TO REDUCE A FRACTION TO AN ENTIRE OR MIXED QUANTITY. ART. 130.--Since the numerator of the fraction may be re garded as a dividend, and merely a case of division. the denominator as a divisor, this is 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 twiec as many, that is, 6; then, 6 thirds plus 1 third, are equal to 7 b thirds; that is, 23 are equal to 3. In the same manner, a+ is 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 tc 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 -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 wil be changed. From this, it also follows, that the signs of either term of a frac tion may be changed, without altering its value, if the sign of the fraction be changed at the same time. ax-x2 ax-x2 a-b to a fractional form. 12ac-(a—b)___12ac—a+b REMARK.-In solving this example, the learner should observe, that 3c is to be subtracted from 4a. We reduce 4a to a quantity whose denominator is 3c; then make the subtraction, and write the result over the common denominator, 3c. Reduce the following quantities to improper fractions. Ans. 3c 3c REVIEW.-130. How do you reduce a fraction to an entire or mixed quantity? 131. How do you reduce a mixed quantity to the form of a fraction? 132. What do the signs prefixed to the terms of a fraction affect? What does the sign placed before the whole fraction, affect? What effect does it have upon the value of a fraction, or upon its sign, to change the signs of both terms? To change the sign or signs of one term, and not of the other? To change the sign of the fraction, and one of its terms? TO REDUCE FRACTIONS OF DIFFERENT DENOMINATORS TO EQUIVALÈNT FRACTIONS, HAVING A COMMON DENOMINATOR. α с ART. 133.-1. Reduce and to a common denominator. d a If we multiply both terms of the first fraction, Τ nominator of the second, we shall have by d, the de a_aXd___ ad bbdbd; and, if we multiply both terms of the second fraction, by b, the denomina tor of the first, we shall have с cxb bc d dxb bď C ď In this solution we observe; first, the values of the fractions are not changed, since, in each fraction, both terms are multiplied by the same quantity; and, second, the denominators in each must be the same, since they consist of the product of the same quantities. a b 2. Reduce and, to a common denominator. m'n' Here, we are at liberty to multiply both terms of each fraction, by the same quantity, since this (See Art. 126) will not change its value. Now, if we multiply both terms of each fraction, by the denominators of the other two fractions, the new denominators in each will be the same, since, in each case, they will consist of the product of the same factors, that is, of all the denominators. It is evident, that the value of each fraction is not changed, and hat they have the same denominators. Hence, the RULE, FOR REDUCING FRACTIONS TO A COMMON DENOMINATOR. Multiply both terms of each fraction by the product of all the denominators, except its own. REMARK.-Since each denominator of the new fractions, will consist of the product of all the denominators of the given fractions, it is unnecessary to perform the same multiplication more than once. EXAMPLES. Reduce the following fractions, in each example, to others, having a common denominator. REVIEW.-133. How do you reduce fractions of different denominators to equivalent fractions having the same denominator? Why is the value of each fraction not changed by this process? Why does this process give to each fraction the same denominator? |