Hence, when a whole number is to be added to a fraction whose denominator is 1 followed by _ciphers_, the number of which is not less than the number of _figures_ in the numerator, the rule is: Write the whole number first, and then the numerator of the fraction, with as many ciphers between them as the number of _ciphers_ in the denominator exceeds the number of _figures_ in the numerator. This is the numerator of the result, and the denominator of the fraction is its denominator. If the number of ciphers in the denominator be equal to the number of figures in the numerator, write no ciphers between the whole number and the numerator. EXERCISES. Reduce the following mixed quantities to fractions: 23707 6 299 2210 1 ------, 2457 ---, 1207 --------, and 233 -----. 100000 10 10000000 10000 116. 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. 117. 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. 118. 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.