will divide both without a remainder; then dividing this result in the same way, and so on, continuing the process till a fraction is obtained, the terms of which are prime to each other; or by dividing both numerator and denominator by their greatest common divisor. 3. A fraction will always be reduced to its lowest terms when there is no number greater than 1 which will divide both its numerator and denominator without a remainder. 1. Reduce to its lowest terms. Solution. Dividing both numerator and denominator by 4, their greatest common divisor, gives, which expresses as many parts, each part 4 times as large as before. Hence, 12 = }. 2. Reduce 38112 to its lowest terms. 9603 Solution. Observing (104, II.) that both numerator and denominator are divisible by 4, we first divide by 4, which gives 127, or 1 as many parts, each 4 times as large as before. Observing (104, IV.) that both numerator and denominator of the last fraction are divisible by 9, we divide by 9, which gives 1897, or as many parts, each 9 times as large as before. 97 Observing (104, V.) that both numerator and denominator of the last fraction are divisible by 11, we divide by 11, which gives 127, or as many parts, each 11 times as large as before. As the numerator and denominator of the last fraction are prime to each other, the reduction can be carried no farther, and 381 reduced to its lowest terms 50292 equals. 12 Second Solution. The greatest common divisor of 38412 and 50292 found by one of the methods of Section IX.) is 396; and dividing both numerator and denominator by it, gives 127, or 6 as many parts, each part 396 times as large as before. Hence, 38252 97 12 = NOTE. The mechanical process by the first solution is merely to divide both terms, first by 4, then by 9, then by 11; while by the last it is to find the G. C. D. of both terms, and divide them by it. The first method will usually be the most convenient, when the divisors can be readily perceived. (d.) The pupil should be careful not to decide that any fraction is incapable of reduction till he has carefully tested it by some of the processes of Section IX. Reduce each of the following fractions to its lowest terms. (a.) When, as is sometimes the case, the factors of the numerator and denominator are given, labor may be saved by reducing the fraction to its lowest terms before multiplying the factors together. (b.) In writing the work, it is well to draw a line through the factors which have been divided, and to place the quotients above those in the numerator, and beneath those in the denominator. (c.) The numbers by which we divide are said to be cancelled, and the process is called cancellation; but it is identical in principle with other cases of reducing fractions to their lowest terms. 1. Reduce Solution. Cancelling 8 from the factors 8 of the numerator and 16 of the denominator, (i. e., dividing each by 8,) gives 1 in place of 8, and 2 in place of 16, and makes the fraction express as many parts, each 8 times as large as before.t Cancelling 3 from the factors 15 of the numerator and 9 of the denominator, gives 5 in place of 15, and 3 in place of 9, and makes the fraction express as many parts, each 3 times as large as before. As no further division can be made, the remaining factors are to be multiplied together, which gives for a result. * Solution. 64 and 81 are prime to each other, and hence cannot be reduced to lower terms. + For multiplying by of a number gives as large a product as multiplying by the number would give. For multiplying by of a number gives as large a product as multiplying by the number would give. 12 X 7 X 25 × 36 × 11 35 X 12 X 4 X 11 X 21 to its lowest terms. Solution. Cancelling the factor 12 from numerator and denominator, gives 1 in place of each. Cancelling 7 from the numerator and from the 35 in the denominator, gives 1 in place of the former, and 5 in place of the latter. Cancelling 5 from the denominator and from the 25 in the numerator, gives 1 in place of the former, and 5 in place of the latter. Cancelling 4 from the denominator and from the 36 in the numerator, gives 1 in place of the former, and 9 in place of the latter. Cancelling 11 from the numerator and denominator, gives 1 in place of each. Cancelling 3 from the 9 in the numerator and from the 21 in the denominator, gives 3 in place of the former, and 7 in place of the latter. As no further reduction can be made, we multiply the remaining factors together, which gives = 24. The written work would be thus: Or, by omitting to write the factors which are equal to 1, as we may do without ambiguity, we have the following more convenient form. 3. Reduce 7 5 X 7 X 11 X 12 X 15 X 18 X 2 4 X 5 X 3 X 11 X 7 X 36 X 5 X 6 its lowest terms. to × 11 × 12 × 15 × 18 × 2 4 × 5 × 3 × 11 × = × 36 × 5 × 6 2 Reduce each of the following to its lowest terms. 7 X 16 X 18 X 5 X 9 5. 20 X 14 X 9 X 9 X 11 8 X 36 X 28 X 48 6. 24 X 72 X 13 × 15 6 X 7 X 8 X 9 X 10 X 11 X 12 7. 7 X 8 X 9 X 10 X 11 X 12 × 13 96 X 65 X 35 8. 25 X 91 X 48 1 14. 4959 X 3487 × 2491 6061 X 53 X 2853 X 47 141. Compound Fractions. (a.) A COMPOUND FRACTION is a fraction of a fraction as of, of of. (b.) A compound fraction is equivalent to a fraction multiplied by a fraction. Thus: of fo. (See 132.) = times; of off = times times (c.) In a compound fraction, the value expressed by one fraction is made the unit of another. of means of the quantity, or 3 such parts as would be obtained by dividing into 4 equal parts. means of the quantity of To, and quantity 17. 9 of of of means of the (d.) In reducing compound fractions to simple ones, it is important to notice which fraction is made the unit of the other, as that is the one on which the operation is to be performed. 1. What is of 5117? Solution. of dividing 5747 by 9, ing 638 by 8, is is 5108 9436 First Form. 17 = 8 times of §117; I of 577, found by 383, and 8 times this result, found by multiply Hence the following forms of written work. 5108 of a. = Hence, of = 8188 $ of 437 = 54, with a remainder of 54, which, reduced 49; 1 of 40 = 4. Hence, of 4374 = 544. |