, a distinct idea is conveyed to the pupil of the meaning of the terms, one-half, one-third, one-fourth, &c., considered in relation to units. When this exercise is introduced to the class, the first column is to be gone over by itself; and then repeated in con*- MULTIPLICATION AND DIVISION. 13 nexion with the second column ; and lastly, the three columns, or the first and third only, may be taken together. The results contained in this exercise are to be given by the children on observing the relations pointed out by the teacher on the Board of Simple Units. As an illustration, let us take the fifth line : When the teacher points to the first square, the children say ** once 5 is 5 ;.5 are contained in 5 once;" then in pointing to the second square, the children say, " twice 5 are 10; 5 are contained in 10 twice; the half of 10 is 5," and so on. If the children do not immediately give these results, the teacher must guide them by such suggestive questions as the following : How many fives are here?. ...Ans, Two fives. What do 2 fives make ?.,..Ans. Ten. How many fives make up 10 ?....Ans, Two fives. How many times is 5 contained in 10 ?....Ans. Twice. If ten-pence be equally divided between two boys, what part of the ten-pence does each receive ?....^W5. €)ne-half. What is meant by the fifth of 10 ?....PupiL That if 10 be separated into five equal parts, then the fifth of 10 is the number of units contained in one of these parts : thus on the second line, I see that 10 is composed of five 2's, therefore 2 is the fifth of 10. What then is the fifth of any numher?.... Pupil, The num- ber of units contained in one of the five equal parts into which the given number is divided. First Line. Fi7'st Column. Twice 1 are 2 3 times 1 are 3 4 times 1 are 4 5 times 1 are 5 And so on to 10 times 1 are 10 Once 2 is 2 Twice 2 are 4 3 times 2 are 6 4 times 2 are 8 •5 times 2 arc 10 And so on to 10 times 2 are 20 Second Column, 1 is contained in 2 twice 1 is contained in 3 three times 1 is contained in 4 four times 1 is contained in 5 five times And so on to 1 is contained in 10 ten times Second Line. 2 are contained in 2 once 2 are contained in 4 twice 2 are contained in 6 three times 2 are contained in 8 four times 2 are contained in 10 five times And so on to 2 are contained in 20 ten times Third Column. The half of 2 is 1 The third of 3.i8 1 The fourth of 4 is I The fifth of 5 is I And so on to The tenth of 10 is 1 The half of 4 is 2 The third of 6 is 2 The fourth of 8 is 2 The fifth of 10 is 2 And so on to Ti^e tenth of 20 is 2 14 SECOND EXERCISE. Third Line. First Column. Once 3 is 3 Twice 3 are 6 3 times 3 are 9 4 times 3 are 12 5 times 3 are 15 And so on to 10 times 3 are 30 Second Column. 3 are contained in 3 once 3 are contained in 6 twice 3 are contained in 9 three times 3 are contained in 12 four times 3 are contained in 15 five times And so on to 3 are contained in 30 ten times Third Column. The half of 6 is 3 The third of 9 is 3 The fourth of 12 is 3 The fifth of 15 is 3 And so on to The tenth of 30 is 3 And SO on through the fourth, fifth, sixth, seventh, and eighth lines. Ninth Line. Once 9 is 9 Twice 9 are 18 3 times 9 are 27 4 times 9 are 36 5 times 9 are 45 And so on to 10 times 9 are 90 9 are contained in 9 once 9 are contained in 18 twice 9 are contained in 27 three times 9 are contained in 36 four times 9 are contained in 45 five times And so on to 9 ai-e contained in 90 ten times The half of 18 is 9 The third of 27 is 9 The fourth of 36 is 9 The fifth of 45 is 9 And so on to The 10th of 90 is 9 Tenth Line. Once 10 is 10 Twice 10 are 20 3 times 10 are 30 4 times 10 are 40 5 times 10 are 50 And so on to 10 times 10 are 100 10 are contained in 10 once 10 are contained in 20 twice 10 are contained in 30 three times 10 are contained in 40 four times 10 are contained in 50 five times And so on to 10 are contained in 100 ten times The half of 20 is 10 The third of 30 is 10 The fourth of 40 is 10 The fifth of 50 is 10 And so on to The tenth of 100 is 10 In this table Division is treated as the reverse operation of MultipHcation ; but since the latter is an extension of the rule of Addition, so, in like manner. Division may be regarded as an extension of the rule of Subtraction, Thus to divide 9 by 3, is the same thing as finding the number of times that 3 may be subtracted from 9 ; first, we say 3 from 9 and 6 remain, then 3 from 6 and 3 remain, and lastly, 3 from 3 and nothing remains ; hence we ascertain that 3 is contained in 9 three times without any remainder. Again, in the division of 7 by 2, we find, that after 3 twos have been taken from 7, there is 1 remaining, that is, 2 is contained in 7 three times, and 1 for the remainder. It will be useful at this stage of the pupil's progress to iMULTIPLICATlON AND DIVISION. 15 show that quantities may be multiplied in any order*. Thus, while the teacher points with one rod to the eighth square of the sixth line, and with another rod to the sixth square of the eighth line, the pupil sees that 48 are 8 times 6, or 6 times 8. Again, on the fourth and sixth lines, 24 are 6 times 4, or 4 times 6. And so on to any other numbers. It will also be instructive to perform the addition and subtraction of different combinations of numbersf ; for example, on the second, third, and fourth lines. 3 twos and 2 twos are 5 twos 4 threes and 3 threes are 7 threes 5 fours and 2 fours are 7 fours 7 fives and 3 fives are 10 fives 4 sixes and 2 sixes are 6 sixes 8 nines and 3 nines are 11 nines 5 twos less by 2 twos leave 3 twos 7 threes less by 3 threes leave 4 threes 7 fours less by 2 fours leave 5 fours 10 fives less by 3 fives leave 7 fives 6 sixes less by 2 sixes leave 4 sixes 11 nines less by 3 nines leave 8 nines (fee. * This important principle may be further demonstrated by the following method. Let it be required to shew that the product of the numbers 3, 4, and 6, will be the same in what- ever order they may be multiplied, that is, that 3X4X6=4X3X6=3X6X4= &c. The product of these numbers will be shewn by the marks in the accompanying figure, where, in each group, 6 is taken 3 times, or 3 is taken 6 times; and the number in each group is repeated four times, making up altogether the number 72. Taking, therefore, the lines horizontally, we have 6 taken 3X4 times or 12 times ; taking them vertically, we have 3 taken 4x6 times^ or 3 taken 24 times ; and so on to every possible combination. + In this manner it is shewn, that any quantity taken a certain number of times, will be the same as the parts of that quantity taken the same number of times ; for example 5 X (6+3) =5 x 6 + (5 X 3). Again 5 is made up of 3 and 2 ; 7 times 5 therefore will be the same as 7 times 3, added to 7 times 2, or 7 X (3+2)=r7 X 3+7 X 2. nil i i i i i i i i i 1 1 i i i n nil Bill E I I B I I BE E B E B 1 B B B B B B B 16 SECOND EXERCISE. As a further and more complete illustration of this useful principle, we have — On the ninth, eighth, and first lines. 9 are 8 and 1 . Twice 9 are twice 8 and twice 1 3 times 9 are 3 times 8 and 3 times 1 4 times 9 are 4 times 8 and 4 times 1 Twice 8 and twice 1 are twice 9 3 times 8 and 3 times 1 are 3 times 9 4 times 8 and 4 times 1 are 4 times 9 And so on to 5, 6, 7 times 9. On the ninth, seventh, and second lines. 9 are 7 and 2. Twice 9 are twice 7 and twice 2 I Twice 7 and twice 2 are twice 9 3 times 9 are 3 times 7 and 3 times 2 j 3 times 7 and 3 times 2 are 3 times 9 And so on to 4, 5, 6 times 9. On the ninth, sixth, and third lines. 9 are 6 and 3. Twice 9 are twice 6 and twice 3 1 Twice 6 and twice 3 are twice 9 3 times 9 are 3 times 6 and 3 times 3 | 3 times 6 and 3 times 3 are 3 times 9 And so on to 4, 5, 6 times 9. On the ninth, fifth, and fourth lines. 9 are 5 and 4. Twice 9 are twice 5 and twice 4 j Twice 5 and twice 4 are twice 9 3 times 9 are 3 times 5 and 3 times 4 j 3 times 5 and 3 times 4 aie 3 times 9 And so on to 4, 5, 6 times 9. We may also operate upon 2, 3, 4, 5, &c. in the same man- ner as we have here done upon 9. The principle here exhibited, so important in the demon- stration of the rule of multiplication according to the decimal notation, is beautifully shewn by the unity board. Thus, for "3 times 9 are 3 times 5 and 3 times 4," &c, we have On the fifth line IIIIB IIBBI IIBII that is, 3 times 5 On the fourth IBIi BB8B BBBB that is, 3 times 4 On the ninth BBBBSBBBB BBBBIlBii BBBBBBBBB „ 3 times 9 Many of the answers given to the following questions may not be such as the pupils would give in the first instance, but are intended to show how the teacher should lead them to apply the method of reasoning employed in the exercise to the solu- tion of any particular question that may be proposed. The same observation will be applicable to the questions appended call the subsequent exercises. MULTIPLICATION AND DIVISION. 17 Questions on the Second Exercise. First Line. i. How many units are contained in 5 ?....Ans, Five. ii. What is the Mihof 5?,.,.Ans. One. Because 1 taken five times produces 5. Second Line. i. Where do you show the product of five times 2 ?.... Ans* The units contained in the first five squares of the second line show that five times 2 are 10. ii. What is the fourth of 8 'i....Ans, Two. Because if 8 be separated into four equal parts, we shall have two units in each part. iii. What operation must be performed upon the number 2 so as to produce \2'^....Ans, Two must be added to itself 6 times, that is, multiplied by 6, to produce 12. iv. What is meant by dividing 14 by 2 '^....Ans. It is finding the number of times that 2 must be repeated to make up 14. Thus 7 times 2 are 14, therefore 2 is contained in 14 seven times. V. Distribute 8 into as many twos as possible An^, 8 = 2 and 2 and 2 and 2, that is, 8 is made up of 4 twos. vi. How many twos can you take out of 6 ? Third Line. i. Knowing that 4 times 3 make 12, how do you find the product of 5 times 3?....^W5. By adding 3 to 12, which gives 15 for 5 times 3. ii. Where are 12 units distributed into four equal collec- tions of units ?....^w,9. In the first four compartments of the third line. iii. What is the sixth part of \^?....Ans. 3: because 6 times 3 are 18 ; and therefore 3 is the sixth of 18. iv. Find a common divisor of 9 and 12 Ans. 3: because 9 and 12 are both formed upon the line of threes. V. What is the common divisor of 15 and 21 ?„„Ans, 3: because 15 are 5 threes, and 21 are 7 threes. 18 Second exercise. vi. Of what numbers is 12 the multiple ?...,Ans. 2, 3, 4, and 6. vii. 9 times 3 are 4 times 3 and how many threes more? Similar questions may be given on any of the other lines. From time to time the teacher will introduce terms com- monly used in arithmetic, taking care always to explain their meaning. Thus in the fourth and sixth questions the terms " common divisor" and *' multiple" are introduced ; and it may be there explained that a common divisor of two or more numbers is a smaller number, which is contained in each of them an exact number of limes. In like manner it may be stated that one number is a " multiple" of another when the former contains the latter two or more times without any remainder; thus, 9, 12, and 15, are multiples of 3; while 3 is a common divisor of 9, 12, and 15. The teacher may also gradually introduce questions relating to money, weights, and measures, such as the following, giving his pupils sufficient instruction in the common tables on those subjects to enable them to give the proper solutions. A com- plete knowledge of such tables would not be requisite early in the course of instruction. Questions on Money, Weights, and Measures. Second Line. i. What is the cost of 2 oz. of coffee at 2c?. per oz. ?.... Ans. Ad. : because twice 2 = 4. ii. Divide Qd. equally amongst 3 boys: how much will each receive ?....^?i5. 2d. each: because 2d. repeated 3 times produces Qd. iii. If 4 lbs. of coffee cost 8*., what is the price of 1 lb. ? ,..,Ans. 2s. : because the cost of 1 lb. will be the fourth of the cost of 4 lbs., and the fourth of 8^. is 2^. Third Line, i. What is the cost of 4 oz. of tobacco at 3c?. per oz.?.... Ans. \2d. : because 4 times 3 are 12. THIRD EXERCISE. 19 ii. What must I pay 'for half a pound of sugar at 6d. per \h.?....Ans. 3c?.: because 6d, divided into two equal parts gives Sd. for each. iii. How many pints of ale can I get for 15c?., when one pint costs M.?....Ans, 5 pints: because 3 is contained in 15 iive times. Fourth Line. i. Divide 20c?. equally amongst 5 persons : how much will each receive ?....Ans. Ad. ii. If 3 lbs. of tea cost 12*., what is the price of 1 lb.?.... Ans. As.', because 1 lb. will cost the third of 12*. which is 4*. iii. What is the fifth of 1 '., or 20*. ?..,.Ans. As. : because 20 separated into 5 equal parts gives 4 units in each part. THIRD EXERCISE. Division lolien there is a Remainder. In the foregoing exercise the nature of Division is rendered evident by considering it as the reverse of Multiplication. In the present exercise the pupil proceeds at once to decompose numbers into certain groups of units vi^ith remainders ; then in the second column he finds the composition of these groups with the remainders ; and lastly, in the third column he finds the number of times, with a certain remainder, which these groups of units are contained in the original number of units. In introducing this exercise, the first column is to be gone over by itself; then the second and third; and, lastly, all three columns are to be recited together. As an example of the mode of teaching this exercise, we will suppose the teacher to be about to show that " twice 3 and 2 are 8." With the right hand rod, he points off eight units on the third line ; then with the left hand rod he shews that these units consist of 2 threes and 2 over, or in other words that " twice 3 and 2 are 8," and conversely, " 3 is contained in 8 twice with 2 remaining." In the course of the exercise the teacher may propose such familiar questions as are likely to lead the pupils to reason on the results obtained. For example: How many twos are here ? (pointing off the seven marks on the second line.).... -4 w*. Three twos. 20 THIRD EXERCISE. And what niore?....^7i*. And one more. Then three twos and one make what number ?,...^y. Three twos and one make 7. Then how many twos are contained in, or can be taken out of,7?....Ans. 2 are contained in 7 three times and 1 remaining. Second Line. First Columti. 2 are once 2 3 are once 2 and 1 4 are twice 2 5 are twice 2 and 1 6 are 3 times 2 7 are 3 times 2 and 1 And so on. 3 are once 3] 4 are once 3 and 1 5 are once 3 and 2 6 are twice 3 7 are twice 3 and 1 8 are twice 3 and 2 And soon. 4 are once][4 5 are once 4 and 1 6 are once 4 and 2 7 are once 4 and 3 8 are twice 4 9 are twice 4 and 1 And so on. Second Column. once 2 are 2 once 2 and 1 are 3 twice 2 are 4 twice 2 and I are 5 3 times 2 are 6 3 times 2 and 1 are7 And so on. Third Column. 2 are contained in 2 , once 2 are contained in 3, once and 1 remaining 2 are contained in 4, twice 2 are contained in 5, twice and 1 remaining 2 are contained in 6, 3 times 2 are contained in 7, 3 times, 1 remaining And so on. Third Line. once 3 are 3 once 3 and 1 are 4 once 3 and 2 are 5 twice 3 are 6 3 are contained in 3, once 3 are contained in 4, once and 1 remaining 3 are contained in 5, once and 2 remaining 3 are contained in 6, twice twice 3 and 1 are 7 ! 3 ai-e contained in 7, twice and 1 remaining twice 3 and 2 are 8 3 are contained in 8, twice and 2 remaining And 60 on. I And so on. Fourth Line. once 4 are 4 once 4 and 1 are 5 once 4 and 2 are 6 once 4 and 3 are 7 twice 4 are 8 twice 4 and 1 are 9 And so on. 4 are contained in 4, once 4 are contained in 5, once and 1 remaining 4 are contained in 6, once and 2 remaining 4 are contained in 7, once and 3 remaining 4 are contained in 8, twice 4 are contained in 9, twice and 1 remaining And so on. In the same way the fifth, sixth, seventh, eighth, and ninth lines are to be gone through, terminating with the Tenth Line. 10 are once 10 11 are once 10 and 1 12 are once 10 and 2 13areoncel0and3 20 are twice 10 And so on. once 10 are 10 once 10 and 1 are 11 oncel0and2arel2 oncel0and3arel3 twice 10 are 20 And so on. 10 are contained in 10, once 10 are contained in 11, once and Iremaining 10 are contained in 12, once and 2 remaining lOare contained in 13,once and 3 remaining 10 are containedin 20, twice And so on. DIVISION WHEN THERE IS A REMAINDER. 21 Questions on the Third Exercise. Second Line. i. What are 4 titnes 2 and l?..,.Ans, Nine. ii. How many twos can you take out of 11 ?...,Ans, Five twos and 1 remaining. iii. If 11 marbles are distributed among" 5 boys, so that each boy may have 2, how many remQ.in ?....Ans» One remains. Third Line. i. Twice 3 and 2 make how many ?....Ans. Eight, ii. How many threes are contained in 1?..,.Ans. Two threes and 1 remaining. iii. If I divide 10 nuts amongst 3 boys, giving 3 nuts to each boy, how many shall I have remaining ?....^?i^. One remaining. iv. If I have 20 apples, among how many boys can I dis- tribute them, supposing I give 3 to each hoy "^ ...,Ans, Among 6 boys ; and 2 apples remaining. Fourth Line, i. Three times 4 and 2 are what number ? . . ..Ans. Fourteen. ii. How many fours are there contained in 17 ?....Ans, Four fours and 1 remaining. iii. If 23 loaves be distributed amongst 5 women, how many loaves remain, after giving 4 to each woman ?....Ans. 3. iv. How many boys can be supplied with nuts from a bag containing 23 nuts at the rate of 4 to each hoy ?....Ans, Five boys ; and 3 nuts remaining. Similar questions may be given on the fifth, sixth, seventh, eighth, and ninth lines. Tenth Line, j. 3 times 10 and 2 are what number ?....