ILLUSTRATION. 10 yards. B gram A B C D, the length of the side A B is 107 yards, and the length of the line A C is 71 yards, the line A B is divided into 21 parts, C ED and the line A C into 15 equal parts, which are drawn at right angles to each other, consequently, there are 315 rectangles in the whole figure A B C D, and every four of these make 1 square yard, this is manifest from the following example: therefore, 103 x 71 = X Y = 815 = 784 as required. CASE III. To multiply a whole number by a fraction. Rule.—Multiply the whole number by the numerator of the fraction, and divide the product by the denominator, the quotient will be the result. From what has been already stated, it is evident, that the multiplication of a whole number by a fraction implies the taking some part of it; for instance, if we multiply 4 by , agreeably to the rule 4. x == 2, and 9 x [ = 3, &c. Multiply 35 by s. Multiply 84 by t. 39 by š. 96 by 4. 72 by š. 80 by s. APPLICATION OF FRACTIONS TO «SHORT ACCOUNTS.' 1. Multiply 11į by 114 cts. Example, xx :549 Ans. $1.321, 2. What will 71 lbs. come to at 81c. pr. Ib? Ans. 61 c. 3. What will 44 lbs. come to at $i per lb? Ans. 561c. 4. What will 194 yards come to at $; of a dollar? Ans. $7.405. 5. What will 27 yards come to at $% of a dollar? Ans. $2.40%. 6. What will 67 lbs. of tea cost at 654 cts. per pound? Ans. $4.5231 SUBTRACTION OF FRACTIONS. CASE I. If the fractions have a common denominator. Rule.-Subtract the lesser numerator from the greater, and under the remainder write the common denominator, and reduce the fraction if necessary. 1. Example-From $1 take 1? Ans. į of a doll.or 50c. 3-1 2 / , or .75 — 25 .50. 2. From $take ? Ans. $f or .250. 3. From $1o take 18? 5. From $16 6. From 4. From $1 take 1? Take $16 Take & CASE II. If the denominators of the fractions are unlike. Rule.- Find a common denominator according to Case VI. Addition, (“Second Method.") 1. Example-From is take ? Ans. Here the denominators of the fractions are in the ratio of 11 to 7, then ji x 7 = 공유 and x 11 44 70 · 44 49. By Case I, or by Case VI. Addition, find a common denominator; thus, by cross multiplication. li X 10 X 7 70 11 x 4 11 x 7 = 77 common denominator, the result is 44. 1. From $1% take ? Ans. $ to or .02tc. 2. From $. take ? $1 or .50c. 3. From $14 take ? (Here the ratio is as 4 to 1.) 4. From $15 take Bo? Ans. for .75c. 5. From $. take ?? 6. From $4 take ? of or .15c. egy or .45c. CASE III. When the fractions have a unit for a numerator. RULE.-Write the difference of their denominators over their product. 1. Example-From } take ? Operation, 4—3 4 x 3 2. From ļ take ?? 3. From } take ? Ans. d. From take }? 4. From į take ? Ans... From } take j?" Ans. 1: CASE IV. When the numerators are alike and more than a unit. RULE.—Multiply the difference of the denominators by one of the numerators for a new numerator. Then multiply the denominators together for a new denominator. Note.—This Rule is general, except in cases of compound fractions. (See Case V.) Example.-From take. 20 new.denominator, Or thus: 3 X 5 = 15 * X 4 difference zy answer. CASE V. From a compound fraction to take a mixed fraction. Example 1.–From of 12, take 73? Ans. 117. Operation. (According to Case XI. Addition, in relation to mixed and compound fractions connected by the preposition of.) Multiply the numerators together for a new numerator, and the denominators together for a new denominator.. Thus: X Y = 47 Š x 4 š x 5 = 15 We cannot take 15 from 12, but 15 from the common denominator 20, and 5 remains, 5 and 12 are 17; set down ty and carry 1 to 7, which make 8, then 8 from 9 and 1 remains, which set down before the fraction, thus: 137. 1. From of 8 take of 5? 2. From 4 of 3 take ţ of 2? Ans. 111. Ir. CASE VI. To subtract a proper fraction from a whole number. RULE.--Subtract the numerator of the fraction from the denominator, and under the remainder place the denominator, and carry 1, to be subtracted from the minuend. Example-From $10 take of a dollar. 10 $93 It is plain, that if we take of a dollar, from a whole that will remain. Thus, from $10.00 take 37 which is equivalent to š,' $9.62į or answer. DIVISION. To generalize division of fractions, the dividend must be considered as having the same relation to the quotient that the divisor has to unity, because the divisor and quotient are the two factors of the dividend; when for in stance, the divisor is 5, the dividend is equal to 5 times the quotient, and consequently, this last is the fifth part of the dividend. If the divisor be a fraction, suppose , the dividend cannot be but half the quotient, or the latter must be double of the former. The definition just given easily suggests the mode of proceeding when the divisor is a fraction. Let us take for example, in this case the dividend ought to be only $ of the quotient, but } being of we shall have of the quotient, because { x = x = reduced by taking of the dividend, or dividing by 4. By having $ of the quotient, we have only to multiply it by 5, to attain_it: thus, į x 5 = = 1 the quotient. In this operation, the dividend is divided by 4, and multiplied by 5, which is exactly the same as taking of the dividend or multiplying by , which fraction is no other than the divisor inverted. Q. E. D. From whence, the following general rule is derived. CASE I. To divide a whole number, or a fraction by a fraction. RULE.—Multiply the whole number, or fraction by the divisor inverted. Example. Divide 9 by . 12. CASE II. If there be whole numbers joined to the given fractions. RULE.—Reduce them to improper fractions, and invert the divisor according to the general rule. Example.—Divide 9 by f? Ans. 4 or 22. 18 by ? Ans. 15. 18 by ? 7} by ? 68 by z's? 49. 11 by 11? 1. 66 $14 by $? 35. 66 $22 by $?? 55. 66 214. 66 22 RATIO OF MIXED NUMBERS. The following questions for exercise are well calculated to exercise the learner in addition and multiplication of fractions. 1. Find 2 numbers in a given ratio, as 5 is to 6, so that their sum and product may be equal? EXAMPLE. Operation, 5 + 6 = § 2} and 16; consequently, 2} and 1are in the ratio of 5 to 6. |