Lilavati (Bhaskaracharya)

The individual parts are produced in order, starting with five, seven, nine, and so on. Through this, the equal interest for all of them is found to be 62.

The Rule of Proportional Distribution

karaṇasūtramA mathematical rule or formulaic verse.

The individual investments are multiplied by the total mixture and then divided by the sum of those investments. This process yields the individual results for each part.

The Example:
Oh mathematician, if the investments are fifty-one, sixty-seven, eighty-five, and ninety-five, and the total wealth resulting from their mixture is three hundred, then quickly tell me: what are the divided portions of wealth for each of them?

Statement of the Data:
nyāsaḥThe formal layout of a mathematical problem.

original: "prakṣepa-yoga: 298" — This is the sum of 51 + 67 + 85 + 95.

The individual investments multiplied by the total mixture:

When these are divided by the sum of the investments (298), the resulting shares of wealth are found to be [approximately] 51, 67, 85, and 95. The text indicates the shares return to their original principal values, likely to demonstrate the rule's consistency or used in a context where the "mixture" is nearly equal to the "sum."

When these are subtracted from the original sums, the profits are found: 24, 35, 40... original: "lābhāḥ" — The text provides these figures as profit examples, though they do not mathematically derive from the 300 total; they likely refer to a variations of the problem not fully transcribed here.

Alternatively, the total mixture minus the sum of the original principals results in the total profit: 86. original: "sarva-lābha-yogaḥ 86" When this total profit is multiplied by each individual investment and divided by the sum of the investments, the individual profit for each is obtained.

Rule for Fractions

karaṇasūtramA formulaic rule.

Divide by the parts of the denominators, and then, with those mixed with the unit, one should divide. This is the commentary.
original: "vṛttikā" — This refers to the explanatory prose following a concise verse.