^w^. Thirty-two. ii. How many tens are contained in 34 ?.,..Ans, Three tens, and 4 remaining. iii. If 42 apples be distributed amongst 4 boys, how many apples remain, after giving 10 to each boy ?....^n5'. Two apples. iv. If I divide 23 apples between 2 boys, how many remain, after giving 10 to each boy ?,.,.Ans. Three remain. 22 THIRD EXERCrSE. Questions on Money, &c. Second Line, i. What have I remaining out of lie?., after paying" for 5 oz. of coffee, at 2d, per oz,?. ...Ans. 1 penny j because if 1 oz. cost 2d., o oz. will cost 5 times 2d. or 10^.; then 11 are 5 times 2 and 1 remaining. ii. How many pence should I receive in exchange for 7 half-pence ?....^w.y. S^d.; because 2 is contained in 7 three times, and 1 remaining. Third Line. i. How much have I remaining out of 10c?., after paying for 3 oz. of tobacco, at Sd. per oz.?.... Ans. 1 penny ; because if 1 oz. cost 3f?., 3 oz. will cost 3 times 3c?. or 9^.; then 10 are 3 times 3 and 1 remaining. ii. How many feet are contained in 4 yards 2 feet?.... Ans. 14 feet. In 1 yard there are 3 feet; 4 times 3 and 2 are 14. iii. How many lemons can I buy for 1 Od., supposing each lemon to cost Sd. ?....A?is. 3 lemons ; and Id. remaining. Fourth Line. i. How many farthings are there in 4}d,?... Ans. 17; because in a penny there are 4 farthings, and 4 times 4 and 1 are 17. ii. In 9 farthings how many pence ?....^w^. 2^d.; because 4 is contained in 9 twice and 1 remaining. iii. How much shall I have left out of 10c?., after paying for 2 oz. of tea, at 4c?. per oz. ?....Ans. 2d. iv. How many fourpenny-pieces are there in 15c?.?.... Ans. 3 pieces and Sd. remaining. Twelfth Line*. i. How many shillings are there in 19c?. ?....Ans. Is. and 7c?. remaining ; because 12 is contained in 19, once and 7 remaining. ' * In the Table of Simple Unity published to accompany this book, the number of lines extends to twelve. |k- division when there is a remainder. 23 P ii. 2*. 5d. contain how many pence ?....^?i^. 29d.; be- ^ cause twice 12 and 5 arc 29. iii. How many pence must I receive for 1^. 9d.?....Ans^ 2W. iv. How many pence have I remaining out of 26c?. after buying 2 lbs. of sugar, at 1*. per lb. ?....Ans, 2d, Miscellaneous Questions on Shillings and Pounds. i. How many shillings are contained in 1/. 7s.?....Ans. 27s.; because, once 20 and 7 are 27. ii. How many pounds sterling are there in S4s.?,...Ans. 1/. and 14.9. remaining; because 20 are contained in 34 once and 14 remaining. iii. How many shillings are contained in 1/. I4s.?..., Ans, S4s. iv. How many shillings have I remaining out of 65s, after paying 3 workmen 1/. each?....^^?^. 5s. remaining. V. How many yards of cloth can I buy with 21, I4s., supposing each yard to cost 20s,?....A7is, 2 yards, and 14^. remaining. FOURTH EXERCISE. Divisio7i^ lohen the Remainder is expressed as a part of the Divisor. In the first column of this exercise the remainder in Divi- sion is expressed as a part of the divisor : thus, when the divisor is 4 and the dividend 9, we say, " 9 are twice 4 and the fourth of 4." The second column gives the reverse opera- tion, " twice 4 and the fourth of 4 are 9." The teacher in this exercise proceeds in the following manner : Let it be required, for example, to show that " 7 are twice 3 and the third of 3 ;" using the right hand rod, he points off seven units on the third line, then with the left hand 24 FOURTH EXERCISE. rod he shows that these units contain 2 threes and one over, or that 7 = 2x3+1; but the unit that is here in excess is the third part of 3 ; therefore the pupils say, " 7 are twice 3 and the third of 3," and then conversely, before the pointer is removed, " twice 3 and the third of 3 are 7." Such questions as the following may form part of the exercise : Teacher, I am going to question you on the line of fours. Here I point to nine marks ; how many fours can you take out of 9 ?.,.. Pupil. Two fours and 1 remaining. Teacher, What part of four is 1 ? ...Pupil. One is the fourth of four, so that 9 is made up of 2 fours and the fourth of 4. Teacher. You now can find out how much twice 4 and the fourth of 4 amount to?. ...Pupil. Nine: because twice 4 are 8, and the fourth of 4 is 1, therefore 8 and 1 are 9. Second Line, or Line of Twos. 1 is the half of 2 2 are once 2 3 are once 2 and the half of 2 4 are twice 2 5 are twice 2 and the half of 2 6 are three times 2_ 7 are three times 2 and the half of 2 10 are five times 2 The half of 2 is 1 Once 2 is 2 Once 2 and the half of 2 are 3 Twice 2 are 4 Twice 2 and the half of 2 are 5 Three times 2 are 6 Three times 2 and the half of 2 are 7 Five times 2 are 10 Third Line^ or Line of Threes. 1 is the third of 3 2 are twice the third of 3 ■ 3 are once 3 4 are once 3 and the third of 3 5 are once 3 and twice the third of 3 6 are twice 3 7 are twice 3 and the third of 3 8 are twice 3 and twice the tbird of 3 30 are 10 times 3 The third of 3 is 1 Twice the third of 3 are 2 Three times the third of 3 ai'C 3 Once 3 and the third of 3 are 4 Once 3 and twice the tliird of 3 are 5 Twice 3 are 6 Twice 3 and the third 3 are 7 Twice 3 and twdce the third of 3 are 8 Ten times 3 are 30 DIVISION WHEN THE REMAINDER, ETC. 25 Fourth Line, or Line of Fours 1 is" the fourth part of 4 2^are twice the fourth of 4 3 are three times the fourth of 4 4 are once 4 5 are once 4 and the fourth of 4 6 are once 4 and twice the fourth of 4 7 are once 4 and three times the fourth of 4 8 are twice 4 9 are twice 4 and the fourth of 4 40 are ten times 4 The fourth of 4 is 1 Twice the fourth of 4 are 2 Three times the fourth of 4 are 3 Four times the fourth of 4 are 4 Once 4 and the fourth of 4 are 5 Once 4 and twice the fourth of 4 are 6 Once 4 and three times the fourth of 4 are 7 Twice 4 are 8 Twice 4 and the fourth of 4 are 9 Ten times 4 are 40 1 is the fifth part of 5 2 are twice the fifth of 5 3 are three times the fifth of 5 4 are four times the fifth of 5 5 are once 5 6 are once 5 and the fifth of 5 7 are once 5 and twice the fifth of 5 8 are once 5 and three times the fifth of 5 50 are ten times 5 Fifth Line, or Line of Fives. The fifth part of 5 is 1 Twice the fifth of 5 are 2 Three times the fifth of 5 are 3 Four times the fifth of 5 are 4 Five times the fifth of 5 are 5 Once 5 and the fifth of 5 are 6 Once 5 and twice the fifth of 5 are 7 Once 5 and three times the fifth of 5 are 8 Ten times 5 are 50 And so on to the sixth, seventh, eighth, &c., lines. This table gives us the reason for the operation in Division, when there is a remainder. For example, let it be required to divide 42 by 5, or to find how many fives are contained in 42. It will be seen, on the fifth line, that 42 contains 8 fives, and 2 units over, or 8 fives and 2 fifths of 5 ; that is, 5 are con- tained in 42, 8f times. 26 FOURTH EXERCISE. Questions on the Fourth Exercise, Second Line. i. How many times are 2 contained in 5?....Ans. Two and a half times : because twice 2 make 4, and the half of 2^ is 1, therefore twice 2 and the half of 2 is the same as 4- and 1, that is, 5. ii. What are three and a half times 2?....Ans. 7: because three times 2 are 6, and the half of 2 is 1 ; therefore three and a half times 2 are 6 and 1, which are 7. Third Line. i. Divide 10 by S,....Ans. 10 contains 3 times 3, and the third of 3 ; that is, three and a third times. ii. How many threes are contained in 8 ?.,..Ans. Twice 3 and twice the third of 3 are 8. iii. What is 4 times 3 and twice the third of 3 ?....Ans. 14 ; because 4 times 3 are 12, and twice the third of 3 are 2 ; therefore, 12 and 2 make 14. Fourth Line. i. How often is 4 contained in 21 ?....Ans, Five times and one fourth: because 5 times 4 and the fourth of 4 are 21. ii. What is 6 times 4 and 3 times the fourth of 4?..., Ans. 24 and 3 = 27. iii. What must I add to 8 times 4 to make up 34?.... Ans. 2. Because eight times 4 are 32 ; 32 and 2 are 34. iv. If 3 times the fourth of 4 be taken from 5 times 4, what remains ?....^/i^. 17: because 5 times 4 are 20, and 3 times the fourth of 4 are 3 ; 3 from 20 leaves 17. V. What remains when twice 4 are taken from 4 times 4 and twice the fourth of 4 ?....Ans. 10. Because twice 4 are 8 ; 4 times 4 and twice the fourth of 4 are 1 8 ; therefore, 8 taken from 18 leaves 10. Fifth Line. i. What is the amount of twice 5 and 3 times the fifth of 5?t,,,Ans. 13: because twice 5 are 10, and 3 times the fifth of 5 is 3, therefore 10 and 3 make 13. DIVISION WHEN THE REMAINDER, ETC. 27 ii. How many fives can you take out of 11 ?....Ans, Two fives and the fifth of five. iii. 4 is what part of 5?....Ans, 1 is the fifth of 5, and therefore 4 is 4 times the fifth of 5. iv. What are 4 times 5 and twice the fifth of 5?..,.Ans» 22 ; because 4 times 5 are 20, and twice the fifth of 5 are 2, and 20 and 2 are 22. Tenth Line, i. How many tens are there in 2S?.„,Ans. Two tens and 3 times the tenth of 10. ii. What are 4 times 10 and 9 times the tenth of 10 ?.... Ans. 49. Second and Third Lines. i. In 3 times 2 and the half of 2, how many threes?.... Ans. 2 threes and the third of 3 ; because, on the second line we find, that 3 times 2 and the half of 2 are 7 ; and on the third line it is shewn that 7 are twice 3 and the third of 3. ii. How many threes are there in 4 times 2 and the half of 2?....Atis. 3 threes. iv. Twice 3, and 3 times 3 and the third of 3, contain how many twos?. ...Ans. 8 twos. Third and Fourth Lines. i. How many fours are contained in 5 times 3, and twice the third of 3 ?....Ans. 4 fours and the fourth of 4. ii. 5 threes and the third of 3 contain how many fours ? ,„,Ans. 4 fours. iii. How many threes can you find in twice 4 and 3 times the fourth of 4?., ..Ans, 3 threes and twice the third of 3. Fourth and Fifth Lines. i. Twice 4 and twice the fourth of 4 contain how many fives?. ...^w*. 2 fives. ii. 3 times 5 and twice the fifth of 5 contain how many fours 7. ...Ans, 4 fours and the fourth of 4. iii. How many fives are contained in 4 times 4 and 3 times the fourth of 4?,„.Ans, 3 fives and. 4 times the fifth of 5. B 2 28 FIFTH EXERCISE. 1. Second and Fourth Lines. 9 times 2 and the half of 2 contain how many fours ?.... ^ns, 4 fours and 3 times the fourth of 4. ii. In 3 times the fourth of 4, how many twos?.... ^w^. One 2 and the half of 2. Second and Fifth Lines. i. 8 times 2 and 4 times the half of 2 contain how many ^\es?....Ans. 4 fives. ii. In 3 times the fifth of 5 and twice 5 how many twos ? .,,.Ans. 6 twos and once the half of 2. Second and Sixth Lines. i. 10 times 2 and 6 times the lialf of 2 are how many sixes ?....^?2.y. 4 sixes and twice the sixth of 6. ii. In 8 times the sixth of 6 and 3 times the half of 2, how many twos ?..,.Ans. Five twos and once the half of two. Similar questions may be proposed on any other two lines. FIFTH EXERCISE. Hattos. In the preceding exercise two numbers are compared, chiefly with the view of ascertaining the result of the division of the one number by the other : in the present exercise is considered the result of such division, or decomposition, as giving the relative magnitude, or ratio, of the two numbers. For ex- ample, in the course of this exercise it is shown that 9 are 3 times the half of 6 ; here, then, the ratio of 9 to 6 is 3 times the half, that is, 9 is produced by repeating the half of 6 three times. As the ratio of two numbers is their relative magnitude, or the number of times that the one number is contained in the other, it follows that this ratio may be given in different forms : ON RATIOS. CASE I. 29 thus the comparison of 12 with 8 may be given in the three following forms : i. 12 are 12 times the eighth of 8. ii. 12 are 6 times the fourth of 8. iii. 12 are 3 times the half of 8. In the first case the ratio is 12 times the eighth; in the second 6 times the fourth ; and in the third 3 times the half. In the last form the ratio is in its least terms, because the num- bers expressing it are the smallest that can be used. Hence, it will be observed that 12 times the eighth, 6 times the fourth, and 3 times the half, mean the same thing, or rather, that they indicate operations which produce the same results. There are three classes of problems which illustrate the nature and use of ratios. These are, 1st. When a number is required which is a given part of a given number; 2nd. When two numbers are given, to find their ratio ; and 3rd. When a given number is a given part of a number required. The peculiarity of these three forms of ratios will be best under- stood by a strict attention to the exercises and questions which accompany them. Case I. When a number is required which is a given part of a given number. This case may be illustrated by the following question : W^hat number is 3 times the fourth of 12 ?....Ans* 9. The teacher calls the attention of the pupils to the first four marks or units on the second line, and then with another rod points to the two units in the first square of that line ; the pupils say " the half of 4 is 2." The teacher then removes the latter pointer to the third square, and the pupils say, " 3 times the half ot 4 are 3 times 2, or 6." Still keeping the first pointer in its place, the teacher places the other on the fourth square, when the pupils say " 4 times the half of 4 are 4 times 2, or 8," and so on. Questions similar to the following may be put in the course of the exercise. What do you understand by 4 times the half of 4 ?...,Ans. That the half of 4, which is 2, is to be taken four times. On what line do you find 7 times the third of 12?....^^^. On the fourth line 12 units are distributed in three squares, which shows that the third of 12 is 4, and seven squares from this line give me 7 times 4, which are 28. 30 FIFTH EXERCISE. First Line. Thehalf of 2is 1. Twice the half of 2 are twice 1, or 2. 3 times the half of 2 are 3 times 1, or 3, 4 times the half of 2 are 4 times 1, or 4. And so on to 10 times the half of 2 are 10 times 1, or 10. The third of 3 is 1. Twice the third of 3 are twice 1, or 2. 3 times the third of 3 are 3 times 1, or 3. 4 times the third of 3, are 4 times 1, or 4. And so on to 10 times the third of 3 are 10 times 1, or 10. And so on, until th€ teacher concludes the line with The tenth of 10 is 1. Twice the tenth of 10 are twice 1, or 2. 3 times the tenth of 10 are 3 times 1, or 3. 4 times the tenth of 10 are 4 times 1, or 4. And so on to 10 times the tenth of 10 are 10 times 1, or 10. Second Line. The half of 4 is 2. 3 times the half of 4 are 3 times 2, or 6. 4 times the half of 4 are 4 times 2, or 8. 5 times the half of 4 are 5 times 2, or 10. And so on to 10 times the half of 4 are 10 times. 2, or 20. The third part of 6 is 2. Twice the third of 6 are twice 2, or 4. 4 times the third of 6 are 4 times 2, or 8. 5 times the third of 6 are 5 times 2, or 10. And so on to 10 times the third of 6 are 10 times 2, or 20. The fourth part of 8 is 2. Twice the fourth of 8 are twice 2, or 4. 3 times the fourth of 8 are 3 times 2, or 6. 5 times the fourth of 8 are 5 times 2, or 10. And so on to 10 times the fourth of 8 are 10 times 2, or 20. ON RATIOS. CASE I. 31 The teacher proceeds in this manner, taking successively the fifth of 10, the sixth of 12, the seventh of 14, the eighth of 16, the ninth of 18, concluding with The tenth of 20 is 2. Twice the tenth of 20 are twice 2, or 4. 3 times the tenth of 20 are 3 times 2, or 6. 4 times the tenth of 20 are 4 times 2, or 8. 5 times the tenth of 20 are 5 times 2, or 10. And so on to 9 times the tenth of 20 are 9 times 2, or 18. Third Line. Thehalf of 6is3. 3 times the half of 6 are 3 times 3, or 9. 4 times the half of 6 are 4 times 3, or 12. 5 times the half of 6 are 5 times 3, or 15. 6 times the half of 6 are 6 times 3, or 18. And so on to 10 times the half of 6 are 10 times 3, or 30. The third of 9 is 3. Twice the third of 9 are twice 3, or 6. 4 times the third of 9 are 4 times 3, or 12, 5 times the third of 9 are 5 times 3, or 15. 6 times the third of 9 are 6 times 3, or 18. And so on to 10 times the third of 9 are ten times 3, or 30. And so on, concluding with The tenth of 30 is 3. Twice the tenth of 30 are twice 3, or 6. 3 times the tenth of 30 are 3 times 3, or 9. 4 times the tenth of 30 are 4 times 3, or 12. 5 times the tenth of 30 are 5 times 3, or 15. And so on to 9 times the tenth of 30 are 9 times 3, or 27. Fourth Line. Thehalf of 8 is 4. 3 times the half of 8 are 3 times 4, or 12. 4 times the half of 8 aie 4 times 4, or 16. 5 times the half of 8 are 5 times 4, or 20. 6 times the half of 8 are 6 times 4, or 24. And so on to 10 times the half of 8 are 10 times 4, or 40, 32 FIFTH EXERCISE. ' The third of 12 is 4. Twice the third of 12 are twice 4, or 8. 4 times the third of 12 are 4 times 4, or 16. 5 times the third of 12 are 5 times 4, or 20. 6 times the third of 12 are 6 times 4, or 24. And so on to 10 times the third of 12 are 10 times 4, or 40. Proceeding in this way through the various multiples of 4, we conclude with The tenth of 40 is 4. Twice the tenth of 40 are twice 4, or 8. 3 times the tenth of 40 are 3 times 4, or 12. 4 times the tenth of 40 are 4 times 4, or 16. 5 times the tenth of 40 are 5 times 4, or 20. And so on to 9 times the tenth of 40 are 9 times 4, or 36. And so on to the 5th, 6th, 7th, &c. lines, concluding with the Tenth Line. The half of 20 is 10. 3 times the half of 20 are 3 times 1 0, or 30. 4 times the half of 20 are 4 times 10, or 40. 5 times the half of 20 are 5 times 10, or 50. And so on to 10 times the half of 20 are 10 times 10, or 100. Proceeding on this way through the different compartments we come to the ninth compartment of this line. The ninth of 90 is 10. Twice the ninth of 90 are twice 10, or 20. 3 times the ninth of 90 are 3 times 10, or 30. 