| Fraction proposed. | Square. | Cube. |
|---|---|---|
| 701 | 491401 | 344472101 |
| 158 | 24964 | 3944312 |
| 140 | 19600 | 2744000 |
| 141 | 19881 | 2803221 |
| 355 | 126025 | 44738875 |
| 113 | 12769 | 1442897 |
From 100 acres of ground, two-thirds of them are taken away; 50 acres are then added to the result, and ⁵/₇ of the whole is taken; what number of acres does this produce?—Answer, (59¹¹/₂₁).
121. In dividing one whole number by another, for example, 108 by 9, this question is asked,—Can we, by the addition of any number of nines, produce 108? and if so, how many nines will be sufficient for that purpose?
Suppose we take two fractions, for example, ⅔ and ⅘, and ask, Can we, by dividing ⅘ into some number of equal parts, and adding a number of these parts together, produce ⅔? if so, into how many parts must we divide ⅘, and how many of them must we add together? The solution of this question is still called the division of ⅔ by ⅘; and the fraction whose denominator is the number of parts into which ⅘ is divided, and whose numerator is the number of them which is taken, is called the quotient. The solution of this question is as follows: Reduce both these fractions to a common denominator (111), which does not alter their value (108); they then become ¹⁰/₁₅ and ¹²/₁₅. The question now is, to divide ¹²/₁₅ into a number of parts, and to produce ¹⁰/₁₅ by taking a number of these parts. Since ¹²/₁₅ is made by dividing 1 into 15 parts and taking 12 of them, if we divide ¹²/₁₅ into 12 equal parts, each of these parts is ¹/₁₅; if we take 10 of these parts, the result is ¹⁰/₁₅. Therefore, in order to produce ¹⁰/₁₅ or ⅔ (108), we must divide ¹²/₁₅ or ⅘ into 12 parts, and take 10 of them; that is, the quotient is ¹⁰/₁₂. If we call ⅔ the dividend, and ⅘ the divisor, as before, the quotient in this case is derived from the following rule, which the same reasoning will shew to apply to other cases:
The numerator of the quotient is the numerator of the dividend multiplied by the denominator of the divisor. The denominator of the quotient is the denominator of the dividend multiplied by the numerator of the divisor. This rule is the reverse of multiplication, as will be seen by comparing what is required in both cases. In multiplying ⅘ by ¹⁰/₁₂, I ask, if out of ⅘ be taken 10 parts out of 12, how much of a unit is taken, and the answer is ⁴⁰/⁶⁰, or ⅔. Again, in dividing ⅔ by ⅘, I ask what part of ⅘ is ⅔, the answer to which is ¹⁰/₁₂.
122. By taking the following instance, we shall see that this rule can be sometimes simplified. Divide ¹⁶/₃₃ by ²⁸/₁₅. Observe that 16 is 4 × 4, and 28 is 4 × 7; 33 is 3 × 11, and 15 is 3 × 5; therefore the two fractions are
| 4 × 4 | and | 4 × 7 | , |
| 3 × 11 | 3 × 5 |
and their quotient, according to the rule, is
- 4 × 4 × 3 × 5
- 3 × 11 × 4 × 7,
in which 4 × 3 is found both in the numerator and denominator. The fraction is therefore (108) the same as