# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३२९

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133
CHAPTER VI-MIXED PROBLEMS.

birds for 5 paņas, purchase, O friend, for 56 paņas 72 birds and bring them (to me)". So saying a man gave over the purchase-money (to his friend). Calculate quickly and find out how many birds (of each variety he,bought) for how many paņas.

150. For 3 paņas, 5 palas of ginger are obtained; for 4 paņas, 11 palas of long pepper; and for 8 paņas, 1 pala of pepper is obtained. By means of the purchase-money of 60 paņas,quickly obtain 68 palas (of these drugs).

The rule for arriving at tho desired numerical value of certain specified objects purchased at desired rates for desired sums of money as their total price:-

151. The rate-values (of the various things purchased are each separately) multiplied by the total value (of the purchase-money), and the various values of the rate-money are (alike separately)

151. The following working of the problem given in 152-153 will illustrate the rule :-

 5 7 9 3 3 5 7 9 500 700 900 300 300 500 700 900 0 0 0 600 200 200 200 0 0 0 0 6 2 2 2 0 0 0 0 36 6 8 10 0 6 4 4 6 6 6 6 4 6 6 6 4 18 30 42 36 30 42 54 12 3 5 7 6 5 7 9 2 9 20 35 36 16 28 45 12

Write down the rate-things and the rate-prices in two rows, one below the other . Multiply by the total price and by the total number of things respectively. Then subtract. Remove the common factor 100. Multiply by the chosen numbers 3, 4, 5, 6. Add the numbers in each horizontal row and remove the common factor 6. Change the position of these figures, and write down in two rows each figure as many times as there are component elements in the corresponding sum changed in position. Multiply the two rows by the rate-prices and the rate-things respectively. Then remove the common factor 6. Multiply by the already chosen numbers 3, 4, 5, 6. The numbers in the two rows represent the proportions according to which the total price and the total number of things become distributed.

This rule relates to a problem indeterminate equations, and as such, there may be many sets of answers, these answers obviously depending upon the quantities chosen optionally as multipliers.

It can be easily seen that, only when certain sets of numbers are chosen as optional multiplier, integral answers are obtained; in other cases, fractional answers are obtained, which are of course not wanted. For an explanation of the rationale of the process, see the note given at the end of the chapter