Therefore, to reduce a fraction to its lowest terms, we have this RULE. Divide both numerator and denominator by their greatest common divisor. How do you reduce a fraction to its lowest terms? EXAMPLES. 1. Reduce to its lowest terms. We have already found (Ex. 1, ART. 35,) the greatest common divisor of 592 and 999 to be 37. Dividing both these terms by 37, we find 16 and 27 for quotients: hence, H, when reduced to its lowest terms, becomes 19. 2. Reduce to its lowest terms. 3. Reduce 38,,, to their lowest terms. 1809 4. Reduce 215 to its lowest terms. Ans. Ans.,,. Ans.. Ans. To Ans. 1. Ans. 4. Ans. . We may frequently discover numbers, by inspection, which will divide both numerator and denominator without a remainder. When this is the case, we need not resort to the rule for obtaining the greatest common divisor, until we have divided by such numbers. In this example, we first divide the numerator and de nominator by 4, which reduces the fraction to 18. We again divide by 4, and obtain 24. Dividing the numerator and denominator of this last fraction by 4, we obtain , which is still further reduced by dividing three successive times by 3. 12. Reduce 194 to its lowest terms. ÷2 ÷3÷3÷3 14=1==== 13. Reduce 20 to its lowest terms. 14. Reduce 1 to its lowest terms. 15. Reduce 175 to its lowest terms. ÷3. Ans. 4. Ans. H. Ans. Ans.. 16. Reduce 472500 to its lowest terms. 37. To reduce an improper fraction to a whole or mixed number. Since the value of a fraction is the quotient arising from dividing the numerator by the denominator, (Art. 34,) we may find the value of 5, by dividing 95 by 13. Performing the division, we find 7 for the quotient, and 4 for a remainder. Hence 17. (Art. 26.) From which we have the following RULE. Divide the numerator by the denominator; che quotient will be the integral part of the mixed number. The remainder being placed over the denominator of the improper fraction, will form the fractional part. Repeat the rule for reducing an improper fraction to a mixed number. EXAMPLES. 1 Reduce to a mixed number. Ans. 2. Ans. 11, 7. Ans. 8135 Ans. 102. 38. To reduce a mixed number to an improper fraction. Reduce the mixed number 373 to an improper fraction. If we multiply the fractional part, , by 8, the product will be 3. (ART. 34.) Multiplying 37 by 8 we obtain 296, to which adding 3, we find 299 for 8 times 373. Hence 373 is equal to 299 divided by 8, that is, to 232 Hence, we have this RULE. Multiply the integral part of the mixed number by the denominator of the fractional part; to the product add the numerator of the fractional part; the sum will be the numerator of the improper fraction; under which place the denominator of the fractional part. This rule is obviously correct, since it is the reverse of the rule, (ART. 37,) where a reverse operation was required to be performed. EXAMPLES. 1. Reduce 4 to an improper fraction. Ars.. 2. Reduce 31, 7%, to improper fractions. Ans. 10, 37. 3. Reduce 81, 74, to improper fractions. Ans. 35, 15 4. Reduce 815 to an improper fraction. 365 Ans. 23877. Ans. 30370 9. Reduce 12343 to an improper fraction. 10. Reduce 77 to an improper fraction. 83 Ans. 38283 Ans. 854. 39. Let us endeavor to reduce the compound frac tion 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 Therefore, to reduce compound fractions to their equiva lent 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 of of 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 X-X X Again, cancelling the 3 and 5 of the numerator against the 15 of the denominator, we finally obtain 1 $ $ $ 2. Reduce of 14 of 3 of 4 of 1 12 to its simplest form. First, eancelling the 7 and 5 of the numerator against the 35 of the denominator, we get 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 |