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115
CHAPTER VI--MIXED PROBLEMS.

The rule for arriving at the value of the prices producing equal sale-proceeds when the price of the remnant is fractional in character:--

107${\displaystyle {\tfrac {1}{2}}}$. When the remnant price is fractional in character, the selling and the buying rates are to be derived as before with (the data consisting of) the (invested) capitals and the remnant-price reduced to the same denominator, which is (however) ignored (for the time being); these selling and buying rates are (then respectively) to be multiplied by (this) denominator and the square of (this) denominator (for arriving at the required selling and buying rates). The value of the equal sale-proceeds is (then obtained) by means of the rule-of-three.

An example in illustration thereof.

108${\displaystyle {\tfrac {1}{2}}}$. (In a transaction) ${\displaystyle {\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}}}$ are the capital amounts (invested respectively by three porsons); the remnant-price is ${\displaystyle {\tfrac {6}{5}}}$. By purchasing and selling at the same prices, they became possessed of equal sale-proceeds. (What is the buying price, what the selling price, and what the equal sale-amount ?)

Again, another rule for arriving at the value of the equal sale-proceeds, when the remnant-price is fractional:-

109${\displaystyle {\tfrac {1}{2}}}$. The continued product of the highest numerator, of two, and of (all) the denominators (to be found in the values of the capital amounts invested), when combined with the (last) denominator belonging to the value of the remnant-price, gives rise to the selling rate. This multiplied by the remnant-price, and then diminished by one, and then multiplied (successively) by two and all the denominators, becomes the purchasing rate. Then the rule-of-three (is to be used for arriving at the common value of the sale-amounts).

{{c|An example in illustration thereof

110${\displaystyle {\tfrac {1}{2}}}$. Having invested ${\displaystyle {\tfrac {1}{2}},{\tfrac {2}{3}},{\tfrac {13}{4}}}$ (respectively), and having bought and sold (the same commodity), and with ${\displaystyle {\tfrac {5}{4}}}$ as the remnant price, three merchants became possessors of equal sale-proceeds