4 times the ninth of 90 are 4 times 10, or 40. And so on to 10 times the ninth of 90 are 10 times 10, or 100. The tenth of 100 is 10. Twice the tenth of 100 are twice 10 or 20. 3 times the tenth of 100 are 3 times 10, or 30. 4 times the tenth of 100 are 4 times 10, or 40. And so on to 9 times the tenth of 100 are 9 times 10, or 90. ON RATIOS. CASE I. 33 The teacher may then propose examples relating to various lines and squares promiscuously : thus, 4 times the fifth of 45 are 4 times 9, or 36. 5 times the seventh of 35 are 5 times 5, or 25. 3 limes the sixth of 24 are 3 times 4, or 12. Twice the fifth of 35 are twice 7, or 14. &c. &c. Questions on Case I. First Line. i. What is 3 times the half of 2?....Jns. 3. Proof. The half of 2 is 1, and 3 times the half of 2 are 3 times 1, or 3. ii. What is 5 times the fourth of 4 ?,...Ans. 5. JProof, The fourth of 4 is 1, and 5 times the fourth of 4 are 5 times 1, or 5. Second Line. i. What is 5 times the half of 4 ?....^7i^. 10. Proof. The half of 4 is 2, and 5 times the half of 4 are 5 times 2, or 10. ii. What is 7 times the fifth of 10 1 ....Ans. 14. Proof The fifth, of 10 is 2, and 7 times the fifth of 10 are 7 times 2, or 14. Third Line. i. What is 5 times the half of 6 1 ..,.Ans. 15. Proof The half of 6 is 3, and 5 times the half of 6 are 5 times 3, or 15. ii. 3 times the fifth of 15 is what number ?....^/i5. 9. Proof The fifth of 15 is 3, and 3 times the fifth of 15 are 3 times 3, or 9. iii. What is 8 times the sixth of \^?..,.Ans. 24. Proof The sixth of 18 is 3, and 8 times the sixth of 18 are 8 times 3, or 24. Fourth Line. i. What is 9 times the third of 12?....^^^. 36. Proof The third of 12 is 4, and 9 times the third of 12 are 9 times 4, or 36. B 3 34 FIFTH EXERCISE. iL What is 6 times the ninth of 36 ?...,Ans, 24. Proof. The ninth of 36 is 4, and 6 times the ninth of 3^ are 6 times 4, or 24. iii. 5 times the sixth of 24 is what number ?.„.Ans. 20,* Proof. The sixth of 24 is 4, and 5 times the sixth of 24 are 5 times 4, or 20. Fifth Line. i. What is 4 times the sixth of SO ?....Ans. 20. Proof The sixth of 30 is 5, and 4 times the sixth of 30 are 4 times 5, or 20. ii. What number is 8 times the seventh of 35 ?....Ans. 40. Proof The seventh of 35 is 5, and 8 times the seventh of 35 are 8 times 5, or 40. iii. 9 times the fifth of 25 is what number ?....Ans. 45. Proof The fifth of 25 is 5, and 9 times the fifth of 25 are 9 times 5, or 45. The Lines taken promiscuously. i. What number is 7 times the sixth of 12? ....Ans. 14. Proof The sixth of 1 2 is 2, and 7 times the sixth of 1 2 are 7 times 2, or 14. ii. What is 3 times the fifth of 20?. ...Ans. 12. Proof The fifth of 20 is 4, and 3 times the fifth of 20 are 3 times 4, or 12. iii. 9 times the seventh of 42 is what number ?,...^?i^. 54. Proof The seventh of 42 is 6, and 9 times the seventh of 42 are 9 times 6, or 54. iv. What number is 8 times the ninth of 63 ? ....Ans. bQ. Proof The ninth of 63 is 7, and 8 times the ninth of 63 are 8 times 7, or bQ. V, What is 5 times the tenth of 90?....^?!^. 45. Proof The tenth of 90 is 9, and 5 times the tenth of 90 are 5 times 9, or 45. ON RATIOS. CASE I. 36 Questions on Money, Weights, and Measures. First Line. i. What istheamountof 3 timesthehalf of 2^.?....^7W. 3^. Proof. The half of 2*. is 1*., and 3 times the half of 2*. are 3 times 1^., or 3^. ii. What is the amount of 5 times the seventh of 7c?.?.... Ans..5d. Proof. The seventh of Id. is Id., and 5 times the seventh of 7c?. are 5 times \d., or 5c?. iii. 5 times the ninth of 9 oz. is what number of ounces ? ,,.,Ans. 5 oz. Proof. The ninth of 9 oz. is 1 oz., and 5 times the ninth of 9 oz. are 5 times 1 oz., or 5 oz. iv. If 15 lbs. cost 1*. 3d., what is the cost of 7 lbs. ?.... Ans. Id. Proof. In 1^. 3c?. there are 15c?. If 15 lbs. cost 1^. 3c?. 1 lb. will cost Ic?., and 7 lbs. will cost 7c?. Second Line. i. What is 5 times the half of 4c?.?....^w^. 10c?. Proof. The half of 4c?. is 2d., and 5 times the half of 4c?, are 5 times 2c?., or 10c?. ii. What is 7 times the half of 4oz.?....^?i.s. 14 oz. Proof. The half of 4 oz. is 2 oz., and 7 times the half of 4 oz. are 7 times 2 oz., or 14 oz. iii. What is the amount of 9 times the third of 6c?.?.... An^. 18c?. Proof. The third of Qd. is 2c?., and 9 times the third of Qd, are 9 times 2c?., or 1 Sc?. iv. What is the amount of 9 times the fourth of 8c?.? .,..-4w*. 1*. Qd. Proof. The fourth of 8c?. is 2c?., and 9 times the fourth of 8c?. are 9 times 2c?., or 18c?. = 1^. Qd. V. If 6 lbs. of coffee cost 12*. what is the price of 8 lbs.? ,,^,Ans. \Qs. Proof If 6 lbs. cost 12*. 1 lb. will cost 2*., and 8 lbs. will cost 8 times 2*., or 16*. 36 FIFTH EXERCISE. Third Line. i. What is 6 times the fifth of Is. Sd.?....Ans. Is. 6d. Proof. The fifth of 1,9. Sd. is 3d., and 6 times the fifth of 1^. Sd. are 6 times Sd., or 1^. 6d. ii. If 1 lb. of tea cost 3^. 3c?. what is the cost of 6 times the thirteenth part of 1 ]h.? ....A7is. I8d. Proof. The cost of the thirteenth part of 1 lb. must be the thirteenth part of 3.s'. Sd., or Sd., and 6 times the thirteenth of 3*. Sd., are 6 times 3^. or 18^. iii. If 1 lb. of sugar cost 9c?. what is the amount of 4 times the third of 1 \h.'^ ....Ans. Is. Proof The cost of 1 lb. is 9c?.; the cost of the third of 1 lb. is the third of 9c/., or Sd,, and 4 times the third of 9c?. are 4 times 3c?., or 1^. iv. If 6 lbs. of tobacco cost 18^. what is the price of 5 lbs.?. ....Afis. 15^. Proof If 6 lbs. cost 18^. 1 lb. will cost 3^., and 5 lbs. will cost 5 times 3*., or 15*. V. What is the value of 3 times the seventh of a guinea? ..,.Ans. 9s. Proof The seventh of a guinea is 3*., and 3 times the seventh are 3 times 3*., or 9*. vi. W^hat is the amount of 3 times the fifteenth of Ss. 9d.?....Ans. 9d. Proof The fifteenth of 3*. 9c?. is 3c?., and 3 times the fifteenth of 3*. 9c?. are 3 times Sd., or 9c?. vii. What is the amount of 3 times the sixth of Is. 6d.? ..,,Ans. 9c?. Proof The sixth of 1*. 6d. is 3c?., and 3 times the sixth of 1*. 6d. are 3 times 3c/., or 9c?. viii. What is the amount of 4 times the fifth of 1*. 3c?.? .,,.Ans. Is. Proof. The fifth of 1*. 3^. is 36?., and 4 times the fifth of Is, Sd. are 4 times 3c?., or 1*. Fourth Line. i. Divide 28?. equally amongst 7 persons Ans. Each would receive 4?. Proof. If 7 persons receive 28/. one person would receive the seventh part of 28?., or 41. ON RATIOS. CASE I. 37 ii. Divide 321. between 2 persons, so that one shall have 3 times the eighth of the money, and the other the remaining part A71S. One would receive 12/. and the other 201. Proof. The eighth of 32/. is 41., and 3 times the eighth of 32/. are 3 times 41., or 12/.; then 32/. less by 12/. = 20/. The Lines taken promiscuously. i. What is the amount of 3 times the fifth of 1^. 8c?.?.... Ans. Is. Proof. Is. 8d. are 20d., the fifth of 20c/. is 4d., and 3 times the fifth of 20d., or 3 times 4d., or 12c?. = 1*. ii. How much is 8 times the seventh of 5s, 3d.?....A7is 6s. Proof The seventh of 63c?. is 9c?., and 8 times the seventh of 63c/. are 8 times 9c/., or 72c?. = 6s. iii. How^ many pounds are there in 8 times the seventh of 6Soz.?....A7is. 4 lbs. 8oz. Proof The seventh of 63 oz. is 9 oz., and 8 times the seventh of 63 oz. are 8 times 9 oz., or 72 oz.=4 lb. 8 oz. iv. What is 9 times the third of 3 quarters of a yard ? ....Ans. 2\ yds. Piroof The third of 3 qrs. is 1 qr.; therefore 9 times the third of 3 qrs. is 9 times 1 qr., or 9 qrs. = 2J yds. V. What is 7 times the sixth of 3^. 6d. ? ....Ans. 4s. \d. Proof 3y. 6d. are 42c?. ; the sixth of 42c?. is 7c?., and 7 times the sixth of 42c?. are 7 times 7, or 49c?. = 4^. Ic?. vi. If 8 oz. of tea cost 2^. Sd., what is the price of 9 oz. ? ....A71S. Ss. P?oof. In 2^. 8^. there are 32 pence. If 8 oz. cost 32 pence, 1 oz. will cost 4c?., and 9 oz. will cost 9 times 4, or 36 pence. 36 pence are 3*. vii. What is the value of 10 yards of calico, if 4 yards cost l.v. 4d.?....A7is. Ss. 4d. Proof In 1*. 4c?. there are 16 pence. If 4 yards cost 16 pence, 1 yard will cost 4 pence, and 10 yards will cost 10 times 4, or 40 pence. 40 pence are 3^. 4c/. 38 FIFTH EXERCISE. Case II. Where two numbers are given y to find their ratio. This case may be illustrated by the following question : What is the ratio between 9 and 12 ? The left hand pointer is to be placed on the third square of the third line, and the pupils say, "9 are 3 times 3;" the right hand pointer is then to be placed on the second square, and the pupils say, " 6 are twice 3, 3 times 3 are 3 times the half of twice 3." The teacher now removes the right hand pointer to the fourth square, and the pupils repeat the next step, "12 are 4 times 3," " 3 times 3 are 3 times the fourth of 4 times 3 ;" and so on, always observing to keep the first pointer in its place. When two numbers are to be compared, we seek to discover their lowest ratio, by finding the highest number which will divide them both. This number gives on the unity board the line from which the ratio is to be taken. For example, let it be required to exhibit the ratio of 12 to 8. On the fourth line, the teacher points with the right hand rod to 12 units or marks which the pupil observes are made up of 3 fours ; then with the left hand rod, he points off 8 units or marks, which are made up of 2 fours ; hence, 12 is 3 times the half of 8, because the half of 8, which is 4, taken 3 times pro- duces 12. By this means we obtain the ratio of 12 to 8, in its lowest terms. As another example, let the ratio of 6 to 9 be required. A line is sought for upon the board, where 6 and 9 are made up of similar groups of units ; this is readily found to be the third line ; then according to the language of the exercise we have 6 are twice 3, 9 are 3 times 3. Twice 3 are twice the third of 3 times 3, that is, 6 are twice the third of 9. In some cases the teacher may find it necessary, at every successive ratio, to repeat the decomposition given at the head of each table, as in the following example: 6 are 3 times 2. 4 are twice 2. 3 times 2 are 3 times the half of twice 2. 6 are 3 times 2. 8 are 4 thnes 2. 3 times 2 are 3 times the fourth of 4 times 2. 6 are 3 times 2. 10 are 5 times 2. 3 times 2 are 3 times the fifth of 5 times 2, It may here be stated, once for all, that whenever two or more numbers are made the subject of comparison, the teacher should use two pointers ; for by this means, the objects of comparison are kept before the eye of the pupil. ON RATIOS. CASE II. 39 As a great proportion of the questions which arise in the business of life involve the principle of ratios, the present exercise should be thoroughly understood. First Line. 1 is the half of twice 1 or 2. 1 is the third of 3 times 1 or 3. 1 is the fourth of 4 times 1 or 4. 1 is the fifth of 5 times 1 or 5. 1 is the tenth of 10 times 1 or 10. 2 are twice 1, or twice the third of 3 times 1 or 3. 2 are twice 1, or twice the fourth of 4 times 1 or 4. 2 are twice 1, or twice the fifth of 5 times 1 or 5. 2 are twice 1, or twice the tenth of 10 times 1 or 10. 3 are 3 times 1, or 3 times the half of twice 1 or 2. 3 are 3 times 1, or 3 times the third of 3 times 1 or 3. 3 are 3 times 1, or 3 times the fourth of 4 times 1 or 4. 3 are 3 times 1, or 3 times the tenth of 10 times 1 or 10. And so on to the ratios of 4, 5, 6, &c. Second Line, or Line of Twos. On this line 2 is first compared with the multiples of itself ; then 4, 6, 8, &c. are successively compared with all the mul- tiples of 2 as far as 10 times 2. 4 are twice 2. 2 are the half of twice 2 or 4. 6 are 3 times 2. 2 are the third of 3 times 2 or 6. 8 are 4 times 2. 2 are the fourth of 4 times 2 or 8. And so on. 4 are twice 2. Twice 2 are twice the half of twice 2. 6 are 3 times 2. Twice 2 are twice the third of 3 times 2. 8 are 4 times 2. Twice 2 are twice the fourth of 4 times 2. 10 are 5 times 2. Twice 2 are twice the fifth of 5 times 2. And so on. 6 are 3 times 2. 4 are twice 2. 3 times 2 are 3 times the half of twice 2. 8 are 4 times 2. 3 times 2 are 3 times the fourth of 4 times 2. 10 are 5 times 2. 3 times 2 are 3 times the fifth of 5 times 2. And so on. 8 are 4 times 2. 4 are twice 2. 4 times 2 are 4 times the half of twice 2. 6 are 3 times 2. 4 times 2 are 4 times the third of 3 times 2. 8 are 4 times 2. 4 times 2 are 4 times the fourth of 4 times 2. 10 are 5 times 2. 4 times 2 are 4 times the fifth of 5 times 2. And so on. 40 FIFTH EXERCISE. 10 are 5 times 2. 4 are twice 2. 5 times 2 are 5 times the half of twice 2. 6 are 3 times 2. 5 times 2 are 5 times the third of 3 times 2. 8 are 4 times 2. 5 times 2 are 5 times the fourth of 4 times 2. 10 are 5 times 2. 5 times 2 are 5 times the fifth of 5 limes 2. And so on. The teacher then goes on to compare 6 times 2, 7 times 2, 8 times 2, &c., with all the multiples of 2. Third Line^ or Line of Threes, 6 are twice 3. 3 are the half of twice 3, or 6. 9 are 3 times 3. 3 are the third of 3 times 3, or 9. 12 are 4 times 3. 3 are the fourth of 4 times 3, or 12. 15 are 5 times 3. 3 are the 5th of 5 times 3, or 15. And so on. 6 are twice 3. Twice 3 are twice the half of twice 3. 9 are 3 times 3. Twice 3 are twice the third of 3 times 3. 12 are 4 times 3. Twice 3 are twice the fourth of 4 times 3. 15 are 5 times 3. Twice 3 are twice the fifth of 5 times 3. And so on. 9 are 3 times 3. 6 are twice 3. 3 times 3 are 3 times the half of twice 3. 9 are 3 times 3. 3 times 3 are 3 times the third of 3 times 3. 12 are 4 times 3. 3 times 3 are 3 times the fourth of 4 times 3. 15 are 5 times 3. 3 times 3 are 3 times the fifth of 5 times 3. And so on. 12 are 4 times 3. 6 are twice 3. 4 times 3 are 4 times the half of twice 3. 9 are 3 times 3. 4 times 3 are 4 times the third of 3 times 3. 12 are 4 times 3. 4 times 3 are 4 times the fourth of 4 times 3. 15 are 5 times 3. 4 times 3 are 4 times the fifth of 5 times 3. And so on. 15 are 5 times 3. 6 are twice 3. 5 times 3 are 5 times the half of twice 3. 9 are 3 times 3. 5 times 3 are 5 times the third of 3 times 3. 12 are 4 times 3. 5 times 3 are 5 times the fourth of 4 times 3. 15 are 5 times 3. 5 times 3 are 5 times the fifth of 5 times 3. And so on. The teacher then goes on to compare 6 times 3, 7 times 3, 8 times 3, &c., with all the multiples of 3. ON RATIOS. CASE II. 41 Fourth Line, or Line of Fours. 8 are twice 4. 4 are the half of twice 4 or 8. 12 are 3 tirres 4. 4 are the third of 3 times 4 or 12. 16 are 4 times 4. 4 are the fourth of 4 times 4 or 16. 20 are 5 times 4. 4 are the fifth of 5 times 4 or 20. And so on. 8 are twice 4. twice 4 are twice the half of twice 4. 12 are 3 times 4. twice 4 are twice the half of 3 times 4. 16 are 4 times 4. twice 4 are twice the fourth of 4 times 4. 20 are 5 times 4. twice 4 are twice the fifth of 5 times 4. And so on. 12 are 3 times 4. 8 are twice 4. 3 times 4 are 3 times the half of twice 4. 12 are 3 times 4. 3 times 4 are 3 times the third of 3 times 4. 16 are 4 times 4. 3 times 4 are 3 times the fourth of 4 times 4. 20 are 5 times 4. 3 times 4 are 3 times the fifth of 5 times 4. And so on. 16 are 4 times 4. 8 are twice 4. 4 times 4 are 4 times the half of twice 4. 12 are 3 times 4. 4 times 4 are 4 times the third of 3 times 4, 16 are 4 times 4. 4 times 4 are 4 times the fourth of 4 times 4. 20 are 5 times 4. 4 times 4 are 4 times the fifth of 5 times 4. And so on. 20 are 5 times 4. 8 are twice 4. 5 times 4 are 5 times the half of twice 4. 12 are 3 times 4. 5 times 4 are 5 times the third of 3 times 4. 1 6 are 4 times 4. 5 times 4 are 5 times the fourth of 4 times 4, 20 are 5 times 4. 5 times 4 are 5 times the fifth of 5 times 4. And so on. The teacher then g-oes on to compare 6 times 4, 7 times 4, 8 times 4, &c., with all the multiples of 4. Fifth Line, or Line of Fives. 10 are twice 5. 5 are the half of twice 5 or 10. 15 are 3 times 5. 5 are the third of three times 5 or 15. 20 are 4 times 5. 5 are the fourth of 4 times 5 or 20. And so on. 10 are twice 5. Twice 5 are twice the half of twice 5, 15 are 3 times 5. Twice 5 are twice the third of 3 times 5. 20 are 4 times 5. Twice 5 are twice the fourth of 4 times 5. 25 are 5 times 5. Twice 5 are twice the fiith of 5 times 5. And so on. ^2 FIFTH EXERCISE. 15 are 3 times 5. 10 are twice 5. 3 times 5 are 3 times the half of twice 5. 15 are 3 times 5. 3 times 5 are 3 times the third of 3 times 5. 20 are 4 times 5. 3 times 5 are 3 times the fourth of 4 times 5. 25 are 5 times 5. 3 times 5 are 3 times the fifth of 5 times 5. 20 are 4 times 5. 10 are twice 5. 4 times 5 are 4 times the half of twice 5. 15 are 3 times 5. 4 times 5 are 4 times the third of 3 times 5. 20 are 4 times 5. 4 times 5 are 4 times the fourth of 4 times 5. 25 are 5 times 5. 4 times 5 are 4 times the fifth of 5 times 5. And so on. And so on to the comparison of 5 times 5, 6 times 5, 7 times 5, &c., wdth all the multiples of 5. The form of the exercise being exhibited in these five lines, the teacher will be enabled to extend the same form to the other lines upon the board. Questions on Case II. First Line, i. What is the ratio of 9 to. 2?... .Ans. 9 times the half. Proof. 9 are 9 times 1 , or 9 times the half of twice 1 or 2. ii. Compare the numbers 3 and 2 Ans. 3 are 3 times the half of 2. Proof. 3 are 3 times 1, or 3 times the half of twice 1 or 2. Second Line, i. What part of 10 is A? ....Ans, Twice the fifth. Proof. 4 are twice 2, 10 are 5 times 2; twice 2 are twice the fifth of 5 times 2. ii. What part of 14 is Q? ...Ans. 3 times the seventh. Proof. 6 are.3 times 2, 14 are 7 times 2; 3 times 2 are 3 times the seventh of 7 times 2. iii. What is the ratio of 18 to ^?....Ans, 9 times the fourth. Proof 18 are 9 times 2, 8 are 4 times 2 ; 9 times 2 are 9 times the fourth of 4 times 2. Third Line. 1. What part of 18 is 12 ?..,.Ans. 4 times the sixth part. Proof 12 are 4 times 3, 18 are 6 times 3 ; 4 times 3 are 4 times the sixth of 6 times 3. ON RATIOS. CASE II. 43 ii. What is the ratio of 21 to 15 ?....Ans. 7 times the fifth. Proof. 