2.3 Ratio & Proportion

Why This Matters

Ratio and proportion questions test your ability to compare quantities and scale them, skills that also feed into profit-loss, time-speed-distance, and mixture problems. RRB typically asks 3 to 4 questions from this family per cycle.

Ratio — Definition and Form

A ratio compares two quantities of the same kind in the form a : b (read "a is to b"). For example, if a class has 15 boys and 20 girls, the ratio of boys to girls is 15 : 20 = 3 : 4 (always simplify).

Key properties:

• a : b = (a * k) : (b * k) for any non-zero k. Multiplying both terms by the same number does not change the ratio.
• The order matters: 3 : 4 is not the same as 4 : 3.
• Compound ratio of a:b and c:d is (a * c) : (b * d). It combines ratios across dimensions.
• Duplicate ratio of a:b is a^2 : b^2; triplicate is a^3 : b^3; sub-duplicate is sqrt(a) : sqrt(b).

Proportion — Equality of Two Ratios

Four numbers a, b, c, d are in proportion if a : b = c : d, written a : b :: c : d. Here a and d are the extremes, and b and c are the means.

Golden rule: product of extremes = product of means, so a * d = b * c.

This is the single most-tested fact in this section. If you know three terms, you can always find the fourth: d = (b * c) / a.

Continued proportion: a, b, c are in continued proportion if a : b = b : c. Then b is the mean proportional between a and c, and b^2 = a * c, so b = sqrt(a * c).

Example: 4, 6, 9 are in continued proportion because 4/6 = 6/9 = 2/3, and 6^2 = 4 * 9 = 36.

Direct and Inverse Proportion

Properties (Componendo, Dividendo)

For a : b = c : d:

• Componendo: (a + b) : b = (c + d) : d
• Dividendo: (a - b) : b = (c - d) : d
• Componendo & Dividendo: (a + b) : (a - b) = (c + d) : (c - d)

The last one is a favourite RRB shortcut. When a ratio equals another and the question asks for the value of an expression in a and b, applying componendo and dividendo often collapses three lines of algebra into one.

Worked Word Problems

Problem 1, distribution. Divide Rs 1,860 between A and B in the ratio 3 : 2. Total parts = 5. Value of one part = 1860 / 5 = Rs 372. A gets 3 * 372 = Rs 1,116; B gets 2 * 372 = Rs 744.

Problem 2, mean proportional. Find the mean proportional between 8 and 18. Mean proportional = sqrt(8 * 18) = sqrt(144) = 12. So 8 : 12 = 12 : 18.

Problem 3, inverse proportion. 12 workers finish a job in 10 days. How many days will 15 workers take? Workers times days is constant: 12 * 10 = 15 * d, so d = 120/15 = 8 days.

Problem 4, mixing. Two alloys contain copper and zinc in ratios 3 : 2 and 5 : 1. Equal masses are melted together. Find the ratio of copper to zinc in the new alloy. Take 5 kg of each. Copper = (3/5 * 5) + (5/6 * 5) = 3 + 25/6 = 43/6 kg. Zinc = (2/5 * 5) + (1/6 * 5) = 2 + 5/6 = 17/6 kg. Ratio = 43 : 17.

Problem 5, componendo and dividendo shortcut. If (x + 7)/(x - 7) = 5/3, find x. Applying componendo and dividendo: [(x + 7) + (x - 7)] / [(x + 7) - (x - 7)] = (5 + 3)/(5 - 3). This simplifies to 2x/14 = 8/2, so x/7 = 4, giving x = 28.

Exam Scenarios and Common Traps

• Wrong order in a ratio: "the ratio of A to B" is A:B, not B:A. Reversing it inverts the answer.
• Forgetting to simplify: answers are expected in lowest terms; 6:8 should be written 3:4.
• Mixing direct and inverse: "more men, more work" is direct; "more men, less time" is inverse. Identify which quantity is fixed before choosing the relation.
• Adding ratios of different wholes: 3:1 in a 100 ml cup and 3:1 in a 500 ml cup are different absolute amounts; combine quantities, not ratios directly.