Rule of Three in Āryabhaṭīyam

Introduction

The Rule of Three is a Mathematical Rule that allows us to solve problems based on proportions.

Verse

त्रैराशिकफलराशिं तमथेच्छाराशिना हतं कृत्वा ।

लब्धं प्रमाणभजितं तस्मादिच्छाफलमिदं स्यात ॥

Translation

Multiply the fruit (pramāṇaphala) by the requisition (icchā) and divide the resulting product by the argument (pramāṇa). Then is obtained the fruit corresponding to the requisition (icchāphala).

$icchaphala={\frac {pramanaphala\ X\ iccha}{pramana}}$

Example 1

If 10 books cost 2000 Rupees , what will 80 books cost ?

Here icchā= 80; pramāṇaphala = 2000; pramāṇa = 10

Hence

$icchaphala={\frac {pramanaphala\ X\ iccha}{pramana}}$

$icchaphala={\frac {2000\ X\ 80}{10}}={\frac {160000}{10}}=16000$ Rupees.

Example 2

If the interest on Rs. 100 for 2 months is Rs. 5, find the interest on Rs. 25 invested for 8 months.?

Here pramāṇa = Rs. 100 and 2 months; icchā = Rs. 25 and 8 months ; pramāṇaphala = Rs. 5

$icchaphala={\frac {pramanaphala\ X\ iccha}{pramana}}$

$icchaphala={\frac {5\ X\ 100\ X\ 2}{25\ X\ 8}}={\frac {1000}{200}}=5$ Rs.

See Also

आर्यभटीयम् में 'तीन का नियम'