What is the seventeenth part of 1237?—Answer, (72-¹³/₁₇).
| What are | 10032 | 663819 | and | 22773399 | ? |
| ———, | ———, | ———— | |||
| 1974 | 23710 | 2424 | |||
| Answer, | 162 | 23649 | and | 2343 | . |
| (5 ——), | (27 ———), | (9394 ——) | |||
| 1974 | 23710 | 2424 | |||
106. By the term fraction is understood a part of any number, or the sum of any of the equal parts into which a number is divided. Thus, ⁴⁹/₅, ⁴/₅, ²⁰/₇, are fractions. The term fraction even includes whole numbers:[14] for example, 17 is ¹⁷/₁, ³⁴/₂, ⁵¹/₃, &c.
The upper number is called the numerator, the lower number is called the denominator, and both of these are called terms of the fraction. As long as the numerator is less than the denominator, the fraction is less than a unit: thus, ⁶/₁₇ is less than a unit; for 6 divided into 6 parts gives 1 for each part, and must give less when divided into 17 parts. Similarly, the fraction is equal to a unit when the numerator and denominator are equal, and greater than a unit when the numerator is greater than the denominator.
107. By ⅔ is meant the third part of 2. This is the same as twice the third part of 1.
To prove this, let a b be two yards, and divide each of the yards a c and c b into three equal parts.
| | | | | | | | | | | | | | |
| A | D | E | C | F | G | B |
Then, because a e, e f, and f b, are all equal to one another, a e is the third part of 2. It is therefore ⅔. But a e is twice a d, and a d is the third part of one yard, or ⅓; therefore ⅔ is twice ⅓; that is, in order to get the length ⅔, it makes no difference whether we divide two yards at once into three parts, and take one of them, or whether we divide one yard into three parts, and take two of them. By the same reasoning, ⅝ may be found either by dividing 5 into 8 parts, and taking one of them, or by dividing 1 into 8 parts, and taking five of them. In future, of these two meanings I shall use that which is most convenient at the time, as it is proved that they are the same thing. This principle is the same as the following: The third part of any number may be obtained by adding together the thirds of all the units of which it consists. Thus, the third part of 2, or of two units, is made by taking one-third out of each of the units, that is,
⅔ = ⅓ × 2.
This meaning appears ambiguous when the numerator is greater than the denominator: thus, ¹⁵/₇ would mean that 1 is to be divided into 7 parts, and 15 of them are to be taken. We should here let as many units be each divided into 7 parts as will give more than 15 of those parts, and take 15 of them.