21 are 7 times 3, 15 are 5 times 3 ; 7 times 3 are 7 times the fifth of 5 times 3. Fourth Line. i. What part of 20 is l6?....Ans. 4 times the fifth. Proof. 16 are 4 times 4, 20 are 5 times 4 ; 4 times 4 are 4 times the fifth of 5 times 4. ii. What part of 28 is 20 ?....Ans, 5 times the seventh. Proof. 20 are 5 times 4, 28 are 7 times 4 ; 5 times 4 are 5 times the seventh of 7 times 4. iii. What part of 8 is 12 ?....Ans. 12 is 3 times the half of 8. Proof. 12 are 3 times 4, 8 are twice 4, and 3 times 4 are 3 times the half of twice 4. iv. What is the ratio of 16 to 28?....^^^. 16 are 4 times the seventh of 28. Proof. 16 are 4 tim.es 4 ; 28 are 7 times 4, and 4 times 4 are 4 times the seventh of 7 times 4. Fifth Line. i. Compare 20 with 25 Arts. 20 are 4 times the fifth of 25. Proof, 20 are 4 times 5, 25 are 5 times 5, 4 times 5 are 4 times the fifth of 5 times 5. ii. Compare 15 with 10 Ans, 15 are 3 times the half of 10. Proof 10 are twice 5, 15 are 3 times 5, 3 times 5 are 3 times the half of twice 5. iii. Compare 35 with SO. ....Ans. 1 times the sixth of 30. Proof. 35 are 7 times 5, 30 are 6 times 5, 7 times 5 are 7 times the sixth of 6 times 5. iv. Compare 40 with 45 Ans, 40 are 8 times the ninth of 45. Proof 40 are 8 times 5, 45 are 9 times 5, 8 times 5 are 8 times the ninth of 9 times 5 . V. Compare 30 with 45 Ans, 30 are 6 times the ninth of 45. Proof 30 are 6 times 5, 45 are 9 times 5, 6 times 5 are 6 times the ninth of 9 times 5. 44 FIFTH EXERCISE. Questions on Money, Weights, and Measures. Hitherto the weights and measures, made use of in the questions, have been of the most simple kind ; but upon pro- posing the following questions, the teacher will find it necessary to instruct his pupils in the more common and useful tables. It is assumed that the teacher has varied the foregoing exercises, by teaching the pupils those rules and operations of common arithmetic, which may be based upon the principles developed in them. At the same time it must be distinctly understood, that the operations contained in these exercises are always intended to precede the corresponding calculations conducted by the use of figures. In proposing questions which contain three or more quan- tities, the teacher may write down the quantities upon the black board, while the pupils proceed to work out the result mentally as usual. It may be advisable to defer the more difficult questions, particularly those of a miscellaneous kind, until the pupils have made some further progress as well in the Exercises on Fractions, as in the remaining parts of the present chapter. First Line. i. How many times S^. are 9cL?....A7is. 9t?. are 9 times the eighth of 8d. Proof. 9 are 9 times 1, or 9 times the eighth of 8 times 1. ii. How many times 9c?. is lie?. ?....^f/5. 1 1 c?. are 1 1 times the ninth of del. Proof. 11 are 11 times 1, or 11 times the ninth of 9 times 1. iii. If 7 articles cost \s. 9c?., what will be the cost of 5 at the same rate ?....J7)s. 5 times the seventh of 1^. 9d. or 1^. 3c?. Prorf. 5 are 5 times the seventh of 7, the cost of 7 is 21c?., therefore, the cost of 5 will be 5 times the seventh of the cost of 7 articles, which is 1^. 3c?. Second Line. i. If 6 articles cost 9d., what will 10 cost? ....Jfis. 15c?. Proof 10 are 5 times 2, 6 are 3 times 2, 5 times 2 are 5 times the third of 3 times 2. The cost of 6 articles is 9c?. ; ON RATIOS. CASE If. 45 therefore the cost of 10, which are 5 times the third of 6, will be 5 times the third of 9d. or 15c?. ii. 8 articles cost 1^., required the value of 14 articles at the same rate A^is. Is. 9d. Proof, 14 are 7 times 2, 8 are 4 times 2, 7 times 2 are 7 times the fourth of 4 times 2. The cost of 8 articles is 1^., therefore the cost of 14 articles, which are 7 times the fourth of 8, will be 7 times the fourth of Is., or 2ld. = Is, 9d. iii. If 10 articles cost 1^. 3^., what will 12 articles cost? .,,,Ans. 6 times the fifth of 1*. Sd. or 1;?. 6d. Proof. 12 are 6 times 2, 10 are 5 times 2, 6 times 2 are 6 times the fifth of 5 times 2; therefore if 10 articles cost Is. 3c?., 12 articles, which are 6 times the fifth of 10, will cost 6 times the fifth of 1^. 3d. or I8d. = Is. 6d, Third Line. i. If 9 pints of ale cost 2*. 3c?., what is the worth of 12 quarts ?....Ans. 8 times the third of 2^. 3c?. or 6s. Proof 12 quarts are 24 pints, 24 are 8 times 3, 9 are 3 times 3, 8 times 3 are 8 times the third of 3 times 3, there- fore if the cost of 9 pints be 2*. 3c?., the cost of 24 pints, which are 8 times the third of 9 pints, will be 8 times the third of 2^. 3c?. or 6s. ii. How much will 15 articles cost, if the price of 12 be 2^. Sd. 7....A71S. 5 times the fourth of 2.v. 8c^. or 3^. 4d. Proof 15 are 5 limes 3, 12 are 4 times 3, 5 times 3 are 5 times the fourth of 4 times 3; therefore the cost of 15 articles, which are 5 times the fourth of 12 articles, will be 5 times the fourth of ^s. Sd. or 3^. 4d. iii. What part of a shilling is ^d.? ....Ajis. 3 times the fourth. Proof. In a shilling there are 12 pence; 9 are 3 times 3, 12 are 4 times 3, 3 times 3 are 3 times the fourth of 4 times 3. iv. 6s. are what part of a guinea '^....Ans. Twice the seventh. Proof 6 are twice 3, 21 are 7 times 3, twice 3 are twice _the seventh of 7 times 3. V. A man earns 26^. per week of 6 days, what are his ges for 9 days '^....Ans. II. ]9s. Proof. 9 are 3 times the half of 6 ; therefore the wages 9 days will be 3 times the half of 26^., that is, 39^. 40. FIFTH EXERCISE. vi. The gross load of a horse is 6000 lbs., the weight of the waggon is 2000 lbs., what part of the gross load is the weight of the material ?..,.Ans. Twice the third. Proof. The net load is 4000 lbs. ; 4 are twice the third of 6; therefore 4000 is twice the third of 6000. Fourth Line. i. If 16 articles be worth 3*. 4c?., what is the value of 20 articles ?....Ans, 20 will cost 5 times the fourth of 3^. 4c?. or 4^. 2d. Proof. 20 are 5 times 4, 16 are 4 times 4, 5 times 4 are 5 times the fourth of 4 times 4; therefore if the cost of 16 be 3^. 4c?., the cost of 20, which are 5 times the fourth of 16, will be 5 times the fourth of 3^. 4c?. or 4^. 2c?. ii. What part of a pound is 8s*?...,Ans. Twice the fifth. Proof. 8 are twice 4, 20 are 5 times 4, twice 4 are twice the fifth of 5 times 4. iii. If a barrel of beer cost Zh 12.9., what will 16 gallons cost ?....^W5. 4 times the ninth of 3?. 12^. or 1/. 12.9, Proof. In a barrel there are 36 gallons, 16 are 4 times 4, 36 are 9 times 4, 4 times 4 are 4 times the ninth of 9 times 4, therefore, if the cost of 36 gallons be Zl. 12.y., the cost of 16, which are 4 times the ninth of 36 gallons, will be 4 times the ninth of 3/. 12^. or 1^. 12^. Fifth Line. i.^ If the cost of 15 articles amount to 2^., what is the value of 40 articles ?....Ans» 8 times the third of 2.9. or 5.9. 4c?. Proof 40 are 8 times 5, 15 are 3 times 5, 8 times 5 are 8 times the third of 3 limes 5 ; therefore, if the cost of 15 articles be 2*., the cost of 40, which are 8 times the third of 15 articles, will be 8 times the third of 2.9. or bs. 4c?. ii. What is the cost of 3 gallons 1 pint, if the cost of 1 gallon 2 pints be 25-. Qd.l ....Aris. 5 times the half of 2.9. Qd* or 6^. 2>d. Proof 3 gallons 1 pint are 25 pints, 1 gallon and 2 pints are 10 pints, 25 are 5 times 5, 10 are twice 5, 5 times 5 are 5 times the half of twice 5 ; therefore if the cost of 1 pints be 2*. 6c/ , the cost of 25 pints, which are 5 times the half of 10, will be 5 times the half of 2*. 6c?. or 6*. 3c?. iii. If 4 cwt. 3 stones of potatoes cost 1 As» 7c?., what will 15 stones cost ?.....4?a^. 3 times the seventh of 14^. Td, or ON RATIOS. CASE II. 47 Proof. 4 cwt. 3 stones are 35 stones ; 15 are 3 times 5, 35 are 7 times 5, 3 times 5 are 3 times the seventh of 7 times 5 ; therefore if the cost of 35 stones be 14*. 7c?., the cost of 15 stones, which are 3 times the seventh of 35 stones, will be 3 times the seventh of 14*. 7c?. or 6*. 3c?. iv. 3 lbs. 7 oz. of tea cost 14*. 8c?., what will 1 lb. 9 oz, cost '^....Ans. 5 times the eleventh of 14*. 8c?. or 6*. 8c?. Proof. 3 lb. 7 oz. are equal to 5b oz. ; 1 lb. 9 oz. are equal to 25 oz.; then 25 are 5 times 5, 55 are 11 times 5, 5 times 5 are 5 times the eleventh of 11 times 5; therefore if 55 oz. cost 14*. Sd., 25, which are 5 times the eleventh of 56, will cost 5 times the eleventh of 14*. 8c?. or 6*. 8c?* V. What cost 3 yds. 3 qrs. of cloth at 21. 10*. for 2 yds. 2qrs. ?....^w*. 3 times the half of 21. 10*. or 3Z. 15*. Proof. 3 yds. 3 qrs. are equal to 15 qrs., 2 yds. 2 qrs. are equal to 10 qrs., 15 are 3 times 5, 10 are twice 5, 3 times 5 are 3 times the half of twice 5; therefore 15, which are 3 times the half of 10, will cost 3 times the half of 2Z. 10*. or 3^. 15*. vi. 10 oz. are what part of 2 lbs. 3 oz.7 ....Ans. Twice the seventh. Proof, 2 lbs. 3oz. contain 35 oz.; 10 are twice 5, 35 are 7 times 5, twice 5 are twice the seventh of 7 times 5. vii. 15*. are what part of a pound ?....^72*. 3 times the fourth. Proof 15 are 3 times 5, 20 are 4 times 5, 3 times 5 are 3 times the fourth of 4 times 5. viii. If a man earn 3^. in 15 days, what will he earn in 25 days at the same rate ?,...^w*. 5/. Proof 25 are 5 times 5, 15 are 3 times 5, 5 times 5 are 5 times the third of 3 times 5 ; then as 25 days are 5 times the third of 15 days, in 25 days the man will earn 5 times the third of 3^., that is, bl, ix. If 10 masons can build 22 yards of walling in 1 day, how many yards will 25 masons build in the same time ? .„.Ans. 55 yards. Proof 25 are 5 times the half of 10; therefore the num- ber of yards built by 25 men will be 5 times the half of 22, that is, 55 yards. I 48 FIFTH EXERCISE. Case III. When a given number is a given part of a number required. Ex, Of what number is 6 three times the fifth ?....^w^. 10. Questions of this kind may be readily solved by a slight variation of the preceding exercise. A few examples will render this apparent. Taking the above example we have " 6 are 3 times 2, 3 times 2 are 3 times the fifth of 5 times 2 or 10." Thus the first step consists in finding what num- ber taken 3 times will produce 6 ; this number must be 2; but the number thus found is the fifth of the required one, and hence the number sought must be 5 times 2 or 10. In going over this process, the teacher first points to the 6 marks upon the line of twos, which will show of what number 6 is 3 times; with another pointer he shows of what number 2 is the fifth part ; so that when the first pointer is placed on the third square, the pupils say "6 are 3 times 2;" when the second pointer is put on the fifth square, then the pupils say, '3 times 2 are 3 times the fifth of 5 times 2, or 10.'* The questions belonging to this Case being peculiar in their form, the teacher may find it necessary to break them up in the manner of which the following is an instance : — Let it be required to show of what number 8 are 4 times the third part. Teacher. (Placing his pointer upon the fourth square of the second line.) In giving the proof of this question, why do I place my pointer upon the fourth square ? Pvpils. Because we have first to find what number of units multiplied by 4 produces 8, which we there find to be 2. Teacher. Upon what square must I now place my pointer, in order to show of what number these two units are the third ? Pupils. Upon the third square, which shows that 2 are the third of 6. Teacher. Then 8 are 4 times Pupils. Two. Teacher. And 8 are 4 times the third of what number ? Pupils. Six. ON RATIOS. CASE 111. 49 Questions on Case III. First Line. i. 3 are 3 times the seventh of what number ?....Ans. 7. Proof. 3 are 3 times 1, 3 times 1 are 3 times the seventh of 7 times 1, or 7. ii. 5 are 5 times the eighth of what number l.,..Ans. 8. Proof, 5 are 5 times 1, 5 times 1 are 5 times the eighth of 8 times 1 or 8. Second Line. i. 6 are 3 times the fifth of what number ?...,Ans. 10. Proof. 6 are 3 times 2, 3 times 2 are 3 times the fifth of 5 times 2 or 10. ii. 10 are 5 times the eighth of what number ?....A7is. 16. Proof 10 are 5 times 2, 5 times 2 are 5 times the eighth of 8 times 2 or 16. iii. 12 are 6 times the seventh of what number ?..,.Ans. 14. Proof 12 are 6 times 2, 6 times 2 are 6 times the seventh of 7 times 2 or 14. Third Line. i. 6 are twice the third of what number ?....^w^. 9. Proof 6 are twice 3, twice 3 are twice the third of 3 times 3 or 9. ii. 15 are 5 times the ninth of what number ?..,.Ans. 27. Proof 15 are 5 times 3, 5 times 3 are 5 times the ninth of 9 times 3 or 27. iii. 18 are 6 times the fourth of what number ?....^;i^. 12. Proof. 1 8 are 6 times 3, 6 times 3 are 6 times the fourth of 4 times 3 or 12. Fourth Line. i. 16 are 4 times the seventh of what number ?..,.Ans. 28. Proof. 16 are 4 times 4, 4 times 4 are 4 times the seventh of 7 times 4 or 28. ii. 32 are 8 times the fifth of what number ?..,.Ans. 20. Proof 32 are 8 times 4, 8 times 4 are 8 times the fifth of 5 times 4 or 20. iii. 12 are 3 times the ninth of what number ?....Ans. 36. Proof 12 are 3 times 4, 3 times 4 are 3 times the ninth of 9 times 4 or 36. c 50 _ FIFTH EXERCISE. Fifth Line. i. 10 are twice the iifth of what number ?....An.o J.1 25 15 11 25 11 14f^^. 92 nence 4-g- — -5-, 1^ — -5-, -g- — -g- —-^-*^ 01 z-^, V. A person sold f of his property, what part had he left?. ...^71*. f. 87 FOURTH EXERCISE, Multiplication and Division of Fractions hy a whole mimher. In this exercise the pupil is led to perform the same opera- tions upon fractional units that he has already performed upon simple units in the Second Exercise of Book I. In all the examples of this exercise the multiplier of the fraction is a whole number ; and the fractional divisor is contained a certain number of times, without a remainder, in the dividend. Other varieties of multiplication and division will be given in the Sixth and Seventh Exercises. The teacher may illustrate this exercise on any one of the lines of the Board of Simple Fractions. Suppose he select the fifth line. When he points off -f , or four fifths, with one rod, and f, or two fifths, with the other, the pupil says "twice ■§- are -f;" and conversely, "-fare contained in ^ twice;" — " the half of |-is f," and so on. In such an instance as this the right hand pointer is removed the space of ■§-, at each step of the table, whilst the other rod is used to show the number of times that ■§- are repeated. Such questions as the following may be put in the course of the exercise. Teacher. What do you understand by 3 times f ? Pupil. That f are repeated 3 times, that is, f and \ and 2. 6. 5 — 5 • Teacher. What is meant by f being contained in f three times ? Pupil, That two-fifths can be taken out of f three times. Teacher. How do you find the third of |-? Pupil. If f be divided into three equal parts, one of these parts is the third of f ; thus f and f and \ or 3 times -I = f ; therefore f are the third of #. 88 ON FRACTIONS. FOURTH EXERCISE. Teacher, Comparing this exercise with the second exercise on unity, do you find any difference between the multipli- cation of units, and the multiplication of fractions by whole numbers ? Pupil. The only difference is, that in the one case the results are units, and in the other fractional parts. Teacher. 3 times -| being -f, how do you obtain the pro- duct of 4 times ^ ? Pupil. By increasing -§• by -| which make f. In this exercise the first column is to be recited by itself; then the first and second columns ; and lastly the first and third columns. First Column. Once one -half is \ Twice I are 2 halves or 1 3 times | are 3 halves or U 4 X i are 4 halves or 2 5 y I are 5 halves r2| And so on. Once I is 3 halves Twice I are 6 : halves or 3 3 times | are | or Second Line^ or Line of Halves. Third Column. Second Column. \ is contained in \ once \ is contained in | twice \ is contained in | 3 times 5 is contained in | 4 times \ is contained in | 5 times And so on. I are contained in | once I are contained in | twice the half of | is ^ the third of | is i the fourth of | is i the fifth of I is i And so on. the half of | is | or 1| the third of | is § or U And so on. I are contained in | 3 times And so on. And so on. And so on to the I^Iultiplication of any number of halves. In like manner we proceed with any other line, such for example as the MULTIPLICATION AND DIVISION. 89 Fifth Line^ or Line of Fifths, First Column. Once I is ^ Twice \ are | 3 times | are | 4 times \ are f And so on. Once I is I Twice I are | 3 X I are f or li I are | or 1 1 4 X And so on. Once I is Twice I are f ( ] are ' or 1 1 >|orn Third Column. the half of | is | the third of | is | the fourth of \ is 5- And so on. j thehalfof |is I ' the third of f is | the fourth of | is | And so on. thehalfof f is | 1 the third of | is | the fourth of '/ is | And so on. the half of | is f the third of V is \ And so on. And so on to the product of any number of fifths. This exercise shows that we multipl^f a fraction hy a whole number when we multiply the numerator hy that whole number : thus to multiply f by 5, the multiplicand ^ must be added 5 times, or f taken 5 times = '^-^. The inverse part of this exercise also shows, that ive divide a fraction by a whole number, when we divide the numerator by that whole number : thus f is the quotient of -^ divided by 5 ; or the fifthof J/isf. It is important also to observe, that any fraction multiplied by its denominator gives the numerator, that is, -f X 7=5. 