Quantitative Aptitude · Ratio and Proportion, Indices and Logarithms

Ratio: Meaning, Types and Properties for CA Foundation

Updated 1 October 2026 · Fact-checked

A ratio compares two quantities of the same kind by division, written a : b or a/b. Types include duplicate (a² : b²), triplicate (a³ : b³), sub-duplicate (√a : √b) and compounded (ac : bd). To compare ratios, convert them to fractions with a common denominator or cross-multiply.

Understand Ratio: Meaning, Types and Properties

A ratio tells you how many times one quantity is of another. It is written a : b, where a is the antecedent (first term) and b is the consequent (second term). It is the same as the fraction a/b, with b ≠ 0.

Both quantities must be of the same kind and in the same unit. You cannot form a ratio of 2 kg to 500 g until you convert both to grams. A ratio has no unit. It is a pure number.

Multiplying or dividing both terms by the same non-zero number does not change the ratio. So 2 : 3 = 4 : 6 = 10 : 15. This is why you can simplify a ratio to its lowest terms by dividing by the HCF.

The exam asks about derived ratios. If the ratio is a : b, then the duplicate ratio is a² : b², the triplicate ratio is a³ : b³, the sub-duplicate ratio is √a : √b, and the sub-triplicate ratio is the cube root of a : the cube root of b. The inverse (reciprocal) ratio is b : a.

A compounded ratio of a : b and c : d is found by multiplying the antecedents together and the consequents together: ac : bd. You can compound any number of ratios this way.

Key formulas to remember

Ratio as a fraction: a : b = a ÷ b = a/b (b ≠ 0); Both terms must be in the same unit. Multiplying or dividing both terms by the same non-zero number keeps the ratio unchanged.
Duplicate ratio: a : b → a² : b²; Square both terms.
Triplicate ratio: a : b → a³ : b³; Cube both terms.
Sub-duplicate ratio: a : b → √a : √b; Take the square root of both terms.
Sub-triplicate ratio: a : b → ∛a : ∛b; Take the cube root of both terms.
Inverse ratio: a : b → b : a; Interchange the terms.
Compounded ratio: (a : b) and (c : d) → ac : bd; Multiply antecedents together and consequents together.
Comparing ratios: a/b > c/d ⇔ ad > bc (for b, d > 0); Cross-multiply. Valid only when both denominators are positive.
Ratio of ratios, a : b and b : c: a : b : c; Make the common term equal by scaling, then join the ratios.

How to solve Ratio: Meaning, Types and Properties questions

Use this order for any ratio question. It keeps you away from sign and unit errors.

1Read what is asked: a derived ratio (duplicate, sub-duplicate, compounded), a comparison, or finding unknown quantities.
2Convert all quantities to the same unit before forming the ratio.
3Write each ratio as a fraction. Simplify by cancelling common factors.
4For derived ratios, apply the operation to each term: square, cube, root, invert, or multiply across.
5For comparison, cross-multiply or bring to a common denominator. Check that the denominators are positive.
6For unknown quantities, take the ratio parts as k times each term, such as 3k and 5k, and use the given total or difference to find k.
7Substitute back and check that your answer reproduces the given ratio.

Quickest way: Cross-multiply and eliminate options

When to use it: Use this for MCQs on comparing ratios and on compounded or derived ratios, where time is short.

For compounded ratios, cancel factors diagonally before multiplying to keep the numbers small.
To find the largest or smallest of several ratios, cross-multiply pairs rather than converting to decimals, unless the decimals are obvious.
To order three or more ratios, convert to decimals only if each fraction is easy, such as 3/4 = 0.75.
For sub-duplicate questions, check whether the terms are perfect squares. If they are not, the option is likely wrong.
Put the final answer back into the question to confirm it. If it takes more than 90 seconds, mark the question and move on, since each wrong answer costs 0.25 marks.

