111. 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 fractions ⅒, ⅚, and ⁷/₉, to others having the same value and a common denominator, begin by finding the least common multiple of 10, 6, and 9, by the rule in (103), which is 90. Observe that 10, 6, and 9 are contained in 90 9, 15, and 10 times. Multiply both terms of the first by 9, of the second by 15, and of the third by 10, and the fractions thus produced are ⁹/₉₀, ⁷⁵/₉₀, and ⁷⁰/₉₀, the same as before.
If one of the numbers be a whole number, it may be reduced to a fraction having the common denominator of the rest, by (106).
EXERCISES.
| Fractions proposed | reduced to a common denominator. | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| 2 | 1 | 1 | 20 | 6 | 5 | ||||
| 3 | 5 | 6 | 30 | 30 | 30 | ||||
| 1 | 2 | 3 | 12 | 3 | 28 | 24 | 18 | 48 | 63 |
| 3 | 7 | 14 | 21 | 4 | 84 | 84 | 84 | 84 | 84 |
| 3 | 4 | 5 | 6 | 3000 | 400 | 50 | 6 | ||
| 10 | 100 | 1000 | 1000 | 1000 | 1000 | 1000 | |||
| 33 | 281 | 22341 | 106499 | ||||||
| 379 | 677 | 256583 | 256583 | ||||||
112. By reducing two fractions to a common denominator, we are able to compare them; that is, to tell which is the greater and which the less of the two. For example, take ½ and ⁷/₁₅. These fractions reduced, without alteration of their value, to a common denominator, are ¹⁵/₃₀ and ¹⁴/₃₁. Of these the first must be the greater, because (107) it may be obtained by dividing 1 into 30 equal parts and taking 15 of them, but the second is made by taking 14 of those parts.