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CHAPTER III --FRACTIONS.

than one for their numerators, when the sum (of those fractions) has either one or (any number) other than one for its numerator:--

87. (Either) numerator multiplied by a chosen (number), then combined with the other numerator, then divided by the numerator of the (given) sum (of the intended fractions) so as to leave no remainder, and then divided by the (above) chosen number and multiplied by the denominator of the (above) sum (of the intended fractions), gives rise to a (required) denominator. The denominator of the other (fraction), however, is this (denominator) multiplied by the (above) chosen (quantity).

Examples in illustration thereof.

88. Say what the denominators are of two (intended fractional) quantities which have 1 for each of their numerators, when the sum (of those fractional quantities) is either ${\displaystyle {\tfrac {1}{4}}}$ or ${\displaystyle {\tfrac {2}{5}}}$; as also of two (other fractional quantities) which have 7 and 9 (respectively) for (their) numerators.

The second rule (is as follows):--

89. The numerator (of one of the intended fractions) as multiplied by the denominator of the sum (of the intended fractions), when combined with the other numerator and then divided by the numerator of the sum (of the intended fractions), gives rise to the denominator of one (of the fractions). This (denominator), when multiplied by the denominator of the sum (of the intended fractions), becomes tho denominator of the other (fraction).

67. Algebraically, if ${\displaystyle {\tfrac {m}{n}}}$ is the sum of two intended fractions with a and b as their numerators, then the fractions are ${\displaystyle {\tfrac {a}{{\tfrac {ap+b}{m}}\times {\tfrac {n}{p}}}}}$ and ${\displaystyle {\tfrac {b}{{\tfrac {ap+b}{m}}\times {\tfrac {n}{p}}\times p}}}$ where p is any number so chosen that ap + b is divisible by m. The sum of these fractions it will be found, is ${\displaystyle {\tfrac {m}{n}}}$.

89. This rule is only a particular case of the rule given in stanza No. 87, as the denominator of the sum of the intended fractions is itself substituted in this rule for the quantity to be chosen in the previous rule.