EXERCISES.
Reduce the following mixed quantities to fractions:
| 1 | 23707 | , |
|---|---|---|
| 100000 | ||
| 2457 | 6 | , |
| 10 | ||
| 233 | 2210 | . |
| 10000 |
- Suppose it required to multiply ⅔ by 4. This by (48) is taking ⅔ four times; that is, finding ⅔ + ⅔ + ⅔ + ⅔. This by (112) is ⁸/₃; so that to multiply a fraction by a whole number the rule is: Multiply the numerator by the whole number, and let the denominator remain.
- If the denominator of the fraction be divisible by the whole number, the rule may be stated thus: Divide the denominator of the fraction by the whole number, and let the numerator remain. For example, multiply ⁷/₃₆ by 6. This (116) is ⁴²/₃₆, which, since the numerator and denominator are now divisible by 6, is (108) the same as ⁷/₆. It is plain that ⁷/₆ is made from ⁷/₃₆ in the manner stated in the rule.
- Multiplication has been defined to be the taking as many of one number as there are units in another. Thus, to multiply 12 by 7 is to take as many twelves as there are units in 7, or to take 12 as many times as you must take 1 in order to make 7. Thus, what is done with 1 in order to make 7, is done with 12 to make 7 times 12. For example,
| 7 | is | 1 + 1 + 1 + 1 + 1 + 1 + 1 |
|---|---|---|
| 7 | times 12 is | 12 + 12 + 12 + 12 + 12 + 12 + 12. |
When the same thing is done with two fractions, the result is still called their product, and the process is still called multiplication. There is this difference, that whereas a whole number is made by adding 1 to itself a number of times, a fraction is made by dividing 1 into a number of equal parts, and adding one of these parts to itself a number of times. This being the meaning of the word multiplication, as applied to fractions, what is ¾ multiplied by ⅞? Whatever is done with 1 in order to make ⅞ must now be done with ¾; but to make ⅞, 1 is divided into 8 parts, and 7 of them are taken. Therefore, to make ¾ × ⅞, ¾ must be divided into 8 parts, and 7 of them must be taken. Now ¾ is, by (108), the same thing as ²⁴/₃₂. Since ²⁴/₃₂ is made by dividing 1 into 32 parts, and taking 24 of them, or, which is the same thing, taking 3 of them 8 times, if ²⁴/₃₂ be divided into 8 equal parts, each of them is ³/₃₂; and if 7 of these parts be taken, the result is ²¹/₃₂ (116): therefore ¾ multiplied by ⅞ is ²¹/₃₂; and the same reasoning may be applied to any other fractions. But ²¹/₃₂ is made from ¾ and ⅞ by multiplying the two numerators together for the numerator, and the two denominators for the denominator; which furnishes a rule for the multiplication of fractions.
- If this product ²¹/₃₂ is to be multiplied by a third fraction, for example, by ⁵/₉, the result is, by the same rule, ¹⁰⁵/₂₈₈; and so on. The general rule for multiplying any number of fractions together is therefore: