We say, then, once for all, that the fifth part of 49 is 9 and ⁴/₅, or 9 + ⁴/₅; which is usually written (9⁴/₅), or if we use signs, 49/5 = (9⁴/₅). EXERCISES. What is the seventeenth part of 1237?--_Answer_, (72-¹³/₁₇). 10032 663819 22773399 What are -----, ------, and -------- ? 1974 23710 2424 162 23649 2343 _Answer_, (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. [14] Numbers which contain an exact number of units, such as 5, 7, 100, &c., are called _whole numbers_ or _integers_, when we wish to distinguish them from fractions. 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.