3X1 -_ 4 X I are V or 2| And so on. Once I is I Twice I are | or 1 § 3 X fare Vor2| And so on. Second Column. \ is contained in \ once \ is contained in § twice I is contained in | 3 times \ is contained in | 4 times And so on. f are contained in f once I are contained in 3 twice I are contained in | 3 times § are contained in | 4 times And so on. I are contained in | once I are contained in | twice I are contained in | 3 times § are contained in V 4 times And so on. I are contained in f once f are contained in | twice f are contained in V 3 times And so on. 90 ON FRACTIONS. FOURTH EXERCISE. Questions on the Fourth Exercise. Second Line^ or Line of Halves. i . How much is 6 times f ? . . . .Ans . ^- or 9 whole numbers . ii. What is 4 times |- ?....Ans. ^ or 10 whole numbers. iii. How often are f contained in ^?..,.Ans, 6 times. iv. What is the fifth part of -^:^?... Ans. f. Because \^- divided into 5 equal parts, gives us f for one of those parts. V. What is meant by |- multiplied by 4 ?....Ans. f repeated 4 times; that isf + |-4-f + f = -V-or6. vi. If "1^ be multiplied by 3, what is the product ?...,^?2^. -^ or lOi. Because ^ repeated 3 times give ^^ or lOy. Third Line, or Line of Thirds. i. Multiply ^ by 6 Ans. The product is ■%^- or 8 whole numbers. ii. When -| are repeated 4 times, what is the result?.... Ans, 4 times -f are ^- or 6f . iii. What is the third part of ^?...A7is. f or 2. Because when ^ are divided into 3 equal parts, each of these parts will be f or 2 whole numbers. iv. What is 7 times | ?....Ans. -^ or 4|. V. How many oranges should I require so as to be able to give two-thirds of an orange to each of 4 boys ?....Ans* 2f . Fourth Line, or Line of Fourths, 1. How much is 8 times f ?....Ans. ^- or 6 whole numbers. ii. Divide ^ into 6 equal parts, what will each of those parts be?....^?2^. | or If. iii. What is 7 times \1 ....Ans. y or 3|-. Because f repeated 7 times are y or 3f iv. How often must % be repeated to make y '^..,,.Am, 5 times. MULTIPLICATION AND DIVISION. 91 Fifth Line, or Line of Fifths. i. Multiply f by 6 Ans. The product is y or 3|-. ii. Find the product when f are multiplied by 5 Ans, ^-^ or 4 whole numbers . iii. How often must f be repeated to make up ^ ?.,,.Ans, 6 times. iv. Divide y into 3 equal parts Ans. for l\. Because f + f + f , or 3 times f = y . V. How many f can be taken out of f ?....Ans. 3. Because y + f + f = f ? that is, f is made up of three f. vi. Divide 2 apples and -§- of an apple equally amongst 3 boys A71S. Each boy will receive f of an apple. Because each apple contains 5 fifths, therefore 2 apples and f will contain -^^-, and then if these ^-^ be divided amongst 3 boys, each boy will receive the third of -y- or f. vii. Eight loaves of bread are divided each into 5 equal parts, how can I distribute them equally amongst 10 poor persons ?....^?i^. By giving to each person f of a loaf. FIFTH EXERCISE. Reduction of Fractions to a Common Denominator. In the first exercise of this Book it was shown that a fraction becomes greater when the numerator is increased, and less when the denominator is increased. The present exercise shows how the denominator of a fraction may be changed without altering the magnitude of the fraction. This operation is important, because it enables the pupil so to adjust two or more fractions that they may have the same denominator, in which state they may be more conveniently added or compared. 92 ON FRACTIONS. FIFTH EXERCISE. This exercise requires the use of the Compound Board. When the teacher is about to introduce any square for the first time he is recommended to construct the figure on a large scale upon the black board in the following manner : The Third Square of the Fourth Line on the Board of Compound Fractions. b c m h S e The vertical lines in the figure are first drawn, and the pupils are shown that each square is thus divided into fourths. Each fourth is then divided into three equal parts by the horizontal lines; so that the pupils cannot fail to observe that each compartment, a, b, c, d, thus formed, is the third part of one-fourth, and at the same time one-twelfth of the whole square or unit. By means of this ^gure fourths are reduced to twelfths. When thi7'ds are required to be reduced to twelfths, the squares must be extended in the vertical direction ; and each square being first divided by the horizontal lines into 3 equal parts, each third, thus formed, is divided by the vertical lines into 4 equal parts. The exercise on this square commences with the reduction of fourths to twelfths, and conversely; and concludes with the reduction of thirds to twelfths. When the construction of the figure has been described by the teacher, the pupil says, " 1 is -^4 ;" ^^^ conversely, " -^ are 1 ." The teacher now places his rod along the upright line d e, and then the pupil says " ^ is y^;" and conversely, "-^ are ^J' The pointer is now removed to the line fog; and the pupil says, "|^ are twice REDUCTION. 93 TT ^^ 1^5" ^^^ conversely, ^ are twice ■ or -I :** when the pointer is laid upon the line d e, the pupil says, ";2. are 7 times TT ^^ IT'" ^^^ conversely, " -f^are 7 times ^g- or ^:" and so on. For the reduction of thirds to twelfths, the teacher places his rod on the first horizontal line dividing the square, the pupil then says, " ^ is ^;'' and conversely, "-3^ are i:" the pointer being removed to the next horizontal line, the pupil says, " -| are twice ^ or -^;' and so on. Seqond Line of Squares. Second Square, in which halves are reduced to fourths; and conversely. 4 are 1 lisi f are twice ^ or j^ f are 3 times |- or f i are 4 X f or f &c. 4" ^^C 2- ^ are twice |- or f f are 3 times f or f f are 4 X f or f &c. Third Square^ in which halves and thirds are reduced to the same denominator, viz. to sixths; and conversely. lisf •| are twice |- or |- f are 3 times f or f f are 4 X f or -^ &c. lis -I f are twice f or f I are 3 times f or ■§■ i are 4 x -§■ or | &c. |- are 1 f arei |- are twice f or f ■§- are 3 times -| or f ■i^ are 4 X f or f &c. I- are 1 f are i |- are twice f or f |- are 3 times f or f -f are 4 times -§- Or -| &c. And so on up to the 94 ON FRACTIONS. FIFTH EXERCISE. Tenth Square, in wliicli halves and tenths are reduced to the same denominator, and conversely. 1 i«; ^^ ■•^ ^^ 2 i is i-^ 2 ^^ 2 I are twice ^ or |§ |- are 3 times -^^ or |-§- 2 :£ yj i are 4 X ■!-§ or f| &c. i% are 1 |-f are twice ^% or | l-g are 3 times -^^^ or f #f are4xi§orf &c. 1 ?a 20 i 1& -20" ft are 1 JL_ is -2- _2_ Q|.p _J_ 10 ^* 2 2 aic jQ -TO are twice -^\ or ^ ^%- are twice 2V or ^ -^3- are 3 times ^\ or ^^o 2^^ are 3 times -^\ or - &c. &c. Third Line of Squares. Second Square, in which thirds and halves are reduced to the same denominator, viz. to sixths, and conversely. lis I 3 IS ^ f are twice |- or f "I are 3 times ■§- or -§- &c. f are 1 f are J f are twice f or ^ f are ~ 3 times -§- or |- &c, lis I •2 IS -5- •2 1S> -g- f are twice f or f f are 3 times f or f -I are 1 "6" are f are twice f or -=- f are 3 times f or f &c. REDUCTION. 95 Third Square, in which thirds are reduced to ninths, and conversely. % are 1 lisf I" are twice f , or |- f are 3 times |, or |^ f are 4 times f, or y &c. |- are twice |^, or f 1^ are 3 times -|, or f y are 4 times |, or f &c. Fourth Square, in which thirds and fourths are reduced to twelfths, and conversely. lis if if are 1 iis^ A are i f are twice ^, or -^^ -^8-2 are twice -^, or f f are 3 times -^-^^ or -[-| if are 3 times y\, or 3 3 f are 4 X ^, or |f if are 4 X y\, or f &c. &c. 1 ic 1 2 1 is yy if are 1 i are ^V T2 are i f are twice y\-, or y\ yV are twice y\, or f f are 3 times y\, or y^g- y\ are 3 times y\, or * &c. &c. And so on with the other squares, concluding with the Tenth Square, in which thirds and tenths are reduced to the same denominator, viz., thirtieths, and conversely. f-S- are 1 are i 1 30 t IS f are twice }%, or |f f are 3 times 1%, or -|§ &c. 1 30 f^ are twice i^, or f f f are 3 times if, or -| &c. lis if If are 1 1 TC 3 To IS 3-0 3^0 are yV TO are twice 3^^, or 3^ 3^ are twice -i^, or y^ yV are 3 times 3^, or ^^ -^ are 3 times ^, or - &c. &c. ^6 ON FRACTIONS. FIFTH EXERCISE. This exercise proves thait if tJie nii7nerator and denominator of a fraction he multiplied by the same number, the fraction is not altered in value. It will also be seen that as by multi- plying the denominator of a fraction by 2, 3, 4, so that I may be able to add ihem? ....Ans. Tenths, as shown on the fifth square of the second line. And so on to the sixth, seventh, eighth, ninth, and tenth squares. Third Line. Second Square. i. What fractions are reduced by this square to the same denominator ?....^?i*. Halves and thirds. ii. In one half how many sixths ?....^n5. Three sixths; because 1 = f ; and therefore \ = the half of -|, that is f. iii. Reduce J and -| to fractions having a common denomi- nator Ans, I" and ^. Proof. This operation must be performed on a square which contains halves and thirds, namely, the second square of the third line ; where we observe that ^ contains f, and f contain ^, Third Square. i. How many ninths are there in \ ?...,A7is, |-; because 1 contains -| ; and therefore -^ contains the third of |, that is, f . ii. What is the common denominator to which fractions are reduced by this squsive? ....Ans. Ninths. iii. Reduce ^- to thirds. ....-^w^. ^, REDUCTION. 99 Fourth Square. i. Reduce f and -| to fractions having the same denomina- tor Ans, -^ and ^. ii. In -J how many twelfths ?....Ans. -|-|. iii. Reduce -^f to thirds Ans, -f . iv. Reduce -^ to fourths Ans^ |-. V. How many twelfths can I take out of f?.... -4^5. 8 twelfths. Fifth Square. i. What fractions are reduced to the same denominator by this square ?....^W5. Thirds and fifths. ii. In y^5- how many thirds ?....^w^. ^, iii. Reduce ^ and |- to the same denominator Ans. |-f and 3^. And so on to the sixth, seventh, eighth, ninth, and tenth squares. Fourth. Line. Second Square. i. What fractions are reduced by this square to the same denominator ?....Ans. Halves and fourths, ii. In |- how many eighths ?....Ans. f . iii. Reduce ^ and f to eighths Ans. f and f , Third Square. i. In J how many twelfths?....-4w^. -f^ ; because 1 contains if, and i contains the fourth of if > that is y\-; therefore f = 3 times -^, or ■^. ii. What IS the common denominator to which fractions are reduced by this square ?...,Ans. Twelfths. iii. Reduce f, ^, and ^ to equivalent fractions having like denominators Ans. -5^, -^, and ^. iv. Reduce -^ to its lowest terms, and tell me on which line and square it is done..,, Ans. -^^ are 3^, and.it is done on the third square of the fourth line. E 2 100 ON FRACTIONS. FIFTH EXERCISE V. How would you bring the third of an orange, and the fourth of an orange, to the same ipa.Yts?....Ans. By dividing the third into four equal parts, which would give -^ ; and then dividing the fourth into three equal parts, which would gi^'e -^. The additions and subtractions performed in the third exercise, were restricted to fractions having the same denomi- nator ; but as the pupil is now enabled to reduce any two or more fractions to that state, it will be desirable to terminate this exercise with a few questions in the Addition and Suhtraction of Fractions having different Denonfiinators. In proposing these questions the teacher is recommended to have the particular square referred to, drawn out upon an enlarged scale and placed before the Class. On the Board of Compound Fractions, Second Line of Squares. Second Square, i. Add I and \ together Ans, f, because i = f ; there- fore f + i = f. ii. Add f and J together Arts, f = 2^, because f = f; and f + f = f , or 2^. iii. AVhat is the difference between l^ and l|?....-4?i^. f. Third Square, i. What is the sum of ^ and \?..,.Ans, -|. ii. What is the sum of \ and ^?....Ans, -|- = H- iii. What is the sum of IJ and \\'^..,.Ans, -V" = ^' iv. What is the difference between f and \'^ ....Ans. \ \ because J are ^; and \ are f ; therefore f less by f = i. V. What is the sum of \ and \'^....Ans. f = i- vi. Two-thirds of a cheese, and the one-sixth of the same cheese, make up what part of the whole cheese ?....^w^. f. ADDITION AND SUBTRACTION. 101 Fifth Square. i. What is the sum of ^ and ^?....Ans. -^-^; because "2 — To ' ^""^ y — To^ » ^"^^ eiore ^q t lo — lo* ii. What is the sum of 1|- and f?....A7is, -f-^ = 2y^-^; be- pin«5p li — ^ — i-A. 3. — _6_. tliprpfore -^^ -I ^ = -^J- = iii. What is the difference between f and ^?....A7is. -f^ because f = to 5 and | = ^^ ; therefore -^-^ — y^^ = y\. iv. What is the sum of |- and -^ ?....Ans, -^ = f. And so on to every square in this line. Third Line of Squares. Third Square, i. What is the sum of ^ and ^ ?.,..Ans. ^. ii. What is the sum of f and ^?....A7is, -|. iii. What is the sum of IJ and l^?..,.Ans. 2-|. iv. What is the difference between f and -| ?.. .^n^. ^; because f = f ; and therefore -| — f = 4. ? Fourth Square, i. What is the sum of -| and ^ ?....Jns, yV ii. What is the sum of ^ and f ? ....Atis, yf = lyV* iii. What is the sum of f and ^?...A7is. -^. iv. What is the difference between 2^ and H?..,.Ans, ^f = 1^ ; because 2^ = -| = f | ; and H = f = yf ; therefore 2 8 J_5 i_3 — 1 1 12 12 12 ^12* And so on to any other square. and -J. Fourth Line of Squares. Second Square. What is the sum of ^ and ^?....Ans, f . ' What is the sum of |- and f ?.,,,Ans. |-. What is the sum and difference of f and ^?...,Aiis. -^ 102 ON FRACTIONS. FIFTH EXERCISE. Fourth Square, i. What is the sum of } and -^?..,.Ans. '^. ii. What is the difiference between f and ■^-^?....Ans, ^. iii. A post is ^ in the mud, and ^ in the water, how much does it stand above the water ?,„.Ans. ^. Fifth Square, i. What is the sum of ^ and ^?....Ans. l|f . ii. What must I take away from -^ to leave ^?.,,,Ans. -/q. &c., &c., &c. Fiftli Line of Squares. Third Square, i. What is the sum of ^ and -^ ?„.,Ans, ^. ii. What is the sum of ^ and ^-^?....Ans, -^^ = ■§-. iii. What is the sum of -J- and -^?....Ans. -^. iv. What must be added to f to make \f ?..,,Ans. ^. Fourth Square, i. What is the sum of ^ and ■^?....Ans, -^ = ^. ii. What is the sum of i and -^-^ ?..,.A7is. -^ = ^, iii. What is the sum of ^ and ^?....Ans. -^. iv. What is the sum of f and f ?..,.Ans. f|- = 1^. &c., &c., &c. Fifth Square, i. What is the difference between ^ and -^ ?....Ans. -^-^, ii. What is the difference between f and ■^?..,.Ans. ^, &c., &c., &c. Sixth Square, i. What is the sum of ^ and 3^ ?...,A7is. ^ = ^. ii. What is the sum of ^ and -^ ?..,.Ans, -Jq, iii. What is the sum of -^ and y ?..,.Ans. ~^, &C.J &c., &c. And so on to other squares in this line. In hke manner questions on the squares contained in the sixth, seventh, eighth, &c. Unes may be given. 103 SIXTH EXERCISE. Multiplication of Fractions. The varieties of multiplication by fractions, to which reference was made in the fourth exercise, may now be studied. For the sake of convenience in teaching, this exercise is divided into three cases. 1st. Multiplication of a whole number by a fraction. 2nd. Multiplication of a fraction by a fraction when the numerator of each fraction is unity. 3rd. Multi- plication of a fraction by a fraction, when the numerators are whole numbers. For the illustration of the first case, the Board of Simple Fractions is required ; for the second case the Board of Compound Fractions; but for the third case both boards will be required. The proper board to be used will be stated before each table. Case I. Multiplication of whole numbers by a fraction^ or the fraction of a whole number. To demonstrate the operations performed with the assistance of this table, it will be found convenient, in certain cases, to draw a few squares, according to the form given in the diagram below, where the space, a, h, c, d, for example, repre- sents "4 times the |^ of 2 whole numbers, or 4 times f, that 104 ON FRACTIONS. SIXTH EXERCISE. On the Board of Simple Fractions. Second Line, or Line of Halves. 1^ of 1 whole number is ^ twice the |^ of 1 w. n. are f 3 times the |^ of 1 w. n. are f And so on. ^ of 2 w.n. is |- twice the ^ of 2 w. n. are twice -|, or 4 3 times the |- of 2 w. n. are 3 times f , or f And so on. 1^ of 3 w. n. is f twice the ^ of 3 w.n. are twice f , or |- 3 times the ^ of 3 w.n. are 3 times f, or -1 4 times the |^ of 3 w. n. are 4 times f , or -^ And so on. The teacher then goes on with the halves of 4, 5, 6, &c., whole numbers. Third Line, or Line of Thirds. ^ of 1 w.n. is ^ twice the ^ of 1 w. n. are f 3 times the J of 1 w.n. are f 4 times the |^ of 1 w. n. are -f And so on. ■J of 2 w. n. is f twice the \ of 2 w.n. are twice f , or ^ 3 times the ^ of 2 w. n. are 3 times f , or |- 4 times the |^ of 2 w. n. are 4 times f , or -| And so on. J of 3 w.n. is f twice the ^ of 3 w. n. are twice f , or 4 And so on. ^ of 4 w.n. is f twice the \ of 4 w. n. are twice ^, or |- And so on to the thirds of 5, 6, 7, &c., whole numbers. MULTIPLICATION. 