Rule of Three in Pāṭīgaṇitam

The Rule of Three is a Mathematical Rule that allows us to solve problems based on proportions. Rule of Three is known as Trairāśika in Samskrita.

Verse

आद्यन्तयोस्त्रिराशावभिन्नजाती प्रमाणमिच्छा च ।

फलमन्यजातिमध्ये तदन्त्यगुणमादिना विभजेत् ॥४३॥

Translation

In (solving problems on) the rule of three, the argument (pramāṇa) and the requisition (icchā), which are of the same denomination, should be set down in the first and last places; the fruit (phala), which is of a different denomination, should be set down in the middle. (This having been done) that (middle quantity) multiplied by the last quantity should be divided by the first quantity.

$required\ fruit={\frac {fruit\ X\ requisition}{argument}}$

Example

If 1 pala and 1 karṣa of sandal wood are obtained for ten and a half paṇas, then for how much will 9 palas and 1 karṣa (of sandal wood of the same quality) be obtained?

Here

argument - 1 pala and 1 karṣa = $1{\frac {1}{4}}$ or ${\frac {5}{4}}$palas

we know 4 karṣas = 1 pala.

fruit = $10{\frac {1}{2}}$ or ${\frac {21}{2}}$paṇas

requisition = 9 palas and 1 karṣa = $9{\frac {1}{4}}$ or ${\frac {37}{4}}$palas

Writing these quantities as directed in the rule, we have

argument (pramāṇa)	fruit (phala)	requisition (icchā)
5 4	21 2	37 4

Then applying the rule, the required result

$required\ fruit={\frac {fruit\times requisition}{argument}}$

$={\frac {{\frac {21}{2}}\times {\frac {37}{4}}}{\frac {5}{4}}}$$={\frac {21\times 37}{2\times 5}}$$={\frac {21\times 37}{2\times 5}}$$={\frac {777}{10}}$paṇas