Ratio and Proportion Cheat Sheet
31 Oct 2026
Quick answer
A quick revision cheat sheet on ratio and proportion for SSC, Banking, Railway and state exams. It covers how to read and simplify a ratio, proportion and the cross-multiplication rule, mean, third and fourth proportionals, duplicate and sub-duplicate ratios, dividing a quantity in a given ratio, and the componendo-dividendo shortcut.
Ratio and Proportion Cheat Sheet
Quick Revision
A ratio compares two quantities of the same kind: a : b means a/b. A proportion states that two ratios are equal: a : b = c : d.
Ready to practise? Take the Ratio and Proportion quiz
1. The Proportion Rule
If a : b = c : d, then the product of the extremes equals the product of the means:
a × d = b × cHere a and d are the extremes, b and c are the means.
2. Special Proportionals
| Term | Definition | Formula |
|---|---|---|
| Mean proportional of a and b | a : x = x : b | x = √(ab) |
| Third proportional of a and b | a : b = b : x | x = b² / a |
| Fourth proportional of a, b, c | a : b = c : x | x = (b × c) / a |
3. Types of Ratios
| Ratio | Meaning |
|---|---|
| Duplicate of a : b | a² : b² |
| Sub-duplicate of a : b | √a : √b |
| Triplicate of a : b | a³ : b³ |
| Sub-triplicate of a : b | ∛a : ∛b |
4. Dividing a Quantity in a Ratio
To divide N in the ratio a : b : c, each share is the whole times its part over the total parts:
Total parts = a + b + c
First share = N × a / (a + b + c)
Second share = N × b / (a + b + c)
Third share = N × c / (a + b + c)Example
Divide ₹6000 in the ratio 2 : 3 : 1.
Total parts = 6
Shares = 6000 × 2/6, 6000 × 3/6, 6000 × 1/6
= ₹2000, ₹3000, ₹10005. Combining Ratios
If a : b and b : c are given, scale b to match and write a : b : c.
Example
A : B = 2 : 3 and B : C = 4 : 5.
Make B common (3 and 4 → LCM 12):
A : B = 8 : 12, B : C = 12 : 15
So A : B : C = 8 : 12 : 156. Componendo and Dividendo
If a/b = c/d, then:
(a + b) / (a − b) = (c + d) / (c − d)This is a fast way to solve many ratio equations.
7. Exam Quick Tips
- Always simplify a ratio to its lowest terms first.
- Mean proportional is
√(ab); do not confuse it with the simple average. - For sharing problems, find the total parts before any single share.
- Keep the order of a ratio;
2 : 3is not the same as3 : 2.
Keep revising with the rest of the Maths section:
How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.