Lilavati (Bhaskaracharya)

The Rule of Supposition (Continued)

Continuing the calculation for the total sum of 1000 at 5% monthly interest for one year.

By the method of the "assumed quantity," the interest on the unit for one year is calculated as 3. That interest added to the unit results in 8.

original: "rūpasya varṣakalāntaraṃ 3 tad-rūpayutaṃ 8" — Here 'unit' refers to a proportional base. If 5 is the monthly interest on 100, then for 12 months the interest is 60. The ratio of principal to interest is 100:60, or 5:3. Adding them (5+3) gives 8.

By this ratio, the desired sum of 1000 is multiplied by 5 and divided by 8, which results in the principal of 625. When this principal is subtracted from the total mixture, the interest is found to be 375.

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Rule for Apportioning Loans

karaṇasūtramA rule-verse or formula for calculation.

The standard principals are multiplied by their respective times, and the interest amounts are multiplied by the elapsed times. These products are then divided by the sum of their own products. When these results are multiplied by the total mixture, the individual portions of the loan are found.

Example:
A mathematician has lent a total of one hundred nishkasA high-value gold or silver coin. in three parts. The monthly interest rates for these parts are five, three, and four percent respectively. The durations for which they were lent are seven months, six months, and five months. If the interest accrued on all three parts is equal, tell me the value of each portion.

Assumed

Statement of the Problem:
khaṇḍa saṃkhyāNumber of parts/portions. 3
Layout: 5/7 3/6 4/5
These represent the rates and times: (5% for 7 months), (3% for 6 months), and (4% for 5 months).

Total Sum: 100

Here, the standard rate is 100. The principal is multiplied by its own time. For the first part: 7 months multiplied by the rate of 5 results in 35.

To find parts that yield equal interest, the portions must be inversely proportional to the product of (rate × time).

In this manner, by making the denominators equal for all fractions, the sum of the portions is 235. By multiplying the total mixture by the reciprocal of these shares, the portions are divided. The resulting shares of the total sum are 24, 18, and 42.

Note: The transcription shows a different example in the numbers (84 and 4 parts), suggesting the manuscript may be moving between the rule-verse and a specific commentary or a second example.