Simple Interest in Līlāvatī

Here we will know how to calculate simple interest as mentioned in Līlāvatī.

Verse 97

प्रमाणकालेन हतं प्रमाणं विमिश्रकालेन हतं फलं च

स्वयोगभक्ते च पृथक् स्थिते च मिश्राहते मूल कलान्तरे स्तः ॥९७॥

To compute simple interest and principal multiply the standard principal (100) by the standard period (1 month or 1 year). Next multiply the given period by the given rate of interest. Keep the two products a, b separately. Multiply a by the amount and divide it by (a+b) to get the principal. Similarly the amount multiplied by b and divided by (a+b) yields the interest.

Comment : A = amount , P = principal, I = interest, R = rate of interest , Y = period . P₀ = standard principal (usually 100)

Y₀ = standard period ( 1 year or 1 month).

$P={\frac {A\ X\ P_{0}\ X\ Y_{0}}{P_{0}Y_{0}+RY}}$

$I={\frac {A\ X\ R\ X\ Y}{P_{0}Y_{0}+RY}}$

Example

पंचकेन शतेनाब्दे मूलं स्वं सकलान्तरम् ।

सहस्त्रं चेत्पृथक् तत्र वद मूल कलान्तरे ॥ ॥

When the interest rate is 5% per month, the amount after one year is 1000 N (niṣkas). Find the principal and the interest.

Comment : The above stanza does not mention 'per month' but it seems that in those times the interest was calculated on monthly basis.

Here A = 1000, R = 5 , P₀ = 100 , Y₀ = 1 month Y = 1 year ( 12 months)

$P={\frac {1000\ X\ 100\ X\ 1}{100\ X\ 1+5\ X\ 12}}=625$ N

$I={\frac {1000\ X\ 5\ X\ 12}{100\ X\ 1+5\ X\ 12}}=375$ N

Alternatively I = A - P = 1000 - 625 = 375 N

Verse 99

अथ प्रमाणैर्गुणिताः स्वकाला व्यतीतकालघ्नफलोद्धृतास्ते ।

स्वयोगभक्ताश्च विमिश्रनिघ्नाः प्रयुक्तखण्डानि पृथक् भवन्ति ॥९९॥

If several parts of a certain principal bear different rates of interest for different periods and yet yield the same interest, to find these parts - Take the product of the standard principal and the standard period, divide this product by the product of respective periods and rates of interest, and write these quotients separately. These quotients multiplied by the given principal and divided by the sum of the quotients written separately are the desired parts of the given principal.

Comment: Using the notation of the previous example with suffixes to denote the parts (we consider three parts):

$P_{1}={\frac {(P_{1}+P_{2}+P_{3})\ X\ {\frac {100}{R_{1}Y_{1}}}}{{\frac {100}{R_{1}Y_{1}}}+{\frac {100}{R_{2}Y_{2}}}+{\frac {100}{R_{3}Y_{3}}}}}$ $P_{2}={\frac {(P_{1}+P_{2}+P_{3})\ X\ {\frac {100}{R_{2}Y_{2}}}}{{\frac {100}{R_{1}Y_{1}}}+{\frac {100}{R_{2}Y_{2}}}+{\frac {100}{R_{3}Y_{3}}}}}$ $P_{3}={\frac {(P_{1}+P_{2}+P_{3})\ X\ {\frac {100}{R_{3}Y_{3}}}}{{\frac {100}{R_{1}Y_{1}}}+{\frac {100}{R_{2}Y_{2}}}+{\frac {100}{R_{3}Y_{3}}}}}$

Example

यत्पंचकत्रिकचतुष्कशतेन दत्तं

खंडैस्त्रिभिर्गणक निष्कशतं षडूनम् ।

मासेषु सप्तदशपंचसु तुल्यमाप्तम्

खंडत्रयेऽपि हि फलं वद खंडसंख्याम् ॥ ॥

94 N (niṣkas) were divided into three parts and were lent at 5 per cent (per month) for 7 months, at 3 per cent for 10 months, at 4 per cent for

5 months. If the three parts yield equal interest, find them.

Comment: Here P₁ + P₂+ P₃ = 94

R₁ = 5 Y₁ = 7

R₂ = 3 Y₂ = 10

R₃ = 4 Y₃ = 5

As per the above formula

$P_{1}={\frac {94\ X\ {\frac {100}{5X7}}}{{\frac {100}{5X7}}+{\frac {100}{3X10}}+{\frac {100}{4X5}}}}={\frac {94\ X\ {\frac {100}{35}}}{{\frac {100}{35}}+{\frac {100}{30}}+{\frac {100}{20}}}}={\frac {94\ X\ {\frac {100}{35}}}{\frac {4700}{420}}}=24$

$P_{2}={\frac {94\ X\ {\frac {100}{3X10}}}{{\frac {100}{5X7}}+{\frac {100}{3X10}}+{\frac {100}{4X5}}}}={\frac {94\ X\ {\frac {100}{30}}}{{\frac {100}{35}}+{\frac {100}{30}}+{\frac {100}{20}}}}={\frac {94\ X\ {\frac {100}{30}}}{\frac {4700}{420}}}=28$

$P_3=\frac{94X\frac{100}{4X5}}{\frac{100}{5X7}+\frac{100}{3X10}+\frac{100}{4X5}}=\frac{94X\frac{100}{20}}{\frac{100}{35}+\frac{100}{30}+\frac{100}{20}}=$