105 Fourth Line^ or Line of Fourths. :^ of 1 w.n. is ^ twice the i^ of 1 w. n. are |- l w.n. are And so on. 3 times the ^ of 1 w. n. are f 1^ of 2 w. n. is f twice the :^ of 2 w. n. are twice f, or ^ 3 times the -^ of 2 w. n. are 3 times |^, or ^ 4 times the ^ of 2 w. n. are 4 times J, or f And so on. :J of 3 w.n. is f twice the :|^ of 3 w. n. are twice J, or ^ 3 times the } of 3 w. n. are 3 times f, or f 4 times the :f of 3 w. n. are 4 times f, or ^ 5 times the ;^ of 3 w. n. are 5 times f , or ^^- And so on to the fractional parts of 4, 5, 6, &c., whole numbers. Fifth Line^ or Line of Fifths. y of 1 w. n. is i twice the y of 1 w.n. are -§- 3 times the -§- of 1 w. n. are |- And so on. y of 2 w.n. is -f- twice the i of 2 w. n. are twice ■§-, or f 3 times the i of 2 w. n. are 3 times f, or f And so on. ■§- of 3 w. n. is f twice the ^ of 3 w. n. are twice f, or |- 3 times the ^ of 3 w. n. are 3 times f, or ■§• And so on. •i of 4 w. n. is f twice the i of 4 w. n. are twice j-, or f 3 times the i of 4 w. n. are 3 times f , or -Jg^ And so on to the fractional parts of 5, 6, 7, 8, &c., whole numbers. E 3 106 [O^ FRACTIONS. SIXTH EXERCISE. Sixth Line, or Line of Sixths. ■^ of 1 w. n. is ^ twice the -^ of 1 w. n. are -§-' 3 times the ^ of 1 w. n. are f And so on. ^ of 2 w. n. is -| twice the -^ of 2 w. n. are twice f, or f 3 times the ■§- of 2 w. n. are 3 times f , or 4 And so on. ■J- of 3 w.n. is f twice the ^ of 3 w. n. are twice f, or f 3 times the -^ of 3 w. n. are 3 times f, or f And so on to the lines of sevenths, eighths, ninths, &c., &c. Questions on Case I. Line of Halves, i. What is the half of S?....Ans, f; because the ^ of 1 = i, and therefore the |- of 3 = 3 times ^, that is f . ii. What is 3 times the -|- of 5 ?....Ans. --f- = 1\ ; because the i of 5 = f , and therefore 3 X ^ of 5 = 3 X |- or -V-. &c. &c. Line of Thirds. i. What is 7 times the |- of 5 whole numbers ?....^7W. ^ or llf ; because the ^ of 5 whole numbers is -|, and 7 times the ^ of 5 whole numbers are 7 times |- or ^. ii. What is f of 4 feet ?..,.Ans. 2 feet and f ; because the third of 4 feet is -| of 1 foot, and twice the ^ of 4 feet are twice 1^ or -| of a foot, which are 2 feet and f . Line of Fourths, i. What is the fourth of 3 ? ....Ans, f ; because the fourth of 1 = :g^, and therefore ^ oi 3=3 times ^^ that is f . ii. What is 3 times the i of 7 ?....^^5. -V- or 5x; because MULTIPLICATION. 107 the i of 7 = ^, and therefore 3 times the ^ of 7 = 3 times J = h'- or 5i. iii. What does it mean to multiply 7 by %'^.,„Ans. That the fourth of 7 is to be taken 3 times. Line of Fifths. i. How many fifths are there in the fifth of 9 '^,.,.Ans. ■§-; or i. because \ of 1 = y : and therefore the i of 9 = 9 times \ What is the ^ of 11 ?....^W5. V-=2i; because i of 1=| and the I of 11 = 11 times | or ^ = 2|. Monejj Weights^ and Measures, Line of Halves. i. What is the ^ of 5L?..,.Ans. 21. 10*.; because the J of 51. is -f. The |^ of H. is 10*., and ^L are 5 times 10*. or 2L 10*. ii. If 2 yards of cloth cost 3*., what will 7 yards cost?.... Ans. 10*. 6d.; because one yard will cost the ^ of 3*., that is, f*., and therefore 7 yards will cost 7 times f or y^s,, which are 10|- shillings = 10*. 6d. Line of Thirds. i. If 3 lbs. cost 7*., what will 11 lbs. cost?..,. Ans, IL 5*. 8d. ; because 3 lbs. cost 7*., 1 lb. will cost the ^ of 7*., that is, -| of a shilling ; 1 1 lbs. therefore will cost 1 1 times -J or -5^-^*., that is, 25|*.= 1/. 5*. Sd. ii. What is the |^ of 1 lb. ?.,..Ans. 5^ ozs. ; because 1 lb. contains 16 oz., and the ^ of 16 is -^^ or 5^ oz. Line of Fourths. i. If 4 articles cost 9c?., what will 5 cost ?....Ans. Hid.; because if 4 cost 9c?., 1 will cost the ^ of 9c?. or f of a penny ; and 5 will cost 5 times f or ^/-, that is, 1 l^c?. ii. What is the } of III. ?....Ans. 21. 15*. ; because the i of 11 is JJ- ; the } of IL is 5*., and ^ f » yS &c., is determined. Fourth Line, or Line of Fourths. Ratio of ^ to any number of fourths. J is the half of twice :| or |^. i is the third of 3 times ^ or f . i is the fourth of 4 times :4 or 4* ^ is the fifth of 5 times J or f. And so on. Ratio of -J to any number of fourths. f are 3 times 4» 4 are 4 times 4» 4 are 5 times ;i^. f are 6 times :i^. 4 are twice 4- Twice ^ are twice the third of 3 times ^. Twice ^ are twice the fourth of 4 times |. Twice ;J are twice the fifth of 5 times i* Twice ^ are twice the sixth of 6 times }* And so on. 134 ON FRACTIONS. EIGHTH EXERCISE. Ratio of f to any number of fourths, f are 3 times ^, -I are twice i» 3 times ^ are 3 times the half of twice i. J are 3 times ^, 3 times } are 3 times the third of 3 times ^, ^ are 4 times ^, 3 times J are 3 times the fourth of 4 times ^. |- are 5 times ^, 3 times ^ are 3 times the fifth of 5 times i» And so on. Ratio of j; to any number of fourths. -f are 4 times i, f are twice i* 4 times ^ are 4 times the half of twice ^. f are 3 times ^. 4 times i are 4 times the third of 3 times ^. f are 4 times ^. 4 times ;|^ are 4 times the fourth of 4 times :J. are 5 times ^. 4 times :j^ are 4 times the fifth of 5 times :|-. And so on. In the same manner the teacher may proceed to exhibit the ratio of j;, |-, ^, &c., to any number of fourths. As in the pre- ceding lines we have the following forms for the reduction of the ratios : Fourth Line^ or Line of Fourths. Ratio of f to the multiples of f . ^ are twice f. f are the half of twice f . ■| are 3 times f . f are the third of 3 times f. ■y- are 4 times f . f are the fourth of 4 times f . And so on. Ratio of f to the multiples of f . f are twice J. f are 3 times f . Twice f are twice the third of 3 times f. -y- are 4 times f . Twice f are twice the fourth of 4 times f . •^ are 5 times f. Twice f are twice the fifth of 5 times f. And so on. RATIOS OF FRACTIONAL NUMBERS. 135 And so on to the ratio of f, ^, ^, &c,, to the multiples of J. As a general illustration of the method the ratio of J/- to the multiples of |- may be g^iven. ^ are 3 times f . ^■£- are twice f . 3 times f are 3 times the third of 3 times |. ■^ are 4 times f . 3 times f are 3 times the fourth of 4 times f . -^''- are 5 times f. 3 times |- are 3 times the fifth of 5 times ^, --£- are 6 times f . 3 times f are 3 times the sixth of 6 times f . And so on. Questions on Case I. On the Board of Simple Fractions. . Second Line, or Line of Halves. i. What is the ratio of ^ to ^?....Ans. \ is the fifth off. ii. What is the ratio of f to ^"^ .,.,Ans. 3 times the fifth. iii. What part of 1 is \^? ....Ans. 3 times the half. Proof, \\ are f , 1 is f , f are 3 times \y f are twice |, 3 times \ are 3 times the half of twice \, Third Line, or Line of Thirds. i. What is the ratio of \ to ^7....A7is, \ is the sixth of -|. ii. What part off is f ?....^7z^. Twice the fifth. Proof f are 5 times \, f are twice \y twice \ are twice the fifth of 5 times \, iii. What is the ratio of f to ^"^ ....Ans, f are twice f . iv. How many times are f contained in b\?..,,Ans. 4 times. Proof b\ are -i^. ^ are 4 times f . V. What is the ratio of -^^ to 8?....^w^. -^ are twice the third of 8. Proof 8 are ^^, ^-^- are 3 times f , -^/ are twice |, twice f are twice the third of 3 times -!• 136 ON FRACTIONS. EIGHTH EXERCISE. vi. What is the ratio of -| to ^?....Ans. -| are twice the third of -if. Proof. -| are twice ^, ^ are 3 times ^, twice ^ are twice the third of 3 times f . vii. What is the ratio of^ to ^?....A7is. ^ are 5 times the half of f . Proof, ^ are 5 times f, f are twice f , 5 times f are 5 times the half of twice f . Fourth Line^ or Line of Fourths. i. What is the ratio of } to ■^?....Ans. ^ is the fifth of f . ii. W^hat is the ratio of |- to f?....Ans. 5 times the third. Proof, |- are 5 times ^, J are 3 times ^, 5 times ;|- are 5 times the third of 3 times ^, iii. What part of f is f ?,...^?25. f is the third of f. iv. What is the ratio of 1|- to ^"^....Ans, Twice the third. Proof \\ are f or -J, |^ are twice f , -J are 3 times f, twice f are twice the third of 3 times f . V. Determine the ratio of |^ of |-to 3|^ ^?25. ^. Proof y of ^ is ;i^, 3|^ are -^^, ^ is the thirteenth of ^. Fifth Line^ or Line of Fifths. i. W^hat part of f is \?....Ans, Twice the third. Proof f are 3 times \, f are twice -^, twice \ are twice the third of 3 times \. ii. What is the ratio of f to f '^....Ans. 3 times the half. Proof f are 3 times f, f are twice f , 3 times f are 3 limps the half of twice -§-. iii. What is the ratio of 2\ to \\'^..,,Ans. 11 times the sixth. Proof 2\ are -V-j H are |, -V" ^^'^ ^^ ^^^^^^ ^^^ ^^^^^ off. iv. What part of 3f is \\'^..,.Ans, The third part. Proof 3f are ^, \\ are f, -V- are 3 times f, f is the third of 3 times f . RATIOS OF FRACTIONAL NUMBERS. 137 Money, Weights, and Measures. Second Line, or Line of Halves, i. If |- a yard of cloth cost 4*., what will f yards cost ? Proof, \ are 3 times \^ the cost of \ is 4^. ; therefore the cost of \ will be 3 times 4*. or 12^. ii. What will be the amount of -| of a gallon of beer, when 1 galbn amounts to 2s,l....Ans. 5s. Proof. 1 gallon contains f, f are 5 times the half of -|, the cost of I" is 2*.; therefore the cost of -f-, which are 5 times the -i- of f , will be 5 times the half of 2*., the half of 2*. is 1^., and 5 times the half of 2^. are 5 times 1^. or os. iii. If 1^ yards of lace cost 9d,, what must I give for 2^ y3iYds?....Ans, Is.Sd, Proof, l^ are f, 2^ are f, |- are 5 times the third off; then as f cost 9c?., |-, which are 5 times the third off, will cost 5 times the third of 9c?. ; the third of 9c?. is 3c?., and 5 times the third of 9c?. are 5 times 3c?. or 15c?. = 1*. 3c?. Third Line, or Line of Thirds. i. What is the value of 3 lb. of tea, when ilb. cost 2^. ?.... Ans. ISs. Proof, 3 are f , f are 9 times i ; then as the cost of i is 2^., the cost of f , which are 9 times i, will be 9 times 2*. or 18^. ii. What is the cost of 41b. of sugar, when l^lbs. cost ls.6d.?.,..A7is. 4s. 6d. Proof 4 are ^, l\ are ^, ^ are 3 times ^; then as the cost of 4. is 1^. 6d,y the cost of ^, which is 3 times ^, will be 3 times 1^. 6d, or 4^. 6d. iii. What is the cost of 2-| yards of cloth, when Ii yards cost l^s,?....A7is. 11.6s. Proof 2f are -|, Ii = A, f are twice f ; then as the cost off is 13*., the cost of -|, which is twice f, will be twice 13*. or 26*. = 1/. 6s. 13& ON fKACTIONS. EIGHTH EXERCISE. Fourth Line, or Line of Fourths. i. If 1 yard of cloth cost 1 2^., what is the cost of f yards ? ,..,Ans, 9s. Proof, 1 is f , f are 3 times the fourth of f , the cost of 1 is 12*.; therefore the cost off, which are 3 times the fourth of 1, will be 3 times the fourth of 12*., the fourth of 12*. is 3*., and 3 times the fourth of 12*. are 3 times 3*. or 9*. ii. If 1 Flemish ell of cloth cost 1 5*., what must be given for 1 English ell ?....^n*. 11. 5s. Proof. I Flemish ell contains *f yard, 1 English ell f yard; f are 5 times the third of f ; then as the cost of J is 15*., the cost of |, which are 5 times the third off, will be 5 times the third of 15*.; the third of 15*. is 5*., and 5 times the third of 15*. are 5 times 5, or 25*. =11. 5s. iii. If 2^ lbs. of sugar cost 2*. Sd., what is the cost of 1 lb.? \.s,Ans. 1*. Proof 2^ are f, 1 is f , f are 4 times the ninth of f ; then the cost of |- being 2*. 3c?., the cost of ^, which is 4 times the ninth of |-, will be 4 times the ninth of 2*. 3c?., 4 times the ninth of 27c?. are 4 times 3c?. or \2d. = Is. iv. If the f of 2ilbs. of tobacco cost 3*. 9c/., what is the value of 5 lbs.?....^w*. 12*. 6d. Proof 2^ are -J, twice the third of ^ are twice f or ^, 5 are -^ ; -^ are 10 times f, ^ are 3 times f, 10 times f are 10 times the third of 3 times f; then as the cost of f is 3*. 9d., the cost of -^, which are 10 times the third of |^, will be 10 times the third of 3*. 9d., the third of 3*. 9c/. is 1*. 3c?. and 10 times the third of 3*. 9c?. are 10 times 1*. 3c?. or 12*. 6d. In the preceding exercise it has been assumed that the fractions, whose ratio is to be determined, are given in the same denominator, or that they have been brought into that form by the process given in the fifth exercise. Although, for all practical purposes, this may be deemed sufficient, it will still be found instructive, in some cases, to give the entire process of demonstration, as in the following table: KATIOS OF FRACTIONAL NUMBERS. 139 Case I.^ continued. On the Board of Compound Fractions. Third Square, Fourth Line. Ratio of J to any given number of thirds. ^ is 3 times -j^. I" is 4 times ^. 3 times -^ are 3 times the fourth of 4 times -j^* ■| are 8 times -^* 3 times yV ^^^ ^ times the eighth of 8 times ^l, ^ are 1 6 times -j^. 3 times -^ are 3 times the sixteenth of 1 6 times y^2-' And so on. Ratio of 1^ to any given number of thirds. |- are 6 time& -j^. i is 4 times -5^. 6 times ^^ are 6 times the fourth of 4 times ^. f are 8 times ^. 6 times ^^ are 6 times the eighth of 8 times -Jg . And so on. Ratio of J to any given number of thirds. f are 9 times ^. i is 4 times ^. 9 times -^ are 9 times the fourth of 4 times ^. I" are 8 times 3^. 9 times ^^2 are 9 times the eighth of 8 times -^. And so on. And so on to the ratio of |^, |^, f , ^, &c. On this square also the ratio of thirds to fourths may be exhibited. Fifth Square, Second Line. Ratio of \ to any given number of fifths. I" is 5 times -^. \ is twice 3^. 5 times -^ are 5 times the half of twice -^, |- are 4 times -^q. 5 times -^ are 5 times the fourth of 4 times -j^. And so on. 140 ON FRACTIONS. EIGHTH EXERCISE. Ratio of |- to any given number of fifths. f are 10 times ^. -I- is twice ■^, 10 times -^ are 10 times the half of twice ^. |- are 4 times -^, 10 times -^ are 10 times the fourth of 4 times ^. And so on. Ratio off to any given number of fifths. f are 1 5 times -^-q. -i- is twice yV 15 times -^ are 15 times the half of twice ■^^. -§- are 4 times ■^, 15 times y^^- are 15 times the fourth of 4 times ^. f are 6 times -5^. 15 times yV ^^^ 1^ times the sixth of 6 times -^. And so on. And so on to the ratio of f , |, f, -|-, &c. The ratio of fifths to halves may also be exhibited on this square. In like manner the teacher may proceed with any other lare unon the board. square upon the board. Questions on Case I.^ continued. On the Board of Compound Fractions. Third Square^ Fourth Line. i. What is the ratio betw^een ^ and \ ?....Ans. :5^ is 3 times the fourth of i. Proof, :^ is 3 times -^, i is 4 times 3^, 3 times y^ are 3 times the fourth of 4 times y^. ii. Compare f with f Ans, f are 8 times the ninth of |. Proof -I are 8 times -^, f are 9 times yV? 8 times yV are 8 times the ninth of 9 times -5^. RATIOS OF FRACTIONAL NUMBERS. 141 iii. f are how many times ^?....Ans, 15 times the eighth. Proof, -J are 15 times -^, -| are 8 times yV> 1^ times -^^ are 15 times the eighth of 8 times -^, Second Square, Fifth Line. i. How many fifths are there contained in f ?....Ans* 7|- fifths. Proof, i is twice -^^, f are J 5 times ^, 15 times -^^ are 15 times the half of twice -^, ii. What part of -^ is f?.,..Ans. f are 8 times the fifth off Proof, ^ is 5 times ^, f are 8 times -j^, 8 times ^^ are 8 times the fifth of 5 times -^, iii. 2|- are how many times f?..,.Ans, 25 times the fourth. Proof. 2\ are 25 times -j^, ■§- are 4 times -^V? ^^ times -^ are 25 times the fourth of 4 times -^, And so on to other squares. Miscellaneous Questions, i. What is the ratio of f to ^l....Ans. Twice the seventh. ii. What is the ratio of f to \^^.„,Ans. f are the half of H. iii. What part of f is \7..,.Ans. The third. iv. What is the ratio of f to ^'^.,„Ans, 6 times the seventh. V. Determine the ratio of -| of f to \ Ans. 4 times the seventh. Proof, Twice the third of -f- are twice -f- or f , f are. 4 times ^, -^ are 7 times i, 4 times \ are 4 times the seventh of 7 times \. vi. What part of ^ is the \ of -|?....-4/i*. Twice the seventh. Proof 1^ of I" are f, |- are twice the seveath of ^. 142 ON FRACTIONS. EIGHTH EXERCISE. vii. Compare f and f Ans. f are the fourth of -|. viii. What is the ratio of ^ to 2} ?....Ans. Twice the ninth. Proof. ^ is 1^, 2^ are f, J are twice the ninth of |-. ix. What is the ratio of ^ of -^^ to |- ?....Ans. \<. Proof, i of i is ^, i is the fifth of f . X. Compare ^ and -^ Ans. -| is 3 times ■^. Proof, i is f, f are 3 times ^. xi. Compare ^ and -|- Ans. -|- is 5 times the third ofi. Proof. \ are ^, -^ is ^, ^ are 5 times the third of ^. xii. What part of 2|- is the i of ^ ?..,.A7is. The tenth. Proo/: The i of -i- is i, 2^ are | or -^j ^ is the tenth of ^. xiii. Compare 2f with li Ans. 7 times the third. Proof. 2f are -i^, 1-|- are f , -L^ are 7 times f ,yare 3 times f, 7 times -§- are 7 times the third of 3 times f. xiv. If I" of a yard cost 6s., what must I give for f yard?....^w*. 4^. Proof. "I are twice the third of f ; then if 6s. are given for |- yards, twice the third of 6s. must be given for -§-; the third of 6s. is 2^., and twice the third of 6s. are twice 2*. or 4^. XV. If ^ lb. cost 6^d. what will 2|-lbs. cost ?....^w^. 5^. 5c?. Proof. 2^ are -I 05* ^ ; -^ are 10 times \ ; therefore the cost of ^ will be 10 times 6\d. or 5^. 5c?. xvi. What is the cost of 2\ ozs. of silver if f of an oz. cost 3^. 9d.?..„Ans. \0s. 1\d. Proof. 2\ are ^, f are f, -i/ are 17 times the sixth off ; then if the cost of -| be 3*. 9c?. or 45^., the cost of -y-, which are 17 times the sixth of-f, wall be 17 times the sixth of 45c?,; the sixth of 45c?. is 7|c?., and 17 times 7|c?. are 127|f?.= \0s. 7 id. xvii. Find the price of 4:^ lbs. of tea, when -|- lbs. cost Id... ..Ans. I9s. IQd. Proof A\ are ^ or -^, ^ are 34 times \ ; then if the cost of \ be 7c?., the cost of -^, which are 34 times \, will be 34 times Id. or 238^. = 19^. 10c?. RATIOS OF FRACTIONAL NUMBERS. 143 xviii. A person possesses f of |- of a ship, and |- of his, share is worth 360/., what is the worth of the whole ship?„t,' A71S. 810/. Proof. T of |- are -|, and f of f are -|, f therefore is sold for 360/.; and ^ will be worth the fourth of 360/. = 90/.; and "I, or the whole ship, will be worth 9 times 90/. = 810/. Case II. When a given Fraction is a given part of a Fraction required. Example. Of what fraction is f five times the seventh ipSivt?....Ans. \f. The principle involved in this case of ratios differs so little from that of the preceding case, that a few illustrations will suffice to exhibit the form of the exercise. On the Board of Simple Fractions. Line of Halves. Ratio of |- to any number of halves. ^ is the half of twice ^ or -| ^ is the third of 3 times |- or -| i is the fourth of 4 times ^ or |^ And so on. Ratio of f to any number of halves. -| are twice i Twice Y ^^6 twice the third of 3 times |- or |- Twice \ are twice the fourth of 4 times |- or -f Twice |- are twice the fifth of 5 times ^ or -| And so on. Ratio of f to any number of halves. 