(c) If of an apple be divided into 7 equal parts, one of those parts will be of the whole apple; and if of is, then of will be, and of 2 will be 1§; i. e. the rule for reducing a compound fraction to a simple one is the same as that for multiplying a fraction by a fraction. 21. Multiply by 4, i. e. reduce of to a simple fraction. 22. Reduce & of 11. 23. Reduce of 3 of 5 Ans. 1. Ans. §§. Ans.. 28 24. Reduce of § of 7 of 3. NOTE. - The principle of canceling can be profitably applied whenever the product of two or more numbers is to constitute a dividend, and the product of other numbers is to constitute a divisor, provided that there are equal factors in the dividend and divisor. (d) The multiplication of a whole number by a fraction is only a modification of the principle of the 3d case; for, if the whole number has 1 placed under it, it becomes a fraction without change of value; thus, 8, 17=47, etc. 30. Multiply 8 by 3. 31. Multiply 35 by 3. X %, or 8 × 3 = 2, Ans. Ans. 21. We may multiply by 3 and divide the product, 105, by 5; or, better, as it keeps the numbers smaller, we may divide by 5 and multiply the quotient, 7, by 3, and the result is the same, 21, by but the divisor, 5, is 7 times,.. (59, f) the quotient,, is only of the quotient sought; hence, 2d, 2 × 7 = 1‡ (125, Rule 1) is the quotient sought; i. e. To divide a fraction by a fraction, RULE. Invert the divisor, and then proceed as in multiplication (127). 4 (a) The reciprocal of is(106), and, multiplying both nu 1 merator and denominator of this complex fraction, by 7, we obtain ; but multiplying both terms of a fraction by the same number does not change its value (60, Cor.), .. reciprocal of 1⁄2 is 7; and, generally, the reciprocal of any frac tion is that fraction inverted. Again, 124=3, and 12 X = 3; so, also,42%, and X = {}; X; i. e. it matters not whether we divide by any number or multiply by its reciprocal. From the above, together with Art. 127, we have another explanation of the rule in Art. 128. (b) If the denominator of the divisor is like that of the dividend, as in Ex. 11, they may both be disregarded; for, evidently, is contained in 24 just as many times as 6 apples are contained in 24 apples, or 6 in 24; i. e. 34÷ 24 6 = numerator of dividendnumerator of divisor; and this is equally true when the numerator of the dividend is not a multiple of the numerator of the divisor; thus, ÷ =53= 5. Ans. 4. Ans. 3. Ans.. Ans. 3. (c) When the numerator and denominator of the divisor are respectively factors of the corresponding terms of the dividend, as in Ex. 18, it is best to divide numerator by numerator, and denominator by denominator. This mode is true in all fractions, but not always convenient. Why true? Why not convenient? 129. To reduce a fraction to its smallest or lowest terms, RULE 1.-Divide each term by any factor common to them; then divide these quotients by any factor common to THEM, and so proceed till the quotients are mutually prime (61, Cor.); or, RULE 2.-Divide each term by their greatest common measure (110). Ex. 1. Reduce 438 to its lowest terms. In the first operation, we divide by 10 by cutting off 0 in each term (58), then divide by 7, then by 2. One unit - 4, and.. 12 will be reduced to units by dividing 12 by ; thus (128, b), ÷ 134 (= numerator÷ 1 4 denominator) = 31. Hence, = To reduce an improper fraction to a whole or mixed number, RULE.-Divide the numerator by the denominator; if there is any remainder, place it over the divisor, and annex the fraction so formed to the quotient. Ex. 1. Reduce § to a whole or mixed number. 896195, Ans. Ans. 125. 2. Reduce 877 to a whole or mixed number. 3. Reduce 144. 4. Reduce 131. 5. Reduce 26. Ans. 93. Ans. 633. Ans. 62 = 61. (a) The fraction obtained by the above rule will not be in its lowest terms unless the improper fraction is in its lowest terms; for the common measure of the numerator and denominator will also be a common measure of the denominator and remainder (113). 6. Reduce 1974 55. 7. Reduce 496870. 25 8. Reduce 17o. 9. Reduce 112. Ans. 98721898723. |