Common mistakes in Ratio: Meaning, Types and Properties

Squaring only one term for the duplicate ratio, or writing 2a : 2b.
Students confuse duplicate with double.
Fix: Duplicate means squared: a² : b². Apply the power to both terms.
Confusing sub-duplicate with duplicate.
The names sound similar.
Fix: Remember that the prefix sub- means the root. Sub-duplicate is the square root, and duplicate is the square.
Adding the terms when compounding ratios.
Students mix up compounding with finding a sum.
Fix: Compounding always multiplies: antecedent × antecedent and consequent × consequent.
Forming a ratio without converting units, such as 2 hours : 40 minutes taken as 2 : 40.
Students rush and read only the numbers.
Fix: Convert first: 120 minutes : 40 minutes = 3 : 1.
Comparing ratios by looking only at the antecedents or only at the sizes of the numbers.
A bigger-looking pair is assumed to be a bigger ratio.
Fix: Cross-multiply. For example 5 : 8 versus 3 : 5 gives 25 against 24, so 5 : 8 is larger.
Joining a : b and b : c without matching the common term.
Students skip scaling when the middle terms differ.
Fix: Make the middle term the same using the LCM, then write a : b : c.

Worked examples

Example 1

The compounded ratio of 2 : 3, 9 : 4 and 5 : 6 is: (a) 5 : 4 (b) 5 : 8 (c) 3 : 2 (d) 5 : 12

Show the solution

Multiply the antecedents: 2 × 9 × 5 = 90.
Multiply the consequents: 3 × 4 × 6 = 72.
The ratio is 90 : 72. The HCF is 18.
Divide both by 18: 5 : 4.

Answer: (a) 5 : 4

Example 2

The sub-duplicate ratio of 4 : 9 is: (a) 16 : 81 (b) 2 : 3 (c) 3 : 2 (d) 3 : 4

Show the solution

The sub-duplicate ratio of x : y is √x : √y.
Here the terms are 4 and 9.
√4 = 2 and √9 = 3.
So the sub-duplicate ratio is 2 : 3.

Answer: (b) 2 : 3

Example 3

Which of the following ratios is the greatest? (a) 2 : 3 (b) 3 : 4 (c) 4 : 5 (d) 5 : 7

Show the solution

Write them as fractions: 2/3, 3/4, 4/5, 5/7.
Use the common denominator 420, since 3 × 4 × 5 × 7 = 420.
2/3 = 280/420, 3/4 = 315/420, 4/5 = 336/420, 5/7 = 300/420.
The largest numerator is 336, which belongs to 4/5.
Check by decimals: 0.667, 0.75, 0.8, 0.714. The largest is 0.8.

Answer: (c) 4 : 5

Exam tips

Learn the four names by their roots: duplicate is square, triplicate is cube, sub-duplicate is square root, sub-triplicate is cube root. Questions test the names directly.
Check the units in word problems before doing any calculation. A unit trap is a common way to build a wrong option.
In comparison questions, expect options that look close. Use cross-multiplication, not guesswork.
When a ratio is given as a : b : c with a total, use the k method and verify that the parts add up to the total.
Compounded ratio questions are often quick. Cancel diagonally and finish in under a minute.

Practice questions from Ratio and Proportion, Indices and Logarithms

Ratio: Meaning, Types and Properties: frequently asked questions

What is the difference between duplicate and sub-duplicate ratio?

The duplicate ratio of a : b is a² : b², which squares each term. The sub-duplicate ratio is √a : √b, which takes the square root of each term. They are inverse operations on the terms.

How do you find a compounded ratio?

Multiply all the antecedents to get the new antecedent. Multiply all the consequents to get the new consequent. Then simplify by dividing by the HCF.

How do you compare two ratios quickly?

Write them as fractions a/b and c/d, with positive denominators. Compare ad with bc. If ad is larger, then a/b is the larger ratio.

Can a ratio have units?

No. A ratio compares two quantities of the same kind, so the units cancel. You must convert both quantities to the same unit before forming it.