What is the quotient4. Of 134818? 5. Of 11441? 6. Of 16284? 7. Of .00425,? 8. Of .067.02? 9. Of 3287.0004? 10. Of of 1 of 3 ÷ 18 of 15% of? 11. Of of 42 of 1% of 1% of 61? 12. Ofofofofofofofofof 1? Martin 13. Of of off of 7 of 13 of 11? 149. Complex Fractions. (a.) A COMPLEX FRACTION is one having a fraction in either numerator or denominator, or in both; as, 7 23 NOTE.-Complex fractions are usually considered as expressions of unexecuted divisions, and are read accordingly. Thus, 7 32 (6.) To show their similarity to other fractions, we may explain them thus:— 7 33 = 7 parts of such kind that 3 of them would equal a unit. (c.) Complex fractions can be reduced to simple fractions by the ordinary process of division. *By reducing 5 to sevenths, and 91 to fourths. (d.) Complex fractions may often be reduced to simple ones, by reducing them to their lowest terms, Reduce each of the following in the same manner:— (e.) Complex fractions may also be reduced to simple ones, by multiplying both numerator and denominator by such a number as will give a whole number in place of each. 20. Reduce to a simple fraction. 43 101 Solution. If 4 be multiplied by 3, or some multiple of 3, and 101 be multiplied by 2, or some multiple of 2, the result will in each case be 4/ 101 a whole number. Hence, if both terms of the fraction be multiplied by some multiple of both 2 and 3, the resulting fraction will be a 4 28 4 simple one. Multiplying by 6 gives = = 101 63 9 In the same way reduce each of the following complex fractions to simple ones:— 150. Other Changes in the Terms of a Fraction. 1. Reduce to an equivalent fraction, having 6 for a numerator. Solution. Observing that the proposed numerator, 6, is two times the given numerator, 3, we multiply both terms by 2, which gives 3 3 X 2 4 X 2 2. Reduce to an equivalent fraction, having 10 for its numerator. Solution.-Observing that the proposed numerator, 10, is the given numerator, 8, we multiply both terms by 5 or of 7 or by 11, which to an equivalent fraction, having 10 for its 4. Reduce to an equivalent fraction, having 9 for its numerator. 5. Reduce to an equivalent fraction, having 6 for its numerator. 6. Reduce to twenty-firsts. Solution.-Observing that the proposed denominator, 21, is three times the given denominator, 7, we have only to multiply both terms 2 2 X 3 = or, omitting to write the inter7 X 3 21' by 3, which gives 7 = 2 6 mediate work, we have = 3 7 21 6 NOTE. The same result might have been obtained thus: = = 21 21' 6 21 21 of is and two times 7 21' 7 21 21' ; but, in practice, the first form will usually be found most given denominator, 7, we have only to multiply both terms by 21 8. Reduce & to halves. Solution. Observing that the proposed denominator, 2, is of the 9 given denominator, 9, we have only to multiply both terms by 151. Reduction to a Common Denominator. (a.) Fractions having their denominators alike are said to have a COMMON DENOMINATOR. Thus,,,, and have a common denominator, but and have not. (b.) In reducing fractions having different denominators to a common denominator, (i. e., to equivalent ones having the same denominator,) we first select a convenient number for the common denominator, and then make the reductions as in the last article. (c.) As far as the denominator is concerned, one number may as well be selected for a common denominator as another; but unless the number selected is a common multiple of all the given denominators, one or more of the resulting numerators will be likely to contain a fraction.* (d.) To avoid such an inconvenience, and at the same time to avoid as far as possible the use of large numbers, it will usually be best to select the least common multiple of the given denominators for a common denominator. 1. Reduce,,, and to a common denominator. *For it will be the product of a whole number multiplied by a frac tional quantity. Solution. We select 72 for the least common denominator, because it is the least common multiple of the given denominators. Then, since 729 times 8, we multiply both terms of the fraction by 9, which gives. Since 726 times 12, we multiply both terms of the fraction by 6, which gives, &c. = Hence, if1⁄2 = 42; § = 43; and 14 = 42. (e.) Many adopt it as a general rule, to select the product of the given denominators for a common denominator; but it usually involves larger numbers than the preceding method, and is hence much less convenient. The following illustrates it. Solution to preceding Example.· The product of 8, 9, 12, and 24, the given denominators, is 20736, which we select for the common denominator. To get this, we multiplied 8, the denominator of, by 9 X 12 X 24, and therefore we multiply the numerator, 7, by the same numbers, which gives = 18. To obtain 20736, we multiplied 12, the denominator of 12, by 8 X 9.X 24, the product of the other denominators, and therefore we multiply the numerator, 5, by the same numbers, which gives 2 = 206. To obtain 20736, we multiplied 9, the denominator of, by, &c. 8640 (f.) When any of the fractions to be reduced are compound or complex, they must first be reduced to simple ones, and the simple fractions should be reduced to their lowest terms, except when to do it would increase the labor of reducing to a common denominator. (g.) Reduce the fractions in each of the following examples to a common denominator: — |
