of the rule under Art. 18, where a reverse operation was required to be performed. EXAMPLES. 1. Reduce 13 to an improper fraction. Multiplying the integer 13 by the denominator 7, we obtain 91; to which, adding the numerator 6, we get 97 for the numerator of the improper fraction; .. the improper fraction equivalent to 13 is 27. 2. What improper fraction is equivalent to 1278 ? 3. What improper fraction is equivalent to 18910 ? Ans. 132874. 4. What improper fraction is equivalent to 4925361? Ans. 6402979. 1 3 4 7 20. Reduce the compound fraction of to its equivalent simple fraction. of can be obtained by dividing the value of the fraction by 4, which (by Prop. II., Art. 16,) can be effected by multiplying the denominator by 4; Again, of is obviously three times as great as 4 of. to obtain of 7, we must multiply by 3, which (by Prop. I., Art. 16,) can be done by multiplying the numerator by 3; hence, we have 91 of 4 Hence, to reduce compound fractions to their equivalent simple ones, we have this RULE. Consider the word of, 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. 12 1. Reduce of of of to its equivalent simple. 3 3 5 15 fraction. Substituting the sign of multiplication for the word of, we get ××× First canceling the 8 of the 8 15 5 12 numerator against the 2 and 4 of the denominator, by drawing a line across them, we get 1 3 $ X 5X 5 A 5 12 Again, canceling the 3 and 5 of the numerator against the 15 of the denominator, we finally obtain 5 2. Reduce of 1 of 7 of of to its simplest form. 3 5 First, canceling the 7 and 5 of the numerator against 3 147 4 $ the 35 of the denominator, we get XX $$ 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. Reduce of of 28 of 333 to its simplest form. 3 1 1 1 3 3 4.9 6 10 4. Reduce of of 1 of 2 of 2 of of to its simplest form. Ans. 15. 5. Reduce of % of 13 of 14 of to its simplest form. Ans. T• 24 70 9 18 2 26 1 6. Reduce 1 of 2 of 7 of 33 to its simplest form. Ans. 62983. 2 3 0 5. 7. Reduce of of 3 of 52 of to its simplest form. Ans. 756. 8 8. Reduce of 2 of 3 of 4 of of of 7 of to its simplest form. Ans. 1. 9. Reduce of of 8 of 19 to its simplest form. 10. Reduce of 1 of 1 of 3 to its simplest form. 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,, 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× 11x9 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 337, 38, 162, and 132. 5949 6 594 2. Reduce of 1,1 of 1, and 4, to equivalent 8 fractions having a common denominator. 2009 9 6 Ans. 88, 188, and 2218. 7 22961 2 2 9 4 0 9 6 3. Reduce, of 3, and 41, to equivalent frac 3 7 tions having a common denominator. Ans. 117, 14%, and 1887. 6149 7667 8 0 4 7 8 0 0 4 4. Reduce, 11, 41, and 43 to equivalent fractions. 31 479 having a common denominator. 53 5. Reduce 3, 3, and, to fractions having a com 6. Reduce, and 11, to fractions having a com |