1^ are 3 times |- 3 times i are 3 times the half of twice ^ or |-, 3 times ^ are 3 times the third of 3 times -j or | 3 times ^ are 3 times the fourth of 4 times i or |- And so on. Then follows the ratio of f , |, f, &c. 144 ON FRACTIONS, EIGHTH EXERCISE. Reduction of Ratios. Line of Halves. Ratio of |- to any multiples of f . •f are twice f Twice -I are twice the third of 3 times f or -| Twice f are twice the fourth of 4 times f or ^- Twice -| are twice the fifth of 5 times |- or -^ And so on. Ratio of I" to any multiples of f . ■f are 3 times |- 3 times -| are 3 times the half of twice f or -| 3 times f are 3 times the third of f or f 3 times |- are 3 times the fourth of 4 times f or -^ And so on. And so on to the ratio of -y-, ^, ■^, See, to any multiples '— . Ratio of ^~ to any multiples of |. ^-^ are twice |- Twice -I are twice the third of 3 times |- or -y- Twice \ are twice the fourth of 4 times -| or ^ Twice f are twice the fifth of 5 times f or ^ And so on. Ratio of -y- to any multiples of f . ^ are 3 times f 3 times -| are 3 times the half of twice \ or ^ 3 times |- are 3 times the third of 3 times |- or ^ 3 times \ are 3 times the fourth of 4 times f or ^- And so on. And so on to any other ratios. REDUCTION OF RATTOS. 145 Line of Thirds. Ratio of i to any number of thirds, i is the half of twice -g- or -| ^ is the third of 3 times ^ ^^ I" ^ is the fourth of 4 times i or -f And so on. Ratio of f to any number of thirds. f are twice i Twice -i- are twice the third of 3 times ^ or f Twice J are twice the fourth of 4 times -g- or |- Twice i are twice the fifth of 5 times ^ or f And so on. Then follows the ratio of f, f, -f, &;c., to any number of thirc^s. Line of Thirds. Ratio of 4 to any multiples of f . -| are twice |- Twice "I are twice the third of 3 times f or -| Twice -| are twice the fourth of 4 times -| or -| Twice -| are twice the fifth of 5 times -| or J^ And so on. Ratio of -I to any multiples of -|. I" are 3 times -| 3 times f are 3 times the half of twice f or f 3 times f are 3 times the third of 3 times f'or f 3 times f are 3 times the fourth of 4 times | or -| And so on. Then follows the ratio of -f, -i/-, -y-, &c., to any' multiples of I. . ' 146 ON FRACTIONS. EIGHTH EXERCISE. Ratio of -y^ to any multiples of -|. ^ are twice -| Twice -| are twice the third of 3 times f or ^ Twice f are twice the fourth of 4 times f or -^ Twice -| are twice the fifth of 5 times f or -^ And so on. Ratio of ^ to any multiples of f. ^ are 3 times -| 3 times -| are 3 times the half of twice f or ^ 3 times f are 3 times the third of 3 times |^ or -^^ 3 times -| are 3 times the fourth of 4 times -f or -^ And so on. Then follows the ratio of ^, -?^, &c., to any multiples of |. And so on to any other ratios. Besides giving the ratio of fractions, this exercise furnishes a demonstrative method for effecting the division of frac- tions. When the divisor is greater than the dividend, it cannot be said, without some modification of language, that the one quantity is contained in the other. To avoid this diffi- culty it may, however, be said, without any impropriety, that the one is contained in the other a certain fraction of once. The most general definition of division is, that the quotient, or result, multiplied by the divisor, is equal to the dividend. For example, the quotient of f divided by \ must be a quan- tity, which multiplied by -§- will give f ; or f X quotient = J ; that is, f is twice the fifth of the quotient or number required ; the \ of the quotient, therefore, will be the half of f , and the quotient itself will be 5 times the half of f , or f of f . Hence we observe, that to divide one fraction hy another^ we must invert the divisor and then proceed as in Multiplication, For instance, to divide -H" ^7 I" ^^ ^^^ same thing as finding a number of which \^ is the ^ part ; the operation of which, by the exercise, is as follows \ \^ d^xe 1 times -^^ 7 times ^t are 7 times the eighth of 8 times -/y or \\ 2 1* RATIOS OF FRACTIONAL NUMBERS. 147 Questions on Case II. On the Board of Simple Fractions. Line of Halves. i. |- are 3 times the fourth of what numhev?....Ans. 2. Proof, 3 times ^ are 3 times the fourth of 4 times ^ or f , 4 are 2 whole numbers. ii. |- are 3 times the fifth of what number ?....A7is. \^- Proof, f are 3 times f , 3 times f are 3 times the fifth of 5 times f or ^2^=7^. Line of Thirds, i. "I are twice the fifth of what number ?....Ans. -| = If. Proof \ are twice i, twice \ are twice the fifth of 5 times JL or ^ = 1^ ii. -| are 3 times the fourth of what number ?....Ans, f — ^3- Proof -| are 3 times -|, 3 times f are 3 times the fourth of 4 times -| or f = 2f . iii. -y- are 3 times the half of what number ?....^?i^. ^-£~ = 31 Proof ^ are 3 times -|, 3 times f are 3 times the half of twice I or -i^ = 3^. iv. ^ are twice the fifth of what n\imber?....A7is. ~ = 8h Proof —- are twice -|, twice -| are twice the fifth of 5 times A or ^ = 8\. Line of Fourths. i. ^ is the third of what number ?....Ans, f . Proof ^ is the third of 3 times ^ or f . ii. f are 3 times the seventh of what number ?....^w^. 7 — 13. Proof f are 3 times ^, 3 times i are 3 times the seventh of 7 times i or ^ = If . G 2 148 ON FRACTIONS. EIGHTH EXERCISE. iii. -y- are 5 times the half of what number ?...,Ans. f = 1|-. Proof. ^ are 5 times J, 5 times f are 5 times the half of twice f or f = 1|^. iv. -|- are 3 times the fifth of what number ? . . . . Ans. ^ = 3f . Proof, \ are 3 times f , 3 times f are 3 times the fifth of 5 times f or -V" = 3f . Case II.;, continued. On the Board of Compound Fractions. Third Square^ Second Line. Ratio of |- to any number of thirds. -|- is 3 times -^ 3 times \ are 3 times the half of twice \ or \ 3 times \ are 3 times the fourth of 4 times ^ or -| 3 times -g- are 3 times the sixth of 6 times -5^ or 1 3 times -J- are 3 times the eighth of 8 times -J- or 1^ And so on. Ratio of -| to any number of thirds, •f are 6 times \ 6 times -5- are 6 times the half of twice -g- or ^ 6 times \ are 6 times the fourth of 4 times \ or f 6 times \ are 6 times the sixth of 6 times -J- or 1 6 times -g- are 6 times the eighth of 8 times ^ or ^ And so on. Ratio of f to any number of thirds. ^ are 9 times \ 9 times ^ are 9 times the half of twice -g- or i 9 times -J- are 9 times the fourth of 4 times \ or -| 9 times \ are 9 times the sixth of 6 times \ or 1 And so on. And so on to the ratio of f, f , f , &c., to any number of thirds. This square also exhibits the ratio of thirds to halves. Any other square may be treated in a similar manner. RATIOS OF FRACTIONAL NUMBERS. 149 'Questions on Case 11.^ continued. On the Board of Compound Fractions. Third Square, Second Line. i. J is 3 times the fourth of what number ?..,.Ans. f . Proof. J is 3 times ^, 3 times ^ are 3 times the fourth of 4 times ^ or f . ii. f are 9 times the fourth of what number ?.„.Ans. f . Proof f are 9 times ^, 9 times ^ are 9 times the fourth of 4 times |- or f . Fifth Square, Third Line. i. -g- is 3 times the fourth of what quantity ?.,,.Ans, -^, Proof y is 3 times -^, 3 times -^ are 3 times the fourth of 4 times -^ or ^. ii. f are 5 times the sixth of what quantity ?..../4^5. \^ = i- Proof f are 5 times -^, 5 times -^ are 5 times the sixth of etimes^^orif, if aref. iii. ^ are 5 times the ninth of what quantity ?..,.Ans, 2f . Proof f are 5 times -^, 5 times -^ are 5 times the ninth of 9 times -^ or ff = 2f . Miscellaneous Questions. i. Of what fraction is -g- the third part ?,...Ans, H, Proof i is the third of 3 times |- or f = 1^. ii. Of what is f the fourth part ?..,.Ans, 2f . Proof I is the fourth of 4 times f or | = 2f • iii. |- are 6 times what number ?....-4n*. -^, Proof f are 6 times the sixth of |, the sixth of f IS y-g. iv. f are 5 times what number ?....^?2^. -^-q. Proof f are 5 times the fifth of f , the fifth of f is ^. 150 ON FRACTIONS. EIGHTH EXERCISE. V. Of what number is f, 4 times the third ?....^7i^. 1-|-. Proof, I" are 4 times |-, 4 times |- are 4 times the third of 3 times |- or f = 1^, vi. Z\ are 4 times the seventh of what number ?....Ans, 6^, Proof. Z\ are -I-, -|- are 4 times |-, 4 times -|- are 4 times the seventh of 7 times |- or ^9- = 6-|-. vii. 1^ is 3 times the fifth of what number ?....^n^. 2|-. Proof l-i- are |^, ^ ^ire 3 times ^, 3 times ^ are 3 times the fifth of 5 times f or -^o- = 2f . viii. 2|- is the fourth of what number '^..,.Ans. 11|-. Proof, 2|- are ^, -^^ the fourth of 4 times -^ or ^= 1 \\. ix. Of what is the third of -f-, the fifth part l...,Ans, ly. Proof The third of f is f , -f- is the fifth of 5 times f or ,10 13 X. Of what number is the |- of 3 the half ?....-4?i5. ^\. Proof The eighth of 3 is -|, and 7 times the eighth of 3 are 7 times f or ^, V" is the half of twice -y- or ^ = 5^. xi. 1\ are 5 times what number ?....-4?i^. 1|-. Proo/. 7i are -^j -^2^ are 5 times the fifth of ^, the fifth of ^ is f = li. xii. 5 times the sixth of 3|- is 6 times what number ?.... Ans. f|. Proof 3-i- are I-, the sixth of |- is ^, and 5 times the sixth of ^ are 5 times ^ or ff, ^4 ^^^ ^ times the sixth of ■ff, the sixth of ff is ||-. xiii. Of what number is J of f , 3 times the eighth part? ..,,Ans, 1. Proof i of f is -|, f are 3 times -|-, 3 times -|- are 3 times the eighth of 8 times -|- or 1 . xiv. The third and fifth of a number is 24 ; required the number Ans, 45. Proof \ and \ are -j^, that is, -^ of the number required is 24 ; then 24 are 8 times 3, 8 times 3 are 8 times the fifteenth of 15 times 3 or 45. XV. f of my property sold for 270/., required the value of the whole Ans. 360/. Proof 270 are 3 times 90, 3 times 90 are 3 times the fourth of 4 times 90 or 360. DIVISION. 151 xvi. What is the worth of a field, when f of f sold for 1801.?.,. .Arts. 450L Proof, f of f = f ; then 1 80 are twice 90, twice 90 are twice the fifth of 5 times 90 or 450. Or thus ; as f is worth 180^., i will be worth the half of 180/., that is, 90/.; and 1 or 1^ will be 5 times 90 or 450/. xvii. 2 are 3 times \ of what number ?....Ans, ^= 3^. Proof, 2 are 3 times -f ; 3 times f are 3 times y of 5 times I or J/. xviii. 5s. are 3 times ^ of what sum?....^n^. 6s. Sd, Proof. 5 are 3 times f; 3 times -f are 3 times :|^ of 4 times ■I or ^, ^s. = G^s. = 6s. Sd. xix. |- of a number is 4 ; what is the number ?....^wa'. ^-f- if- Proof. 4 are 7 times f ; 7 times f are 7 times y of 5 times j. or ^-^ = 2f . Or thus ; as 7 times \ of the number required = 4; i of the number will be the -^ of 4 or f ; and therefore the number itself = 5 times j- or %P: — 96 ADDENDA TO THE EIGHTH EXERCISE. Division of Fractions. 1. Division of Whole Numbers by a Fraction. i. Divide 1 by^ Ans. 4. ' Proof. Because 1 = ^; therefore 4. -s_ i = 4. ii. Divide 2 by \ Ans. 6. I Proof. 2 = 1; therefore f -J- i = 6. iii. Divide 1 by f Ans. f = 1 J. Proof. 1 = f , and f H- i = 3 ; therefore f -r- f = the half of 3, that is, f . i^ iv. Divide 4 by ^ Ans. ^ = If. Proof. 4 = ^, and ^ -^ i = 1 2 ; therefore if ^ | = the seventh of 1 2, that is, -y . 152 ON FRACTIONS. EIGHTH EXERCISE. V. Divide 7 by i Ans. 21. Proof. Because each unit contains \ three times ; therefore 7 will contain seven times 3 or 21 thirds. vi. How many halves are there contained in 5 1 ,.,,Ans» 10. Because each unit contains 2 halves, and therefore 5 will contain 5 times 2 or 10 halves. vii. How often will f, then, be contained in 5?....-4w^. The third of 10 times, or ■^. Because |- will be contained the third of the number of times which \ is contained in the given fraction. viii. Divide 5 by -| Ans. 9. ix. Divide 3 by 1\ Ans. xr = f. X. Divide 9 by f Ans. ^ = 13^. 2. Division of a Fraction by a Whole Number. i. Divide \ by 1 Ans, \. Because 1 = f ? and -| -t- i =: 1 ; therefore i -f- f = the third of 1 or \. ii. Divide \ by 4 Ans. -^. Proof i -f- i = 1, but 4 = -1/; therefore | -r- ^ = the -^ of 1, that is, -j^. iii. Divide \ by 5, by another method Ans. -^. Proof. Because the quotient x 5 = ^J ; therefore the quotient = i of ^, that is, -^q. iv. Divide f by 7 Ans. ■^■^. ^ Because f -t- i = 3, but 7 = •^; therefore J -f- -^ = the twenty-eighth part of 3, that is 3 2 8' 3. Division of a Fraction by a Fraction. i. Divide ^ by ^ Ans. 2. Proof. Because i = f , and therefore ^-^ i=2, ii. Divide f by i Ans. -V^- = 3^. Proof Reducing the fractions to the same denominator, i=:if i==i^' then i|-3V= 10; therefore ii^^ = the third of 10, or ^. iii. Divide f by i, by another method Ans. -^/ = 2|-. Proof i X quotient = f ; therefore quotient = 4 times 3 12 ON FRACTIONS. NINTH EXERCISE. 153 iv. Divide f by ^ Ans, If. Proof. Reducing the fractions to the same denominator, f = if » 5- = TT- ^^^ iV is contained in |f , ten times; there- fore T§- -J- -fi = the twenty-first part of 10, that is, if-. NINTH EXERCISE. Proportion of Fractional Numbers . This exercise is intended to show that the relations of unity, demonstrated in the Sixth Exercise of Book I. are also appHcable to fractional units or fractional parts having the same name. After what has been proved in that exercise, a few illustrations, in each case, will render this truth sufficiently obvious. This exercise comprises three cases : 1st. When the second term of the proportion is divisible by the first; 2nd. When the first term is divisible by the second ; and 3rd. When the first and second terms have any ratio. The Board of Simple Fractions only is used for the exercises, but a few questions appended to the third case are given in connection with the Board of Compound Fractions. Case I. When the second term of the proportion is divisible by the first. On the Board of Simple Fractions. Third Line^ or Line of Thirds. ■| are twice i I is to 2 X I or I as I is to 2 X I or -I ■| are twice ^ |:2x|orf::|:2x|orf ■| are twice ^ f:2x|or-|::|:2xf or|. And so on. G 3 154 ON FRACTIONS. NINTH EXERCISE. f are 3 X i i:3xiorf::f:3x|or| -I are 3 X | |:3x|or|::f:3xiorf ■I are 3 X f f:3xforf::A:3xAorY. And so on. Then follow the tables beginning with f are 4 X i, f are 5 X i, &c. Fifth Line^ or Line of Fifths. f are 2 X i i:2xior4::f:2xf or| f are 2 X -§- f:2x|orf::|:2xf orJ^ f are 2 X f f:2xf orf::^:2x^orY. And so on. f are 3 X i i:3xiorf::f:3xf orY f are 3 X f f:3x|orf::^:3x^ori^ I- are 3 X f f:3xf orf::^-3x-i^or^«. And so on. Then follow the tables beginning with f are 4 X \, ^ are 5 X |, &c. Similar tables may be formed for the proportion of halves, fourths, sixths, &c. PROPORTION. 155 Second and Fourth Lines, or Lines of Halves and Fourths. ■| are twice ^ ^ is to twice ^ or -| : : :y : twice ^ or •|- are twice |- |- : twice f or -| : : f : twice f or |- |- are twice ■§- f : twice f or |- : : f : twice f or f . And so on. f are 3 times |- ■J : 3 times |^ or f : : ^ : 3 times J or f •| are 3 times f f:3xforf::f:3xf orf f are 3 X f i:3xf orf::f:3xf orf. And so on. •f are 4 times i J:4xiorf ::i:4x-i-orf •f are 4 times |- f:4xf or|::f:4xf or|- ~ are 4 times |- f :4 X f or -i^^ :: f :4 X f or -^/. ' And so on. And so on, as before. Other combinations of hnes may be treated in the same manner. 156 ON FRACTIONS. NINTH EXERCISE. Questions on Case I. Second Line, or Line of Halves. i. ^ :^:: ^iwhsit rmmher 7... .Ans. -3-?-= 6. Proof, f are 3 X 1-3 therefore f:3 xf or f ::-|:3 X for -V". ii. |. :^:: I : what number ?....^W5. -^=6. Proof. 2_o are 4 X f ; therefore f : 4 X | or -^g^. : : f : 4 X f Second and Third Lines^ or Lines of Halves and Thirds, i. A :|-::f : what number ?....^?i5. |-=2. Proof fare 3 x f ; therefore f :3 xf or f :: f :3 xf or |. ii. i: 2i::f : what number ?....^n^. ^=H- Proof 2i = f , f are 5 X |- ; therefore | : 5 X ^ or | : : f : 5x|or-^. Case IL When the first term of the proportion is divisible by the second. Third Line, or Line of Thirds. i is the i^ of -1 2 3 :ioff ori::A:iof|or| 1 are i of A A' 3 :iofior|::f:iof|or| 4 are i of -1 3.'JL of -^ or ■^••-^•1 of -^ or A 3^*2 '^^ 3 ^^ 3**3*2 ^^ 3 ^^ 3* And so on. ^ is the "5^ of f f:ioff ori::f:ioff orf ■| are the -g- of f f:ioff or|::f:ioff orf. And so on. And so on, as before. PROPORTION. - 157 Fifth Line, or Line of Fifths. i is the i of I f:ioff ori::|:ioff orf -f- are the i of f f are the -J of f f:ioff orf::Y:iofYorf And so on. i is the i of f |:ioff ori::i/:iof-^/orf f are the -§■ of -f |:ioffor|::XL:^of^or| ■§■ are the ^ of |- ftioff orf::-^/:iof^orJ/. And so on. And so on, as before. Similar tables may be formed for the proportion of halves, fourths, sixths, &c. Second and Fourth Lines^ or Lines of Halves and Fourths. I is the -1^ of f -2.2 OI 2 Ol 2 • . 4 . T OI 4. Ol 4 f are the i of f A • i of ^ or 2. . • A • i of ^ or ^ 2 . 2 ^A 2 "^ 2 • • 4 • 2 ^^ 4 "* 4. f are the i of f 6 • i of A or A • • 6. • i of A ov -^ 2" • 2 "^ 2 ^^ 2 • • 4 • 2 01 4 OX 4. And so on. Y is the ^ of f 3. . JL of -1 or i • * ^ • i of ^ or J- 2 . 3 ui 2 "A 2 • • 4 • 3 "^ 4 "^ 4 f are the i of f A-j. of A or 2..- 6.-1. of 6 or -^ 2 . 3 Ul 2 "^ 2 • • 4 • 3 "^ 4 ^^ 4 •| are the -5- of -| 9. -i of 9. or A ••5- -X of ^ or A- 2 . 3 ui 2 ^^ 2 • • 4 • 3 '^^ 4 '-'^ 4 • And so on. 158 ON FRACTIONS. NINTH EXERCISE ^ is the fourth of 4 i:lofiovi::i:iofiovi f is the i of l JL'JL of A or-?^"-^*-! of -^ or-^- 'a-.^ oi -2 oi 2 •• 4 -4 "^ 4 "^4 f are the i of -^ i_2 -i of i_2 or J-"l-2.»i of iA or 3- -2" . 4 oi -g- or "2 . . -J- . 4 oi -4- or ^. And so on. And so on, as before. Other combinations of lines may be treated in the same Questions on Case II. Second Line, or Line of Halves. i. -^ : -|- :: I" : what number ?.,..Ans. f = 2. Proof, i. are i of Y; therefore -^^ : i of ^ or 1 : : f : | of f or f. ii. 4i : f :: -y- : what number ?....^w*. f = 2|-. Proof. 