The Rule of Five 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

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

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

$P_{3}={\frac {94\ X\ {\frac {100}{4X5}}}{{\frac {100}{5X7}}+{\frac {100}{3X10}}+{\frac {100}{4X5}}}}={\frac {94\ X\ {\frac {100}{20}}}{{\frac {100}{35}}+{\frac {100}{30}}+{\frac {100}{20}}}}={\frac {94\ X\ {\frac {100}{20}}}{\frac {4700}{420}}}=42$

Compute shares of profit

Here we will compute the shares of profit when total profit and individual investments are given.

प्रक्षेपका मिश्रहता विभक्ता प्रक्षेपयोगेन पृथक् फलानि ॥ ॥

An individual's share (after business) is the individual's investment multiplied by the total output and divided by the total investment.

Comment :If a, b and c are investments and x is the output, then the shares are ${\frac {ax}{a+b+c}}$ , ${\frac {bx}{a+b+c}}$ , ${\frac {cx}{a+b+c}}$ respectively.

Example

पंचाशदेकसहिता गणकाष्टषष्टिः पंचोनिता नवतिरादिधनानि येषाम् ।

प्राप्ता विमिश्रितधनैस्त्रिशती त्रिभिस्तैः वाणिज्यतो वद विभज्य धनानि तेषाम् ॥ ॥

Three grocers invested 51, 68 , 85 N (niṣkas) respectively. Skillfully they increased their total assets to 300 N. Find the share of each .

Comment : Total investment = $51+68+85=204$ N.

Here a = 51; b = 68 ; c = 85 ; x = 204

Using the above formula

Their shares are ${\frac {51\ X\ 300}{204}}=75$ N ${\frac {68\ X\ 300}{204}}=100$ N ${\frac {85\ X\ 300}{204}}=125$ N

Their profits are $75-51=24$ N ; $100-68=32$ N ; $125-85=40$ N

Formula for filling up of tanks

Here we will know the formula for filling of reservoirs (pools, lakes, tanks).

भजेच्छिदोंऽशैरथ तैर्विमिश्रै रूपं भजेत् स्यात् परिपूर्तिकालः ॥॥

One divided by the sum of the reciprocals (of the times taken by the sources to fill up a pool) is the time of filling (the pool when the sources are used simultaneously).

Comment: Suppose the sources take times t₁, t₂,... to fill up a reservoir. If they are simultaneously used then

the time taken = ${\frac {1}{{\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}+...}}$

Example

ये निर्झरा दिनदिनार्धतृतीयषष्ठैः सम्पूर्णयन्ति हि पृथग्पृथगेव मुक्ताः ।

वापीं यदा युगपदेव सखे विमुक्ताः ते केन वासरलवेन तदा वदाऽशु ॥ ॥

Four streams flow into a pool and separately they take $1$, ${\frac {1}{2}}$, ${\frac {1}{3}}$ , ${\frac {1}{6}}$ days respectively. If all the four are used simultaneously, find the time required to fill up the pool.

Comment: As explained in the previous stanza.

Time = ${\frac {1}{1+2+3+6}}={\frac {1}{12}}$th day

The four streams can fill up 1, 2, 3, 6 pools in one day and so together

they can fill up 12 pools in one day. So time for one pool = ${\frac {1}{12}}$day.

See Also

लीलावती में 'पाँच का नियम'