Trairasika, the Rule of Three

A school cooks 15 kg of rice for 120 students. On a rainy day only 80 come. How much rice should be cooked so none is wasted?

Students and rice must stay proportional: 120 : 15 :: 80 : ?. The factor on the first term is 80/120 = 2/3, so the rice is 15 × 2/3 = 10 kg. (Or simplify first: 120 : 15 = 8 : 1, one kg for every 8 students.)

That shape of problem is everywhere: four quantities linked proportionally, three known, one to find. It has a name and a general method.

Write a : b :: c : d. Being proportional means there's one factor f with c = fa and d = fb. So f = c/a and also f = d/b, which gives c/a = d/b.

Multiply both sides by ab: ab × c/a = ab × d/b, which cancels to bc = ad. That's cross multiplication: when a : b :: c : d, then ad = bc. And it gives you the unknown directly: d = bc/a.

Ancient Indian mathematicians called these Rule of Three problems (Trairāśika). Āryabhaṭa and others named the three known numbers:

The answer is the ichchhāphala, the yield.

His instruction: 'Multiply the phala by the ichchhā and divide the resulting product by the pramāṇa.' In our symbols, ichchhāphala = (phala × ichchhā) ÷ pramāṇa: exactly d = bc/a.

One caution before you use it: the units must match. A car goes 90 km in 150 minutes; how far in 4 hours? Writing 150 : 90 :: 4 : ? is wrong, because 150 is minutes and 4 is hours. Convert first: 4 hours is 240 minutes, so 150 : 90 :: 240 : x, and x = (240 × 90) ÷ 150 = 144 km.

The same method answers many everyday questions: a school with 240 students and 8 teachers has 30 : 1; keeping that ratio for 300 students needs 300 ÷ 30 = 10 teachers.