4^ are f, f are -| of f; therefore f : 3- of f or f . . iA . i of -i-^ or -^ • • 2 • 3 ^^ 2 "^ 2 • Third Line, or Line of Thirds. i. -^ : "I : : f : what number ?....^?i5. -I. Proof f are i of ^; therefore^ : i of -y- : : | : ^ of A or -^ ii. -^ : I : : 6| : what number ?.,..Ans. f = 1|. Proof 6| are ^, f are 1 of -i^; therefore -i^ : i of -i^ or 2. . . 2_0 . i. Qf 2_0 Qj, 4 Fourth Line, or Line of Fourths. i. f : f : : ^ : what number ?....Ans. ^ = If. Proof J are ^ of f; therefore f : |^ of f or J : : ^ : ^ of ii. 2^ : f : : 3f : what number ? ....Ans. f = l^. Proof 2i are f, f are i of f ; therefore f : i of f or 3. . . j_5 . i. of x_5 or -^ 4 •• 4 • 3 "^ 4 "^ 4* PROPORTION. 159 Second and Third Lines, or-Lines of Halves and Thirds. 1. Q ' -i^ : what number ?.,,.Ans, 4 = It- 3 Proof, i is i of f ; therefore f : -i- of f or 1 : : i/ : i of • or ^. ii. 4|- : 1 J : : 7-|- : what number ?....Ans. f = 2^*. Whole Numbers and Fifths. i. 9 : 3 : : f : what number ?....Ans, f. Proof. 3 are i of 9 ; therefore 9 : i of 9 : : f : ^ of f or f. ii. 16 : 4 : : f : what number ?....^^,^. -|. Proof 4 are i of 16; therefore 16 : ^ of 16 or 4 : : | : i- off orf. Case III. When the first and second terms have any ratio. Third Line^ or Line of Thirds. Where the ratio is 5 X |-. f are 5 X ^ of f f:5xiofforf::f:5xiof4or-Lo ■LQ. are 5xi off A : 5 X^ of f or i^::| : bx\ of f or J/ -V-are5xiof-| A : 5 X i of f or -V- : : f : 5 X i- of I r ^^ And so on. And so on to any other ratio. 160? ON FRACTIONS. NINTH EXERCISE. Third and Fifth Lines, or Lines of Thirds and Fifths. fare Sx^of -| -|:5xiof|or|::|:5xiof|- or| -i^areSxioff i : 5 xi of i or ^::i: 5xi of i ov ^ J^ are 5 X i of 4 And so on. : 5 X ^ of f or -i^ : : f : 5 X i of f or Y • And so on to any other ratio. The preceding tables are constructed to find the fourth term of a proportion when three terms are given ; but if two fractions be given to find their proportion, the following form may be used with advantage. 3 T are 3x^ off 3 are 3xi of 2 2 T . 3 .. •y • :2:3 f are 3xi off 3 are 3xi of 2 i :f: :2:3 fare 3x^off 3 are 3 X ^ of 2 f :f ::2:3. And so on. The form of these tables may be varied, to suit particular questions. For instance, 2 : 2-i- :: | : w^iat number ?.. ..In this case 2| are once 2 and- the -J of 2, and 2 : once 2;and | of 2 : : -I : once f and l of f , that is, ^. Again, i : ^ : : i is to what number ?....Here } is the |- of |-; and then, i • i nf i nv i • • X • i of i or -i-. PROPORTION. 161 Questions on Case III, On the Board of Simple Fractions. Third Line^ or Line of Thirds. i. f : 1| : : 2| : what number ?....Ans. ^ = 6|. Proof. If = f, I are 5 X J of I; therefore | : 5 X i of | orf::-|:5 X 1 of | or -^/. ii. 3i : 2i : ; -^ : what number ?.,.,Ans, ^—7, ill. f : -^ : : 7 : what number ?.,..Ans. -^ = 17-J-. Seventh Line^ or Line of Sevenths. i. What proportion does f bear to ^ ?.,,.A72s, 2 to 3. Proof |- are 3 X ^ of f , 3 are 3 X i of 2 ; therefore f :f :: 2: 3. ii. What proportion does j- bear to 2^7. ,..Ans* I to 3. iii. What proportion does |- bear to -y- ?...Ans\ 3 to 4. iv. What proportion does f bear to 2f ?....Ans. 2 to 5. V. What proportion does 1 bear to -f- ?.,..Ans. 7 to 9. On the Board of Compound Fractions. Second Line, Third Square. i. f : i :: f : what number ?....Ans. f = 1. Proof f are f, i is f , f are 3 X ^ of -1; therefore ^ : 3 Xiof f orf ::A:3xi ofAorf. ii. 1 : J : : -^ : what number ?....^/i5. \. iii. |- : -g- : : f : what number ?....Ans> f. iv. -J : f : : 1 : what number ?..„Ans. -f = 2|-, 162 ON FRACTIONS. NINTH EXERCISE. Miscellaneous Questions. i. "I is to -I as ■§- is to what number ?..,.Ans, f. Proof, f are twice f ; therefore f : twice f or f : : |^ : twice "• ^ ' i '-' -Y' ' ^^ unknown number ?....-4w^. f . Proof i-areiof-y-; therefore ^ : i of -^i or J :: ^ : i ofijS orf iii. f : f :: f : what number ?.,..Ans. f or 1-f-. Proof f are f or 3 times f . Hence f : 3 X f : : -f- : 3 X forf iv. 4 • i • • "^ • ^^^^ number ?.,.,Ans. -|. v» "O" J ^ • I f • an unknown number ?....Ans. ^ or 2^, vi. -f- : f : : "l^ : an unknown number ?....Ans. -^ = -f^. vii. What number has the same proportion to ^ that \ has to -f^?....^w^. 2. viii. y : 2^ : : ^ : what number ?..,,Ans, -f. ix. 1-|- : f : : 1-| : what number ?....Jns, -f . X. f i z • • if • what number ?....-4w5. -|. xi. f • T • • xf • ^^ unknown number ?....^«^. ij, Proo/: f are if, f are i|, if are 1 5 X -Jg- of if ; there- fore if : 15 X -i^ of if or if :: if = 15 X ^\ of if or ^. xii. i : f : : f : what number ?...Ans. -gV Proo/ i is I, f are 3 X i of f ; f : 3 X i of I or f : : f :3 xioff or^V xiii. f : ^ : : 8 : what number ?.,..Ans, 12. xiv. i *• y : I "1^ • what number ?....Ans, -g^. XV. i : f • • f * what number?.... ^72^. ^. xvi. IJ : 2i : : 2i : what number ?....Ans. ^ or 2f . xvii. 7i : 1 : : 9i : what number ?....^w^. ff or 1-^^. xviii. 8 : 4 : : I" : what number ?....Ans. -f. Proof. 4 are i of 8; therefore 8 : i of 8 or 4 : : f : i of • -A or -a . PROPORTION. 163 XIX. 4 : 6 : : "I : what number ?....Ans. 1. Proof, 6 are 3 X i of 4; therefore 4 : 3 X ^^ of 4 or 6 : : 3 X i of for 1. XX. i of i : an unknown number :: 2 : 8 Ans. 1. xxi. What proportion does 3^ bear to \ ?..,.Ans, 21 to 2, Questions on Money, Weights, and Measures. i. If 15 yards cost 7*. 6d,, what is the amount of 1^ yards ? „,,Ans, 9d, Proof. 15 = ^, and H = hi are -^ of ^, 7^. 6d. are 90 pence ; hence ^ : ^ of -^ :: 90^;. : -^ of 90^^. or 9d. ii. If 20 yards cost 15^., what is the price of 1|- yards ? ,...Ans. Is. Sd. iii. If 3f yards cost 3^. 9c?., what is the price of -f- of a yard?....^w5. y^d. iv. If 2i yards cost 11. 10^., what is the cost of f of a yard?.... -4725. 9^. V. If goods be bought for 7^d. and sold for 9d., what is the gain upon 60^. worth of the same goods ?....Ans. 12/. Proof. 7^d. taken from 9c?. leaves l^c/. for the gain on 7-|c?. ; hence 7|- : 1^ : : 60/. : the gain required, but 7^ = ^, and li = f , f are the \ of \^- ; hence ^ : \ of \^- or f :: 60/. :-i-of 60/.or 12/. vi. If on a shilling I gain 3c?., what do I gain per cent. ? .„.Ans. 25/. vii; A ton of tallow cost 20/., and was sold for 22/. 10,9., how much per cent, was gained ?....Ans. 12^ per cent. viii. If I buy cloves for 6s. 3c?. per lb., and sell them for 6s., how much per cent, is lost ?,...Ans. 41. ix. If 7 lbs. cost 3^. 6d.y how much will 17-5- lbs. cost ?.... Ans. 8s. 9d. 164 ON FRACTIONS. NINTH EXERCISE. X. If 6 men working for 9 days earn 41., how much would 3 men earn in 12 days at the same rate of wages ? ,,.,Ans. '21. l^s.Ad, Proof, 6 men working for 9 days produce 54 days' work, 3 men for 12 days produce 36 days' work, 36 are 2 X i of 54; therefore 56 : 2 X i of 56 or 36 :: 4/. : 2 X i of Al or |/. ; f/. = 2|/.=: 2/. 13^.4^. xi. A cistern can be filled by 3 pipes : by the first in 2 hours, by the second in 3 hours, and by the third in 4 hours ; in what time will the cistern be filled, when the 3 pipes are opened at once ?....y4w*. -i-f hours. Proof, As the first pipe would evidently fill \ the cistern in 1 hour ; the second \ part in 1 hour ; and the third \ part in 1 hour; they must, together, fill i -f i -\- \ part in 1 hour, that is, if- in 1 hour : from this it follows that the ^^ of the cistern may be filled in -J3 of an hour, and therefore ^, or the whole, in 12 times -^ of an hour, or \^ hours. xii. How much tea at Qs. per lb. must be given for 40 pairs of stockings, at 2^. per pair ?....24/z5. 13^ lbs. Proof 6 are 3 times 2, that is, a lb. of tea costs 3 times the price of a pair of stockings ; therefore, the number of lbs. of tea to be given will equal \ the number of pairs of stockings ; that is, ^ of 40 is ^ = 13^. Or we have by proportion, 6 ; i of 6 or 2 : : 40 : i of 40 or ^. xiii. A and B commenced trade with 200/., of which A advanced 150/. and B the remainder; they gained 40/.; required each man's share of the profit ?.... Ans, A's share = 30/., and B's share = 10/. Proof, 150/. are 3 X i of 200/., that is, A's money is | of the whole ; but each man's share of the profit must be in pro- portion to his share of the principal ; therefore A's share of the profit will be f of 40/. = 30/., and therefore B's share = 40/.— 30/. = 10/. 165 TENTH EXERCISE. Square and Solid Measure. In this exercise, duodecimals or cross multiplication, will be considered to belong to the subject of fractions. In the first part of the exercise the pupil is shown, how a square unit is formed from a lineal one ; and the latter part shows how to obtain the product of two quantities, not expressed in the same lineal unit. In teaching this exercise, a table of square measure will be found useful. This table is supposed to represent a square foot divided into 144 parts representing square inches. 12 3 4 5 6 7 8 10 11 12 1 1 2 5 4 5 6 7 8 9 10 11 1 ^ Table of Square Measure. 166 ON FRACTIONS. TENTH EXERCISE. One square foot contains 144 square inches. 1 in. by 1 in. contains 1 sq. in. 1 in. by 2 in. contains 2 sq. in. 1 in. by 3 in. contains 3 sq. in. 1 in. by 4 in. contains 4 sq. in. And so on. 1 in. by 1 in. contains 1 sq. in. 2 in. by 1 in. contain 2 sq. in. 3 in. by 1 in. contain 3 sq. in. 4 in. by 1 in. contain 4 sq. in. And so on. 2 in. by 1 in. contain 2 sq. in. 2 in. by 2 in. contain 4 sq. in. 2 in. by 3 in. contain 6 sq. in. 2 in. by 4 in. contain 8 sq. in. And so on. 1 in. by 2 in. contains 2 sq. in. 2 in. by 2 in. contain 4 sq. in. 3 in. by 2 in. contain 6 sq. in. 4 in. by 2 in. contain 8 sq. in. And so on. 3 in. by 1 in. contain 3 sq. in. 3 in. by 2 in. contain 6 sq. in. 3 in. by 3 in. contain 9 sq. in. 3 in. by 4 in. contain 12 sq. in. And so on. 1 in. by 3 in. contains 3 sq. in. 2 in. by 3 in. contain C sq. in. 3 in. by 3 in. contain 9 sq. in. 4 in. by 3 in. contain 12 sq. in. And so on. 4 in. by 1 in. contain 4 sq. in. 4 in. by 2 in. contain 8 sq. in. 4 in. by 3 in. contain 12 sq. in. 4 in. by 4 in. contain 16 sq. in. And so on. And I 1 in. by 4 in. contains 4 sq. in. 2 in. by 4 in. contain 8 sq. in. 3 in. by 4 in. contain 12 sq. in. 4 in. by 4 in. contain 16 sq. in. And so on. In tbe same manner it may be shown that feet by feet produce square feet, yards by yards produce square yards^ &c. 2 ft. 1ft. 1 in. 2 in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 10 in. 11 in. 12 in. SQUARE AND SOLID MEASURE. 167 In the preceding table feet by inches, or inches by feet, are shown to produce twelfths of feet : 12 sq. inches are 1 tAvelftli of a I 1 twelfth of a sq.foot contains 12 sq. foot. I sq. inches. 1 ft. by 1 in. contains 1 twelfth of a sq. ft. 1 ft. by 2 in. contains 2 ditto I ft. by 3 in. contains 3 ditto ^ 1 ft, by 4 in. contains 4 ditto And so on. 2 ft. by 1 in. contain 2 twelfths of a sq. ft. 2 ft. by 2 in. contain 4 ditto 2 ft. by 3 in. contain 6 ditto 2 ft. by 4 in. contain 8 ditto And so on. 3 ft. by 1 in. contain 3 twelfths of a sq. ft. 3 ft. by 2 in. contain 6 ditto 3 ft. by 3 in. contain 9 ditto 3 ft. by 4 in. contain 12 ditto And so on. 1 in. by 1 ft. contains 1 twelfth of a sq. ft. 2 in. by 1 ft. contain 2 ditto 3 in. by 1 ft. contain 3 ditto 4 in. by 1 ft. contain 4 ditto And so on. 1 in. by 2 ft. contains 2 twelfths of a sq. ft. 2 in. by 2 ft. contain 4 ditto 3 in. by 2 ft. contain 6 ditto 4 in. by 2 ft. contain 8 ditto And so on. 1 in. by 3 ft. contains 3 twelfths of a sq. ft. 2 in. by 3 ft. contain 6 ditto 3 in. by 3 ft. contain 9 ditto 4 in. by 3 ft. contain 12 ditto And so on. And so on. 168 ON FRACTIONS. TENTH EXERCISE. Questions on Square Measure. In proposing the following questions, the teacher is recom- mended to place before the class the figure, upon which each question is given. i. How many square feet are contained in 200 square inches ?....Ans, 1 square foot and 56 square inches. ii. How many square inches are contained in a surface which measures 3 inches by 5 inches ?..,.A7is, 15 square inches. Proof, 3 in. by 1 in. contain 3 square in.; therefore 3 in. by 5 in. will contain 5 times 3 square in., that is, 15 square in, iii. How many square feet are contained in a surface which measures 4 feet by 3 feet ?....Ans. 12 square ft. iv. What amount of surface is contained by a rectangle, whose length is 2 ft. and breadth 5 inches ?..,.Ans. 10 twelfths of a square ft. Proof, 1 ft. by 5 in. contains 5 twelfths ; and therefore 2 ft. by 5 in. will contain twice 5 twelfths or 10 twelfths. V. What is the superficial area of a deal board 2 feet by 11 inches ?,.,.Ans, 1 square foot and 10 twelfths of a square foot. Proof 1 ft. by 11 inches contains 11 twelfths ; and there- fore 2 ft. by 1 1 inches, contain twice 1 1 twelfths or 22 twelfths, 22 twelfths are 1 square ft. and 10 twelfths of a square ft. vi. The length of a desk is 10 ft., the breadth 2 ft. 3 in., what is its surface ?„„Ans, 22 square ft. and 6 twelfths. Proof 10ft. by 2 ft. contain 20 square ft., 10 ft. by 3 in. contain 30 twelfths, or 2 square ft. and 6 twelfths ; then 20 square ft. + 2 square ft. and 6 twelfths = 22 square ft. and 6 twelfths. vii. 10 in. by 4 ft. 2 in Ans. 3 square ft. 5 twelfths and 8 square inches. Proof 10 in. by 2 in. contain 20 square in., or 1 twelfth and 8 sq. in.; 10 in, by 4ft. contain 40 twelfths or 3 square SQUARE MEASURE. 169 feet and 4 twelfths ; then 3 square ft. and 4 twelfths -f 1 twelfth and 8 square in. = 3 square ft., 5 twelfths, and 8 square in. viii. The length of a door is 6 feet, and the breadth 3 ft. 5 in., what would it cost at the rate of 1*. 6d. per square ft. ? ....Ans, \lA0s,9d. Proof, 6 ft. by 3 ft. 5 in. contain 20 square ft. and 6 twelfths; then 20 square ft. at 1*. 6d. = IL lOs.; 6 twelfths are ^ of a square foot, ^ of 1^. 6d, is 9d.; therefore the cost would be ll.lOs.Sd. ix. The school yard is 20 feet long by 12 ft. 3 in. broad, what would be the value of the pavement at 1*. per square foot ?....Ans. 121. OS. X. What would be the cost of flooring a room 21 feet long by 15 feet broad, at 6s, per square yard ?..,.Ans, lOl, lOs, Proof, 21 ft. = 7 yards, 15 ft. = 5 yards, 7 yards by 5 yards, contain 35 square yards, then 35 at 6s, = lOZ. 10^. xi. Prove by a figure that 1 square yard contains 9 square feet. xii. Represent the product of 2 feet 3 in. by 3 feet Ans, The space abed represents the product of 2 feet by 3 feet = 6 square feet : c d ef the product of 3 in. by 3 ft. = 9 twelfths, so that the whole rectangle a h ef^ which is 2 ft. 3 in. by 3 ft. = 6 square ft. and 9 twelfths. a CO h f 2 ft. d\\m. 170 ON FRACTIONS. TENTH EXERCISE. xiii. Give a representation of the product of 2 feet 4 in. by 3 ft. 5 in Ans. The space abed represents the product of 2 ft. 4 in. by 3 ft. = 6 sq. ft. and 12 twelfths = 7 sq. ft. ; b » 2 it. 4 in. name the product of 2 ft. 4 in. by 5 in. = 10 twelfths and 20 sq. in. = 11 twelfths and 8 sq. in.; and the whole rectangle nhmd, which is 2 ft. 4 in. by 3 ft. 5 in. = 7 sq. ft. 1 1 twelfths and 8 sq. in. xiv. What is the area of a board whose length is 3 ft. 2 in. and breadth 2 ft. 5 \n.'^....Ans, 7 sq. ft. 7 twelfths and 10 sq. in. Proof, 3 ft. 2 in. by 2 ft. = 6 sq. ft. and 4 twelfths; 3 ft. 2 in. by 5 in. = 15 twelfths and 10 sq. in. = 1 sq. ft. 3 twelfths and 10 sq. in.; and therefore the whole area will be 7 sq. ft. 7 twelfths and 10 sq. in. SOLID MEASURE. 171 Questions on Solid Measure. In proposing' the following questions, it is desirable that some of the more important solids should be placed before the class. i. A rectangular block is 5 in. by 3 in. in the base, and 8 in. long, how many inch cubes could be cut out of it ?.... Ans. 120. Proof, The base contains 15 sq. in.; and if the block were an inch in height it would contain exactly 15 inch cubes,, so that in every inch of length there would be 15 inch cubes; therefore for 8 inches in length we have 8 X 15, or 120 inch cubes. ii. How many cubic inches are contained in a cubic foot? ....Ans. 1728 cubic inches make one cubic foot. Proof. The base of the cube being 12 in. by 12 in., contains 144 sq. in., so that each section of an inch in height contains 144 inch cubes; and therefore 12 inches in height contains 12 times 144, or 1728 inch cubes. iii. How many foot cubes are contained in a cube whose side is a yard ?....Ans, 27 cubic feet make one cubic yard. Proof, The base of the cube being 3 ft. by 3 ft., contains 9 sq. ft. ; if the solid were a foot high, it would contain 9 foot cubes ; and therefore when the solid is 3 ft, high, it contains 3 times 9, or 27 foot cubes. 172 ON FRACTIONS. TENTH EXERCISE. iv. The base of a block of stone is 2 ft. by 3 ft., and the length 5 ft. ; how many cubic feet does it contain ?....Ans. 30 cubic ft. Proof. The area of the base contains 6 sq. ft., which, multipHed by the length, gives 30 cubic ft. for the contents of the solid- V. The base is 2 in. by 5 in., and the height 4 feet; required the solid contents — Ans. 480 cubic inches. Proof. The base contains 10 sq. in. ; 4 feet = 48 inches ; then 10 sq. in. X 48 in. = 480 cubic inches. vi. How many cubic yards are contained in a cutting of earth 10ft. by 9ft. in the base, and 12ft. high?... .Ans. 40 cubic yards. Proof. The area of the base is 90 sq.ft. ; and the solid contents = 90 sq. ft. X 12 = 1080 cubic ft. ; but 1 cubic yd. contains 27 cubic ft. ; therefore the twenty-seventh part of 1080 = 40 cubic yards. vii. What is the cost of a log of timber 20 ft. long, and 1 ft. 6 in. by 1 ft. in the end or section, at 1^. 6d. per cubic foot ?....Ans. 2L 5s. Pi'oof The area of the end= 1 ft. 6 in. by 1 ft. = 1 sq. ft. and 6 twelfths = \\ sq. ft. ; and the solid contents = 1^ X 20 = 30 cubic ft. ; then 30 at 1*. Qd. — 11. bs. viii. What is the cost of a block of marble 6 feet long, 2 ft. broad, and 6 in. thick, at 6,y. ^d. per cubic foot?.... Ans, \l. 19^. ix. 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