108. The value of a fraction is not altered by multiplying the numerator and denominator by the same quantity. Take the fraction ¾, multiply its numerator and denominator by 5, and it becomes ¹⁵/₂₀, which is the same thing as ¾; that is, one-twentieth part of 15 yards is the same thing as one-fourth of 3 yards: or, if our second meaning of the word fraction be used, you get the same length by dividing a yard into 20 parts and taking 15 of them, as you get by dividing it into 4 parts and taking 3 of them. To prove this,
let a b represent a yard; divide it into 4 equal parts, a c, c d, d e, and e b, and divide each of these parts into 5 equal parts. Then a e is ¾. But the second division cuts the line into 20 equal parts, of which a e contains 15. It is therefore ¹⁵/₂₀. Therefore, ¹⁵/₂₀ and ¾ are the same thing.
Again, since ¾ is made from ¹⁵/₂₀ by dividing both the numerator and denominator by 5, the value of a fraction is not altered by dividing both its numerator and denominator by the same quantity. This principle, which is of so much importance in every part of arithmetic, is often used in common language, as when we say that 14 out of 21 is 2 out of 3, &c.
109. Though the two fractions ¾ and ¹⁵/₂₀ are the same in value, and either of them may be used for the other without error, yet the first is more convenient than the second, not only because you have a clearer idea of the fourth of three yards than of the twentieth part of fifteen yards, but because the numbers in the first being smaller, are more convenient for multiplication and division. It is therefore useful, when a fraction is given, to find out whether its numerator and denominator have any common divisors or common measures. In (98) was given a rule for finding the greatest common measure of any two numbers; and it was shewn that when the two numbers are divided by their greatest common measure, the quotients have no common measure except 1. Find the greatest common measure of the terms of the fraction, and divide them by that number. The fraction is then said to be reduced to its lowest terms, and is in the state in which the best notion can be formed of its magnitude.
EXERCISES.
With each fraction is written the same reduced to its lowest terms.
| 2794 | = | 22 × 127 | = | 22 |
| 2921 | 23 × 127 | 23 | ||
| 2788 | = | 17 × 164 | = | 17 |
| 4920 | 30 × 164 | 30 | ||
| 93280 | = | 764 × 122 | = | 764 |
| 13786 | 113 × 122 | 113 | ||
| 888800 | = | 22 × 40400 | = | 22 |
| 40359600 | 999 × 40400 | 999 | ||
| 95469 | = | 121 × 789 | = | 121 |
| 359784 | 456 × 789 | 456 | ||
110. When the terms of the fraction given are already in factors,[15] 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′ | |||