Problem Solving with Proportions

The Rule of Three (Trairasika)

Ancient Indian mathematicians called proportionality problems the “Rule of Three”. You are given 3 numbers and need to find the 4th.

Model: $a : b :: c : x$

Method 1: Factor of Change

Problem: To make 6 glasses of lemonade, you need 10 spoons of sugar. How many spoons for 18 glasses?

• Ratio: $6 : 10$
• Target: $18 : x$
• Factor: $6 \rightarrow 18$ is times 3 ($6 \times 3 = 18$).
• Apply factor to sugar: $10 \times 3 = 30$.
• Answer: 30 spoons.

Method 2: Cross Multiplication

Problem: A car travels 90 km in 150 minutes. How far does it go in 240 minutes? $150 : 90 :: 240 : x$

$\frac{150}{90} = \frac{240}{x}$

$150 \times x = 240 \times 90$

$x = \frac{240 \times 90}{150}$

$x = \frac{21600}{150} = 144 \text{ km}$