fraction by 4, which (by Prop. II., Art. 16,) can be effected by multiplying the denominator by 4; Again, of TT is obviously three times as great as 7 of; to obtain of TT, we must multiply 4×11 by 3, which (by Prop. I., Art. 16,) can be done by 3 multiplying the numerator by 3; hence, we have of 4 Hence, to reduce compound fractions to their equivalent simple ones, we have this RULE. Consider the word or, which connects the fractional parts as equivalent to the sign of multiplication. Then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator, always observing to reject or cancel such factors as are common to the numerators and denominators, which is the same as dividing both numerator and denominator by the same quantity, and (by Rule under Art. 17,) does not change the value of the fraction. EXAMPLES. 8 1. Reduce of of of to its equivalent simple fraction. Substituting the sign of multiplication for the word of, we get××× First canceling the 8 of the numerator against the 2 and 4 of the denominator, by drawing a line across them, we get 13 $ 5 2 512 X-X Again, canceling the 3 and 5 of the numerator against the 15 of the denominator, we finally obtain 1 $ $ $ 1 112 12 2. Reduce 윽 of of of of TT to its simplest form. First, canceling the 7 and 5 of the numerator against 3 14 745 the 35 of the denominator, we get×××9×1. Again, canceling the 7 of the denominator against a part of the 14 of the numerator, and the 3 of the numerator against a part of the 9 of the denominator, we obtain Finally, canceling the 2 and 4 of the numerator against 8 of the denominator, we get NOTE. We have written our fractions several times, in order the more clearly to exhibit the process of canceling. But in practice, it will not be necessary to write the fraction more than once. It will make no difference which of the factors are first canceled. When all the common factors have, in this way, been stricken out, the fraction will then appear in its lowest terms. The student will find it to his interest to perform many examples of this kind, as this principle of canceling will be extensively employed in the succeeding parts of this work. 3 3. Reduce of of of to its simplest form. Ans. 4. Reduce of of of of of off to its Ans.. simplest form. form. 2 5. Reduce of of of of to its simplest Ans. T 6. Reduce of 묶음 of of 옳을 to its simplest form. 6292 Ans. 7. Reduce of of of of to its simplest 5 23 5 form. Ans. 이 8. Reduce of of of of of of of to its simplest form. Ans.. 9. Reduce of of of to its simplest form. 91 Ans. T 10. Reduce of of 높 of to its simplest form. Ans. 136. 21. To reduce fractions to a common denominator, we have this RULE. Reduce mixed numbers to improper fractions-compound fractions to their simplest form. Then multiply each numerator by all the denominators, except its own, for a new numerator, and all the denominators together for a common denominator. It is obvious that this process will give the same denominator to each fraction, viz: the product of all the denominators. It is also obvious that the values of the fractions will not be changed, since both numerator and denominator are multiplied by the same quantity, viz: the product of all the denominators except its own. EXAMPLES. 1. Reduce, of, r, and of, to equivalent fractions having a common denominator. These fractions, when reduced to their simplest form, are,,, and . The new numerator of the first fraction is 1 × 3 × 11 × 9-297. The new numerator of the second fraction is 2×2× 11×9=396. The new numerator of the third fraction is 3×2×3× 9-162, The new numerator of the fourth fraction is 2×2×3 ×11=122. The common denominator is 2×3×11×9=594. Therefore, the fractions, when reduced to a common 94 594 594, 32 denominator, are 중음, 중음, 중음을, and 중용을 2. Reduce of, of, and, to equivalent fractions having a common denominator. Ans. 음,, and 22969 3. Reduce, 1 of 7, and 41, to equivalent frac 439 tions having a common denominator. Ans., and 7. 80419 80419 4. Reduce 용, 중구, 4, and to equivalent fractions having a common denominator. Ans. 2463599 2048821 2961799 2754623 656197 662257 682 5. Reduce 주유,, and or, to fractions having a common denominator. Ans. 무음음 6. Reduce 공극, and, to fractions having 977 mon denominator. a com Ans. 194279 1083493 22. To reduce fractions to their least common denominators, we have this RULE. Reduce the fractions to their simplest form. Then find the least common multiple of their denominators, (by Rule under Art. 10, or Rule under Art. 11,) which will be their least common denominator. Divide this common denominator by the respective denominators of the given fractions; multiply the quotients by their respective numerators, and the products will be the new numerator. The correctness of the above rule may be shown in the same way as was that of the preceding rule. EXAMPLES. 3 7 1. Reduce of of T', 음, and T5, to equivalent fractions having the least common denominator. These fractions, when reduced to their simplest form, become,, and 3. The least common multiple of the denominators 8, 20, and 15, is 120=common denominator. New numerator of the first fraction is 12×1=15. of the second fraction is×3=18 of the third fraction is ×7=56. 120 Hence, the fractions, when reduced to their least com 8 mon denominator, become, T', and 2. Reduce of, 41⁄2, and, to equivalent fractions having the least common denominator. Ans. 응, 응, and 3. Reduce of of of 응, 금, and 71⁄2, to fractions having the least common denominator. Ans. 금융, ᄒᄒ, and |