Ratio & Proportion • Topic 3 of 5

Proportion

A proportion is the statement that two ratios are equal: a:b = c:d. Here a and d are the extremes and b and c are the means, and the defining identity is product of extremes = product of means, i.e. a×d = b×c. This cross-multiplication is the single most useful tool — it converts any proportion into a clean linear equation. Four numbers a, b, c, d are said to be in proportion when a:b = c:d. CAT also tests the derived forms built from a:b = c:d: componendo (a+b):b = (c+d):d, dividendo (a−b):b = (c−d):d, and the very handy componendo–dividendo (a+b):(a−b) = (c+d):(c−d), which lets you find a:b instantly when you know (a+b):(a−b). Recognising these saves a full page of algebra in equation-heavy questions.

✅ Solved examples

1. If 6:8 = 9:x, find x.
Extremes × = means ×: 6x = 8×9 = 72 ⇒ x = 12.
2. Are 4, 6, 10, 15 in proportion?
Check 4×15 vs 6×10: 60 = 60. Yes, 4:6 = 10:15, so they are in proportion.
3. Find the fourth proportional to 5, 8, 15.
Fourth proportional d satisfies 5:8 = 15:d ⇒ 5d = 8×15 = 120 ⇒ d = 24.
4. If (a+b):(a−b) = 7:3, find a:b.
By componendo–dividendo, a:b = (7+3):(7−3) = 10:4 = 5:2.

✏️ Practice — try these, take hints as needed

1. If 3:5 = x:20, find x.
Cross-multiply.
5x = 3×20.
x = 60/5.
12
2. Are 2, 3, 8, 12 in proportion?
Compare 2×12 and 3×8.
24 and 24.
Equal means yes.
Yes (2:3 = 8:12)
3. Find the fourth proportional to 4, 9, 12.
4:9 = 12:d.
4d = 9×12.
d = 108/4.
27
4. If x:7 = 12:21, find x.
21x = 7×12.
21x = 84.
x = 84/21.
4
5. If (a+b):(a−b) = 5:1, find a:b.
Use componendo–dividendo.
a:b = (5+1):(5−1).
6:4 reduced.
3 : 2

📝 Topic test — 8 questions

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