80. The denominator (of the given sum , when combined with an optionally chosen quantity and then divided by the numerator of that sum so as to leave no remainder, becomes the denominator related to the first numerator (in the intended series of fractions); and the (above) optionally chosen quantity, when divided by this (denominator of the first fraction) and by the denominator of the (given) sum, gives rise to (the sum of) the remaining (fractions in the series). From this (known sum of the remaining fractions in the series, the determination) of the other (denominators is to be carried out) in this very manner.
Examples in illustration thereof.
81. Of three (different) fractional quantities having 1 for each of their numerators, the sum is ; and of 4 (such other quantities, the sum is) . Say what the denominators are.
The rule for arriving at the denominators (of certain intended fractions) having either one or (any number) other than one for their numerators, when the sum (of those fractions) has a numerator other than one:--
82. When the known numerators are multiplied by (certain) chosen quantities, so that the sum of these (products) is equal to the numerator of the (given) sum (of the intended fractions), then, if the denominator of the sum (of the intended fractions ) is divided by the multiplier (with which a given numerator has) itself (been multiplied as above), it gives rise to the required denominator in relation to that (numerator).
80. Algebraically, if is the sum, the first fraction is ; and the sum of the remaining fractions is mentioned in the role to be , where p is the optionally chosen quantity . This is obtained obviously by simplifying
We must here give such a value to p that n + p becomes exactly divisible by a.
