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 |
- 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.
It is evident that of two fractions which have the same denominator, the greater has the greater numerator; and also that of two fractions which have the same numerator, the greater has the less denominator.
Thus, ⁸/₇ is greater than ⁸/⁹, since the first is a 7th, and the last only a 9th part of 8. Also, any numerator may be made to belong to as small a fraction as we please, by sufficiently increasing the denominator. Thus, ¹⁰/₁₀₀ is ¹/₁₀, ¹⁰/₁₀₀₀ is ¹/₁₀₀, and ¹⁰/₁₀₀₀₀₀₀ is ¹/₁₀₀₀₀₀₀ (108).
We can now also increase and diminish the first fraction by the second. For the first fraction is made up of 15 of the 30 equal parts into which 1 is divided. The second fraction is 14 of those parts. The sum of the two, therefore, must be 15 + 14, or 29 of those parts; that is, ½ + ⁷/₁₅ is ²⁹/₃₀. The difference of the two must be 15-14, or 1 of those parts; that is, ½-⁷/₁₅ = ¹/₃₀.
- From the last two articles the following rules are obtained:
I. To compare, to add, or to subtract fractions, first reduce them to a common denominator. When this has been done, that is the greatest of the fractions which has the greatest numerator.