267. There were four merchants. Each of them obtained from the others half of what he had on hand (at the time of the respective transfer of money). Then they all became possessed of equal amounts of money. What is the measure of the money (they respectively had) on hand (to start with) ?
The rule for arriving at the gain derived (equally) from success and failure (in a gambling operation):-
268-269. The two sums of the numerators and denominators of the (two fractional multiple) quantities (given in the problem) have to be written down one below the other in the regular order,and (then) in the inverse order. The (summed up) quantities (in the first of these sets of two sums) are to be multiplied according to the vajrāpavartana process by the denominator, and (those in the second set) by the numerator, (of the fractional quantity) corresponding to the other (summed up quantity). The results (arrived at in relation to the first set) are written down in the form of denominators, and (those arrived at in relation to the second set are written down) in the form of numerators : (and the difference between the denominator and numerator in each set is noted down). Then by means of these differences the products obtained by multiplying (the sum of) the numerator and the denominator (of each of the given multiple fractions in the problem) with the denominator of the other are (respectively) divided. These resulting quantities, multiplied by the value of the desired gain, give in the inverse order the measure of the moneys on hand (with the gamblers to stake) .
An example in illustration thereof.
270–272. A great man possessing power of magical charm and medicine saw a cock-fight going on, and spoke separately in
268-269. Algebraically,
, where x and y are the moneys on hand with the gamblers, and , the fractional parts taken from them, and p the gain. This follows from .
