Inverse Proportion in Līlāvatī

Introduction

Inverse proportion is a type of proportionality relationship. If two quantities are inversely proportional then as one quantity increases, the other decreases. An example of inverse proportion would be the hours of work required to build a wall.If there are more people building the same wall, the time taken to build the wall reduces.

Verse 84

इच्छावृद्धौ फले ह्रासो ह्रासे वृद्धिश्च जायते ।

व्यस्तं त्रैराशिकं तत्र ज्ञेयं गणितकोविदैः ॥८४॥

If, in a given situation, the desired result decreases (respectively increases) as the requisition increases (respectively decreases) then the inverted rule of three is applied by an adept mathematician.

Comment: Common sense (which is uncommon!) should be used to determine whether the fourth term is going to be smaller or larger than the second term. Then use the rule of three. At present unitary method has replaced the rule of three and in the West, the rule of three has become obsolete. Naturally rules of five and higher numbers have also disappeared. The authors are of the firm opinion that the rule of three is simpler than the unitary method.

Verse 85

जीवानां वयसो मौल्ये तौल्ये वर्णस्य हेमनि ।

भागहारेच राशीनां व्यस्तं त्रैराशिकं विदुः ॥८५॥

Vigorous age of a creature is inversely proportional to its price. Same is true of the relations between the desired fineness of gold and its weight, and given unit of measure and the quantity of grain.

Example 1

प्राप्नोति चेत् षोडशवत्सरा स्त्री द्वात्रिंशकं विंशतिवत्सरा किम् ।

द्विघूर्वहो निष्कचतुष्कमुक्षा प्राप्नोति धूः षट्कवहस्तदा किम् ॥ LXXXVI ||

If the price of a sixteen-year-old female slave is 32 niṣkas, find the price of a twenty-year-old one. An ox which has been yoked for two years cost 4 niskas, what will an ox which has been yoked for six years cost ?

Comment : i) Price decreases as the age increases

16 ⇒ 32

20 ⇒ ?

Here is the price is : ${\frac {32\ X\ 16}{20}}={\frac {128}{5}}=25{\frac {3}{5}}$ N

ii) 2 ⇒ 4

6 ⇒ ?

Here is the price is : ${\frac {4\ X\ 2}{6}}={\frac {4}{3}}=1{\frac {1}{3}}$ N

Example 2

दशवर्णं सुवर्णं चेद्गद्याणकमवाप्यते ।

निष्केण तिथिवर्णन्तु तदा वद कियन्मितम् ॥ LXXXVII ||

One gadyāṇaka (= 48 guñjās) of 10 carat gold can be bought for 1 niṣka. How much 15 carat gold can be bought for the same price ?

Comment : Finer the gold , less can be bought for the same price.

1 10 ⇒ 48

1 15 ⇒ ?

So quantity of 15 carat gold =

${\frac {10\ X\ 48}{15}}=32$ guñjās.

Example 3

सप्ताढकेन मानेन राशौ सस्यस्य मापिते ।

यदि मानशतं जातं तदा पञ्चाढकेन किम् ॥ LXXXVIII ॥

With a measure of 7A (āḍhakas), a certain quantity of grain measures 100 units. How many units will there be if the measure is 5A?

Comment: This is also an example of inverse proportion.

7 ⇒ 100

5 ⇒ ?

The answer 140 units.

${\frac {100\ X\ 7}{5}}=140$ units

See Also

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