Introduction to Ratio and Proportion

Ratio and Proportion

A ratio compares two quantities of the same kind by division. The ratio of a to b is written a : b. A ratio has no unit.

A proportion says two ratios are equal: if a : b = c : d, then a × d = b × c (product of extremes = product of means).

Key rules

• Multiplying or dividing both terms by the same non-zero number does not change a ratio (4 : 6 = 2 : 3).
• To divide N in the ratio a : b: first part = N × a/(a+b), second part = N × b/(a+b).
• Fourth proportional of a, b, c: x = (b × c)/a.
• Third proportional of a, b: x = b²/a.
• Mean proportional of a and b: √(a × b).
• Combining ratios: a : b = 2 : 3 and b : c = 4 : 5 → make b equal → a : b : c = 8 : 12 : 15.

Worked example

Divide ₹360 between A and B in the ratio 4 : 5. Total parts = 9, one part = 40. A gets ₹160, B gets ₹200.

Exam tips (RRB NTPC)

• Convert quantities to the same unit before forming a ratio.
• Use the "one part" method: total ÷ sum of ratio terms.
• Check: the parts must add up to the total.

Analogy

Think of a recipe for tea. If one cup needs 2 spoons of sugar for every 1 spoon of tea leaves, sugar : tea = 2 : 1. Making 4 cups? Keep the same taste by keeping the same ratio — 8 spoons sugar to 4 spoons tea. That equal taste is a proportion: 2 : 1 = 8 : 4. Sharing 9 laddoos in the ratio 4 : 5 works the same way — count total parts (9), one laddoo per part, then hand out 4 and 5.