1. Example-From į take ? Operation, 4-3 4 X3 =11: 2. From štake ?? 3. From s take d? Ans. d. From į take ? “ 4. From į take f? Ans. d. From } take 3? “ Ans. • zo 3 40 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 i take . 20 new denominator. Or thus: i X 5 등 5 x 4 if difference answer, From a compound fraction to take a mixed fraction. Example 1.-From of 12, take 78? 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 1 46 93 $ X 5 = 6. 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: 117. = 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 03 $9. It is plain, that if we take of a dollar, from a whole that will remain. Thus, from $10.00 374 which is equivalent to š, $9.624 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 instance, 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 to 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 { xf == } 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, } 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 . CASE II. If there be whole numbers joined to the given fractions. ROLE.-Reduce them to improper fractions, and in- Ans. 15. 63 by 's? 1. “ $14 by $3? 35. “ $22 by $3? 55. 66 215. 66 225 66 49. 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 = = 24 and 1 16; consequently, 2} and 15 are in the ratio of 5 to 6. 1 Operation, 24 added to 437 and 2} multiplied by 14 = 43o as required ; hence, the ratio of numbers can be proved as above, as well as mixed numbers, whose sum and product may be equal. 2. Find 2 numbers in a given ratio, as 3 to 5, whose sum and product are equal? Ans. 2 and 13. 3. Find 2 numbers in the ratio of 5 to 9, whose sum and product are equal? Ans. 15 and 24. 4. Find 2 numbers in a given ratio as 2 to 3, whose sum and product may be equal? Ans. 13 and 21. 5. Find 2 numbers in the ratio of 3 to 8, whose sum and product are equal? Ans 13 and 341. RULE OF THREE DIRECT IN FRACTIONS. 1. If į of a yard of velvet cost 62} cts. what will 2 yards cost? Operation, x 145 25 = 118.00 The Rule for the common method is to invert the first term, then after preparing the fractions, to multiply the numerators together for a new numerator, and the denominators for a new denominator. OR THUS : Let a line be drawn in all statements representing equality, placing multipliers on the right, and divisors on the left, and transpose the nụmerators when cancelled on both sides. This is a general principle in all proportional operations. 125 125 x 23 = 2875 4) 1 7.183. 2 2 4 4 2 Answer as above. 2. At 18 cts. per pound, what will 631 lbs. come to? Ans. $11.903. 3. If | of a lb. of cinnamon bring of a dollar, what will 15 lbs, cost? Ans. $2.744. 2 INVERSE PROPORTION BY FRACTIONS, RULE. When the fractions are prepared and the third term inverted for a divisor, as in division of fractions, then agreeably to the Rule in multiplication of fractions, multiply the numerators for a new numerator, and the denominator for a new denominator. 1. How many yards of brown Holland, 5 quarters wide will line 20 yards, that is 3 quarters wide? Ans. 12 yds. 2. How many yards of matting 2 feet, 6 inches broad, will cover a room that is 27 feet long and 20 feet broad? Ans. 72 yds. 3. How much shalloon of a yard wide, will line 43 yards of cloth 11 yards wide? Ans. 9 yds. 4. What quantity of shalloon yard wide, will line 71 yards of cloth 14 yards wide? Ans. 15 yds. 5. If 3 men can do a piece of work in 45 hours, in how many hours will 10 men do the same? Ans. 170 6. How many pieces of cloth at 20; dollars per piece, are equal in value to 2404 pieces at 124 dollars per piece? Ans. 1492 pieces. PROMISCUOUS EXAMPLES. 1. What will } of 21 cwt. of chocolate come to when 63 lbs. cost of of a dollar? Ans. $10.761$. 2. If of a yard of cloth cost $2%, what will 515 yards cost at the same rate? Ans. $16.25. 3. If 45 of a yard cost $9.75, what will 13 yards cost? Ans. $29.25. 4. If $1.75 will buy 7 lbs. of loaf sugar, how much will $213.50 buy? Ans. 7 cwt. 2 qrs. 14 lbs. 5. How many yards of carpeting that is half a yard wide, will cover a room that is 30 feet long and 18 feet wide? Ans. 120 yds. |