- When the terms of the fraction given are already in factors,2 any one factor in the numerator may be divided by a number, provided some one factor in the denominator is divided by the same. This follows from (88) and (108). In the following examples the figures altered by division are accented.
| 12 × 11 × 10 | = | 3′ × 11 × 10 | = | 1′ × 11 × 5′ | = 55 |
|---|---|---|---|---|---|
| 2 × 3 × 4 | 2 × 3 × 1′ | 1′ × 1′ × 1′ | |||
| 18 × 15 × 13 | = | 2′ × 3′ × 1′ | = | 1′ × 1′ × 1′ | = ¹/₁₆ . |
| 20 × 54 × 52 | 4′ × 6′ × 4′ | 2′ × 2′ × 4′ | |||
| 27 × 28 | = | 3′ × 4′ | = | 3′ × 2′ | = ⁶/₅. |
| 9 × 70 | 1′ × 10′ | 1′ × 5′ |
- As we can, by (108), multiply the numerator and denominator of a fraction by any number, without altering its value, we can now readily reduce two fractions to two others, which shall have the same value as the first two, and which shall have the same denominator. Take, for example, ⅔ and ⁴/₇; multiply both terms of ⅔ by 7, and both terms of ⁴/₇ by 3. It then appears that
| ⅔ is | 2 × 7 | or ¹⁴/₂₁ |
|---|---|---|
| 3 × 7 | ||
| ⁴/₇ is | 4 × 3 | or ¹²/₂₁ |
| 7 × 3 |
Here are then two fractions ¹⁴/₂₁ and ¹²/₂₁, equal to ⅔ and ⁴/₇, and having the same denominator, 21; in this case, ⅔ and ⁴/₇ are said to be reduced to a common denominator.
It is required to reduce ⅒, ⅚, and ⁷/₉ to a common denominator. Multiply both terms of the first by the product of 6 and 9; of the second by the product of 10 and 9; and of the third by the product of 10 and 6. Then it appears (108) that
| ⅒ is | 1 × 6 × 9 | or ⁵⁴/₅₄₀ . |
|---|---|---|
| 10 × 6 × 9 | ||
| ⅚ is | 5 × 10 × 9 | or ⁴⁵⁰/₅₄₀ . |
| 6 × 10 × 9 | ||
| ⁷/₉ is | 7 × 10 × 6 | or ⁴²⁰/₅₄₀ . |
| 9 × 10 × 6 |
On looking at these last fractions, we see that all the numerators and the common denominator are divisible by 6, and (108) this division will not alter their values. On dividing the numerators and denominators of ⁵⁴/₅₄₀, ⁴⁵⁰/₅₄₀, and ⁴²⁰/₅₄₀ by 6, the resulting fractions are, ⁹/₉₀, ⁷⁵/₉₀, and ⁷⁰/₉₀. These are fractions with a common denominator, and which are the same as ⅒, ⅚, and ⁷/₉; and therefore these are a more simple answer to the question than the first fractions. Observe also that 540 is one common multiple of 10, 6, and 9, namely, 10 × 6 × 9, but that 90 is the least common multiple of 10, 6, and 9 (103). The following process, therefore, is better. To reduce the