The Rule of Three in Līlāvatī

The rule of three is a form that allows the resolution of problems of proportionality between three known values and an unknown, in other words, the rule of three is an operation that allows us to find a fourth term with respect to a given proportion.

Verse 79

प्रमाणमिच्छा च समानजातिः

आद्यन्तयोस्तत्फलमन्यजातिः ।

मध्ये तदिच्छाहतमाद्यहृत्स्यात्

इच्छाफलं व्यस्तविधिर्विलोमे ।। LXXIX ।।

Translation

There are three quantities involved herein. The first one on the left (a) is called pramāṇa (scale), the second (b) phala (result or fruit) and third (c) icchā (desire or requisition). The answer to be found (d) is called icchā-phala (desired result). Here a and c must be of the same kind and b should be different from a and c. The formula is $d={\frac {b\ X\ c}{a}}$ . d is of the same kind as b.

Example 1

कुंकुमस्य सदलं पलद्वयं निष्कसप्तमलवेत्रिभिर्यदि ।

प्राप्यते सपदि मे वणिग्वर ब्रूहि निष्कनवकेन तत्कियत् ॥८१॥

If $2{\frac {1}{2}}$ pala of saffron costs ${\frac {3}{7}}$niṣkas O you expert businessman tell me quickly what quantity of saffron can be bought for 9 niṣkas .

Comment:

This is direct proportion, since more money buys more saffron.

As per rule of three.

Cost in niṣkas ⇒ Qty of Saffron

${\frac {3}{7}}$⇒ ${\frac {5}{2}}$

9 ⇒ d

Hence $d={\frac {9\ X{\frac {5}{2}}}{\frac {3}{7}}}$ =${\frac {105}{2}}$palas.

Example 2

द्रम्मद्वयेन साष्टांशा शालितण्डुलखारिका ।

लभ्या चेत् पणसप्तत्या तत्किं सपदि कथ्यताम् ॥८३॥

$1{\frac {1}{8}}$ khārikās of rice can be bought for 2 drammas, then how much rice can be bought for 70 paṇas ?

Comment: This is also an example of direct proportion.

16 paṇas = 1 dramma

Hence 2 drammas = 32 paṇas

Cost in paṇas ⇒ Qty of rice

32 ⇒ $1{\frac {1}{8}}$

70 ⇒ d

$d={\frac {70\ X{\frac {9}{8}}}{32}}$ = ${\frac {630}{256}}$=${\frac {315}{128}}$=$2{\frac {59}{128}}$khārikās

See Also

लीलावती में 'तीन का नियम'