4. Reduce 81 to an improper fraction. Ans. 2872. 5. Reduce 377 to an improper fraction. Ans. 1554. Ans. 13 Ans. 150. Ans. 3.0370. Ans. 38283 9. Reduce 12343 to an improper fraction. 10. Reduce 77 to an improper fraction. Ans. $54. 39. Let us endeavor to reduce the compound fraction of to an equivalent simple fraction. of can be obtained by dividing the value of the fraction by 4, which (by PROP. II., ART. 34,) can be effected by multiplying the denominator by 4; therefore, Again, of is obviously three times as great as of; therefore, to obtain of, we must multiply 7 by 3, which (by PROP. I., ART. 34,) can be done by 4× 11 multiplying the numerator by 3; hence we have 4 of = 3x7 21 4x11 44° Therefore, 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 numbers, and which (by PROP. VI, ART. 34,) does not change the value of the resulting fraction. Repeat the Rule for reducing a compound fraction to a simple one. EXAMPLES. 1. Reduce of 4 of off to its equivalent simple fraction. Substituting the sign of multiplication for the word of, we get XXX First, cancelling the 8 of the numerator against the 2 and 4 of the denominator, by drawing a line across them, we get 1 3 $ 5 Again, cancelling the 3 and 5 of the numerator against the 15 of the denominator, we finally obtain 2. Reduce of 14 of 7 of 4 of Fr to its simplest form. First, eancelling the 7 and 5 of the numerator against the 35 of the denominator, we get 3 14 74 X X-X X Again, cancelling the 7 of the denominator against the same factor of the 14 of the numerator, and the 3 of the numerator against the same factor of the 9 of the denominator, we obtain 2 Finally, cancelling the 2 and 4 of the numerator against the 8 of the denominator, we get NOTE. We have written our fractions several times, in order the more clearly to exhibit the process of cancelling. But in practice, it will not be necessary to write the fractions more than once. It will make no difference which of the factors is first cancelled. When all the common factors have in this way been stricken out, the fraction will then appear in its lowest terms. 3. Reduce of of of to its simplest form. Ans.. 4. Reduce of of 4 of to its simplest form. Ans. 28. 5. Reduce of of of of to its simplest form. Ans. . 6. Reduce of 21 of 34 of 6 to its simplest form. Ans. 25. 7. Reduce of of of to its simplest form. Ans. 8. Reduce of 4 of 4 of 4 of 4 to its simplest form. Ans. 1. 9. Reduce of 4 of 8 of of of 4 to its simplest form. Ans. 10. Reduce of of 4 of of of of 7 of & of to its simplest form. Ans. 1o. 40. To reduce fractions to a common denominator We know (ART. 34, PROP. III.,) that the value of a fraction is not changed by multiplying both numerator and denominator by the same number. If, then, we multiply the numerator and denominator of each fraction by the product of the denominators of all the other fractions, we shall retain the values of the respective fractions, and at the same time they will have a common denominator. Let it be required to reduce, of %, f, and 7 of 4, to equivalent fractions having a common denominator. These fractions, when reduced to their simplest form, ,,, and . become For first fraction, Multiply the numerator and denominator, each by 3× 11x9, the product of the denominators of the other fractions, and we find 1×3×11x9=297 for new numerator. 2×3×11x9=594 " 66 denominator. For second fraction, Multiply the numerator and denominator, each by 2× 11x9, the product of the denominators of the other fractions, and we find 2×2×11x9=396 for new numerator. 3x2x11x9=594 66 66 denominator. For third fraction, . Multiply the numerator and denominator, each by 2× 3×9, the product of the denominators of the other fractions, and we find 3 ×2×3×9-162 for new numerator. 11 ×2×3×9=594 66 (6 denominator. For fourth fraction, . Multiply the numerator and denominator, each by 2× 3x11, the product of the denominators of the other frac tions, and we find 2×2×3×11-132 for new numerator. Hence the fractions become 37, 389. 194, and 134. 594: It will be seen that each numerator is multiplied by the product of all the denominators except its own. It will also be seen that in obtaining each new denominator, the factors are the same, namely, all the denominators. Hence the following RULE. Reduce mixed numbers to improper fractions, and 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. Repeat this Rule. EXAMPLES, 1. Reduce,, to equivalent fractions having a common denominator. Ans. 15, 105, 105. 2. Reduce, to equivalent fractions having a common denominator. Ans. 14, 24, 24. 3. Reduce,,,, to equivalent fractions having a common denominator. Ans. 48, 78, 388, 308. 4. Reduce of 3, 4, 5, to equivalent fractions having a common denominator. Ans. 15, ¡¡, 18. 5. Reduce of 2 of 5, 7, 5, to equivalent fractions having a common denominator. Ans. 15, 220, 156. 30 41. In most cases fractions may be reduced to equiv alent ones having a smaller common denominator than is given by the above rule. Before showing how to find the |