for a result, which expresses the same number of parts, each 4 times as large as before. Therefore, = 4 times, or, dividing the denominator of 12 by 4 has multiplied the fraction by 4. Explain the effect of dividing the denominator Hence, dividing the denominator of a fraction multiplies the fraction, by multiplying the size of each part without affecting the number of parts expressed. 137. Recapitulation and Inferences. (a.) Multiplying the numerator multiplies the fraction, by multiplying the number of parts considered without affecting their size. (b.) Dividing the numerator divides the fraction, by dividing the number of parts considered without affecting their size. (c.) Multiplying the denominator divides the fraction, by dividing each part without affecting the number of parts considered. (d.) Dividing the denominator multiplies the fraction, by multiplying each part without affecting the number of parts considered. (e.) Hence, 1. A fraction may be multiplied either by multiplying the numerator or by dividing the denominator. 2. A fraction may be divided either by dividing the numerator or by multiplying the denominator. 3. Multiplying both numerator and denominator of a fraction by any number both multiplies and divides the fraction by that number, and, therefore, does not alter its value. 4. Dividing both numerator and denominator of a fraction by the same number both divides and multiplies the fraction by that number, and, therefore, does not alter its value. 138. Multiplication and Division of both Numerator and Denominator by the same Number. 1. Explain the effect of multiplying both numerator and denominator of the fraction by 6. Answer. Multiplying both numerator and denominator of the frac tion by 6 gives for a result, which expresses 6 times as many parts, each as large as before. Therefore, the value of the fraction is unaltered, and = 22. Explain the effect of multiplying both numerator and de 10. Explain the effect of dividing both numerator and denominator of the fraction by 3. Answer. — Dividing both numerator and denominator of the fraction by 3 gives for a result, which expresses as many parts, each part 3 times as large as before. Therefore, the value of the fraction is unaltered, and 12 = }. Explain the effect of dividing both numerator and denominator of (a.) The numerator and denominator are called TERMS of the fraction. (b.) A fraction is said to be reduced to its LOWEST TERMS when its numerator and denominator are the smallest entire numbers which will express its value. (c.) From the preceding explanations, it follows that, 1. A fraction may be reduced to lower terms by dividing both numerator and denominator by any number which will divide both without a remainder. 2. A fraction may be reduced to its lowest terms by dividing both numerator and denominator by any number which 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 3812 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 1257, or as many parts, each 4 times as large as before. 067 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. 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 8 reduced to its lowest terms equals 17. 38412 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 396 as many parts, each part 396 times as large as before. Hence, 252 97 38412 97 = 127. 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. 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. Solution. 8 × 15 9 × 16 2 116 12 X 7 X 25 × 36 × 11 to its lowest terms. 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. = 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 4 X 5 X 3 X 11 X 7 X 36 × 5 × its lowest terms. 6 to |
