The Rule of Three
Three numbers are known. Multiply the phala by the ichchhā, divide by the pramāṇa.
A rainy day at school
For the mid-day meal in a school of 120 students, the cook makes 15 kg of rice. On a rainy day only 80 students came.
How much rice should the cook make so that food is not wasted?
What this lesson covers
The idea
When four quantities are linked proportionally as a : b :: c : d and three are known, the fourth is found by cross multiplication, d = bc/a; such problems are called Rule of Three (Trairasika) problems.
A rainy day at school
For the mid-day meal in a school of 120 students, the cook makes 15 kg of rice. On a rainy day only 80 students came.
How much rice should the cook make so that food is not wasted?
- 8 kg
- 10 kg
- 12 kg
Āryabhaṭa's rule
Three numbers are known, one is missing. Multiply the phala by the ichchhā, then divide by the pramāṇa. Follow the hints.
Cross multiplication finds the fourth term
When four quantities are linked proportionally as a : b :: c : d and three are known, the fourth is found by cross multiplication, d = bc/a; such problems are called Rule of Three (Trairasika) problems.
"Multiply the phala by the ichchhā and divide the resulting product by the pramāṇa." That is d = (b × c) ÷ a.
- Āryabhaṭa's name | Meaning | Our letter
- pramāṇa | measure | a
- phala | fruit | b
- ichchhā | requisition | c
- ichchhāphala | yield | d
Notes
When four quantities are linked proportionally as a : b :: c : d and three are known, the fourth is found by cross multiplication, d = bc/a; such problems are called Rule of Three (Trairasika) problems.
Check yourself
A tailor uses 6 m of cloth for 4 shirts. How many metres of cloth for 10 shirts?
Answer: 15 m
4 : 6 :: 10 : d, so d = 6 × 10 ÷ 4 = 60 ÷ 4 = 15 m.
To find the missing term in 5 : 8 :: 15 : ?, which calculation is right?
A shopkeeper sells 3 kg of sugar for ₹135. What is the price of 8 kg at the same rate? (in ₹)
Answer: 360
3 : 135 :: 8 : d, so d = 135 × 8 ÷ 3 = 1080 ÷ 3 = ₹360.
In 12 : 18 :: 8 : d, find d using d = bc/a.
Answer: 12
d = bc/a = 18 × 8 ÷ 12 = 144 ÷ 12 = 12.
- 5 × 15 ÷ 8. That puts the pramāṇa 5 on top. The phala 8 and the ichchhā 15 are multiplied.
- 8 × 15 ÷ 5 — correct. Yes! d = (b × c) ÷ a = (8 × 15) ÷ 5 = 24.
- 5 × 8 ÷ 15. That multiplies the two terms of the first ratio. Multiply the phala by the ichchhā instead.
- 5 × 8 × 15. That multiplies all three and never divides.