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 14 ÷ 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 x 4) or 12: 68. It therefore follows that 14 ÷ 4 = ᄒ.

It may be remarked, as in Art. 8, that the two results, and 1, 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—

Here it is convenient to divide the numerator and denominator by the common factor 4 (Art. 7).

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

53

Again, the ratio

will not be altered in value if we multiply both its terms by the same quantity. Let us multiply them by 9 and it becomes × 9 × 9. Now, by Art. 8,16 × 9 = 16X9 and 47 × 9 47 =

5

then becomes 16x9 : 47.

5

99

=

= 47. The ratio

We will again multiply the terms of this ratio by the same quantity, viz. by 5, and we get the ratio 16x9 × 5:47 × 5. Now, by Art. 8, 16X9 X 5 Hence the ratio 16:47 is equi

5

16X9 16X9 =

5÷5

1

=

16 x 9.

valent to the ratio 16

5

=

x 9: 47 × 5, and hence the

Now 16 and 9 are called the extreme terms of the complex

fraction and 5 and 47 are called its mean terms.

16 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 denominator.

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.

RULE.-Multiply together the several numerators for a new numerator, and the several denominators for a new denominator.

1. Reduce the following to improper fractions34, 43, 100, 351, 1, 113.

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

28, 11, 4, 43, 27, 7.

4. Multiply the following fractions each by 10, 11, 12— 4, T‰, ra, 13, 71, 31.

5. By how much does 8 times the fraction exceed the quotient of by 3?

6. Divide the following fractions each by 6, 7,

37, 14, 18, 373, 103, 11.

18

7. Diminish the following ratios respectively 6, 7, 8-fold— 12:5, 3:4, 97:24.

8. Simplify the expressions—

(1.) of of of 18. (2.) of 12 of 11 of

(3.) † of 34 of 3 of 11.

(4.) 1 of 2 of 1 of 1.

of of. (5.) 24 of of 14 of 4.
음

10. What is the difference between 2 of 31 of 21

4층

8호

to their

12. Add the sum of the fractions difference.

13. To reduce a fraction to its lowest terms.

DEF. A fraction is in its lowest terms when its numerator and denominator have no common factor, or are prime to each other. (Among such common factors, we include either the numerator or denominator itself, when one of them happens to be a divisor of the other.)

When the numerator and denominator have a common factor, we may divide them both by it (Art. 7) without altering the value of the fraction. Now, the highest common factor of two or more numbers is called their greatest common measure, usually written G.C.M.

:

Hence we have the following rule :

RULE. To reduce a fraction to its lowest terms, divide the numerator and denominator by their G.C.M.

Ex.-Reduce to its lowest terms.

4;

We can easily see that 12 is the G.C.M. of 24 and 84 hence, dividing each by 12, we get

24

84

2412 = 8412

=

, the fraction required.

14. It is not, however, always easy to tell by inspection the G.C.M.; but, before giving a general method for determining it, it will be useful to make a few remarks as to the divisibility of numbers in certain cases.

A number is divisible as follows:

By 2, when it is even.

By 3, when the sum of its digits is divisible by 3.

By 4, when the number formed by the last two figures is divisible by 4.

By 5, when it ends in 5 or 0.

By 6, when it is even, and is also divisible by 3.

By 8, when the number formed by the last three figures is divisible by 8.

By 9, when the sum of its digits is divisible by 9.

By 10, when it ends in 0.

By 11, when the sum of the digits in the odd places (that is, the sum of the 1st, 3rd, 5th, &c.) is equal to the sum of the digits in the even places, or the one exceeds the other by a multiple of 11.

By 12, when the number formed by its last two figures is divisible by 4, and the sum of its digits is a multiple of 3. We may add also:

(1.) A number is divisible by 37, when it is composed of digits which are repeated three times, or any multiple of three times, as 111, 333, 444444, &c.

(2.) A number which has three figures repeated in the same order is divisible by 7, 11, 13.

Thus 271271, 165165, 23023 are divisible by 7, 11, and 13; for the last may be written 023023.

(3.) A number which has four figures repeated in the same order is divisible by 73 and 137.

Thus 53245324, 2760276 are both divisible by 73 and 137; for the last may be written 02760276.

Hence a fraction may often then be reduced to its lowest terms by gradually striking out factors determined by inspection.

Now 792 and 2244 are each divisible by 4, for the numbers 92 and 44, which are formed by the last two figures of each, are evidently so. Hence, dividing numerator and denominator by 4, we have

792 = 7924 = 198 each

2244

2244 4

;

561

= 66 187

is evidently divisible by 3. Hence 792 = 198÷÷3

2224

5613