9. To divide a fraction by a whole number, we may either multiply the denominator by the number, or we may divide the numerator by it. The ratio of 12 : 17 will evidently be diminished 4 times if we divide its antecedent 12 by 4. We thus get the ratio (124): 17 or 3:17; and it therefore follows that 1 ÷ 4 = 137. As to the second method The ratio of 12:17 can also be diminished 4 times by increasing its consequent or divisor 4 times, so that we thus get the ratio 12: (17 × 4) or 12:68. It therefore follows that ÷ 4 = }}• It may be remarked, as in Art. 8, that the two results, and, have exactly the same value, for the latter can be obtained from the former by multiplying each of its terms by 4 (see Art. 7). And again, in actual practice, we usually take the first method when the numerator contains the divisor as a factor, but not otherwise. Thus- 11 18÷6 = 19 (1.) 18 ÷ 6 (3.) 12 ÷ 8 = U 13 X 5 = 12 17X8 = Here it is convenient to divide the numerator and denominator by the common factor 4 (Art. 7). We then have 1 ÷ 8 12 4 = 17X(8÷4) = 17 X2 = 31. 10. To reduce a mixed number to an improper fraction. Looking at our definition of a mixed number (Art. 6), the following rule is evident: Multiply the integral part by the denominator of the fractional part, and add in the numerator; this gives the required numerator, and the denominator of the fractional part is the required denominator. Ex.-Reduce 53, 73 to improper fractions. 11. To reduce a complex fraction to its equivalent simple fraction. Before stating a rule, let us take an example. Suppose we have to reduce 3 to an equivalent simple fraction. Now, by the last Art., Again, the ratio 16: = 5 53 = 47 = = Let us multiply 47. The ratio will not be altered in value if we multiply both its terms by the same quantity. them by 9 and it becomes 16 × 9:47 × 9. 8, 16 × 9 16X9 and 47 then becomes 16x9:47. of this ratio by the same ratio 16X9 x 5:47 x 5. 16X9 16X9 = 55 5 1 = 5 16 x 9. valent to the ratio 16 × 9 47 We will again multiply the terms quantity, viz. by 5, and we get the Now, by Art. 8, 16x9 × 5 Hence the ratio 16: 47 is equix 9: 47 x 5, and hence the 5 = Now 16 and 9 are called the extreme terms of the complex 16 fraction 3 and 5 and 47 are called its mean terms. 47' We arrive then at the following rule : RULE.-Bring the numerator and denominator to the form of simple fractions, then multiply together the extreme terms for a new numerator, and the mean terms for a new denomi 12. To reduce a compound fraction to its equivalent simple fraction. Let it be required to find the simple fraction equivalent to the compound fraction of 4. Now of is the ratio 3:4, where the unit of this ratio is . It is therefore, from the definition of ratio, equal to 3 times this unit divided by 4. × 3 5 × 3 (Art. 8), 7 3 times ÷ 4 = 5×3 ÷ 4 1= 7 Hence we arrive at the result several numerators for a Hence the rule :— RULE.-Multiply together the new numerator, and the several denominators for a new denominator. Ex. 31 of 2 of 6 = 3 × 8 +1 of 2 × 2+1 of † 8 25 of 5 of 2 = 25 × 5 × 6 = 8 X 2 X 1 Ex. II. 1. Reduce the following to improper fractions31, 43, 100, 351, 1, 112. 2. Reduce the integer 19 to sixths, tenths, thirteenths, eighteenths, nineteenths, and twentieths. 3. Bring the following fractions to integers, and reduce them respectively to fourths, sixths, eighths, tenths, twelfths, and fourteenths 4. Multiply the following fractions each by 10, 11, 12 5. By how much does 8 times the fraction exceed the quotient of, by 3? 6. Divide the following fractions each by 6, 7, 1837, 14, 18, 315, 103, 115. 7. Diminish the following ratios respectively 6, 7, 8-fold— 12:5, 34, 97:21. |