PROPOSITION III. ART. 124.-If we multiply the denominator of a fraction, without changing the numerator, the value of the fraction is diminished, as many times as there are units in the multiplier. If we take the fraction, and multiply the denominator by 2, without changing the numerator, we get . Thus: 3 3 Now, each of the fractions, and 3, have the same numerator, and, therefore, express the same number of parts; but, in the second, the parts are only one half the size of those in the first; consequently, the whole value of the second fraction, is only one half that of the first. And the same may be shown of any fraction whatever. PROPOSITION IV. ART. 125.—If we divide the denominator of a fraction, without changing the numerator, the value of the fraction is increased as many times as there are units in the divisor. If we take the fraction, and divide the denominator by 3, without changing the numerator, we get. Thus: 2 2 9÷3 3 Now, each of the fractions, and, have the same numerator, and, therefore, express the same number of parts; but, in the second, the parts are three times the size of those of the first; consequently, the whole value of the second fraction is three times that of the first. And the same may be shown of other fractions. PROPOSITION V. ART. 126.-Multiplying both terms of a fraction by the same number or quantity, changes the form of the fraction, but does not alter its value. If we multiply the numerator of a fraction by any number, its value (by Prop. I.) is increased, as many times as there are units in the multiplier; and, if we multiply the denominator, the value (by Prop. III.) is decreased, as many times as there are units in the multiplier. Hence, if both terms of a fraction are multiplied by the same number, the increase from multiplying the numerator, REVIEW.-124. How is the value of a fraction affected by multiplying only the denominator? How is this proposition proved? 125. How is the value of a fraction affected by dividing the denominator only? How is this proposition proved? 126. How is the value of a fraction affected by multiplying both terms by the same quantity? Why? is equal to the decrease from multiplying the denominator; consequently, the value remains unchanged. PROPOSITION VI. ART. 127.-Dividing both terms of a fraction by the same number or quantity, changes the form of the fraction, but does not alter its value, If we divide the numerator of a fraction by any number, its value (by Prop. II.) is decreased, as many times as there are units in the divisor; and if we divide the denominator, the value (by Prop. IV.) is increased, as many times as there are units in the divisor. Hence, if both terms of a fraction are divided by the same number, the decrease from dividing the numerator is equal to the increase from dividing the denominator; consequently, the value remains unchanged. CASE I. TO REDUCE A FRACTION TO ITS LOWEST TERMS. ART. 128. Since the value of a fraction is not changed by dividing both terms by the same quantity (See Art. 127), we have the following RULE. Divide both terms by their greatest common divisor. Or, Resolve the numerator and denominator into their prime factors, and then cancel those factors common to both terms. REMARK. The last rule will be found most convenient, when one or both terms are monomials.. REVIEW.-127. How is the value of a fraction affected by dividing both terms by the same quantity? Why? 128. How do you reduce a fraction to its lowest terms? NOTE. In the preceding examples, the greatest common divisor in each is a monomial; in those which follow, it is a polynomial; but, by separating the quantities into factors, or by the rule (Art. 106,) the greatest common divisor is readily found. This is equal to ART. 129.-Exercises in Division (See Art. 76,) in which the quotient is a fraction, and capable of being reduced to lower terms. In a similar manner, when one polynomial can not be exactly divided by another, the division may be indicated, and the result reduced to its most simple form. TO REDUCE A FRACTION TO AN ENTIRE OR MIXED QUANTITY. ART. 130. Since the numerator of the fraction may be regarded as a dividend, and the denominator as a divisor, this is merely a case of division. Hence, the RULE. Divide the numerator by the denominator, for the entire part, and, if there be a remainder, place it over the denominator for the fractional part. The fractional part should be reduced to its lowest terms. NOTE. Reduce the following fractions to entire or mixed quantities. CASE III. TO REDUCE A MIXED QUANTITY TO THE FORM OF A FRACTION. ART. 131.-1. In 23 how many thirds? In 1 unit there are 3 thirds; hence, in 2 units, there are twice as many, that is, 6; then, 6 thirds plus 1 third, are equal to 7 b thirds; that is, 2 are equal to 3. In the same manner, a+ is с FOR REDUCING A MIXED QUANTITY TO THE FORM OF A FRACTION. Multiply the entire part by the denominator of the fraction; then add the numerator with its proper sign to the product, and place the result over the denominator. REMARK.-Cases II. and III., are the reverse of, and mutually prove each other. Before proceeding further, it is important for the learner to consider THE SIGNS OF FRACTIONS. ART. 132. It has been already stated (See Art. 121,) that in every fraction the numerator is a dividend, the denominator a divisor, and the value of the fraction the quotient. The signs prefixed to the terms of a fraction, affect only those terms; and the sign placed before a fraction, affects its whole value. Thus, in the a2-b2 fraction the sign of a2, the first term of the numerator, x+y' is plus; of the second, b2, minus; while the sign of each term of the denominator, is plus. But the sign of the fraction, taken as a whole, is minus. By the rule for the signs in Division, Art. 75, we have +a a +ab -ab =+b; or, changing the signs of both terms, =+b. And, if we change the sign of the denominator, we have + a +ab -a Hence, the signs of both terms of a fraction may be changed, without altering its value, or changing its sign; but, if the sign of either term of a fraction be changed, and not that of the other, the sign of the fraction will be changed. |