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Fundamentals of Business Mathematics and Statistics · Ratios, Variations and Proportions

Ratio: Meaning and Types Explained for CMA Foundation

Updated 10 October 2026 · Fact-checked

A ratio compares two quantities of the same kind by division, written a : b or a/b. To solve questions, convert each ratio to a fraction, then apply the rule: duplicate squares both terms, triplicate cubes them, sub-duplicate takes square roots, compounded multiplies ratios, inverse reverses the terms.

Understand Ratio: Meaning and Types

A ratio compares two quantities of the same kind and in the same unit. It tells you how many times one quantity contains the other. If Ravi earns ₹30,000 and Meena earns ₹20,000, the ratio of their incomes is 30,000 : 20,000 = 3 : 2.

In a ratio a : b, a is the antecedent (first term) and b is the consequent (second term). A ratio has no unit, because the units cancel. You cannot compare 2 kg with 500 g until you convert both to the same unit. A ratio stays the same if you multiply or divide both terms by the same non-zero number. So 3 : 2 = 6 : 4 = 9 : 6. This is why you can reduce a ratio to its lowest terms.

The exam also tests special types of ratio. The duplicate ratio of a : b is a² : b². The triplicate ratio is a³ : b³. The sub-duplicate ratio is √a : √b. The sub-triplicate ratio is ∛a : ∛b. The inverse (reciprocal) ratio of a : b is b : a. The compounded ratio of a : b and c : d is ac : bd, found by multiplying antecedents together and consequents together.

Ratios are also classed by size. If a > b, a : b is a ratio of greater inequality. If a < b, it is a ratio of lesser inequality. If a = b, it is a ratio of equality.

To compare two ratios, turn each into a fraction and compare the fractions. Use a common denominator or cross-multiplication. a : b is greater than c : d when ad > bc, for positive terms.

Key formulas to remember

Ratio
a : b = a ÷ b (b ≠ 0)
a is the antecedent, b the consequent. Both must be in the same unit.
Equivalent ratio
a : b = ka : kb = (a ÷ k) : (b ÷ k), k ≠ 0
Multiply or divide both terms by the same non-zero number.
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
Square root of both terms.
Sub-triplicate ratio
a : b → ∛a : ∛b
Cube root of both terms.
Inverse ratio
a : b → b : a
Interchange the terms.
Compounded ratio
(a : b) and (c : d) → ac : bd
For three ratios, multiply all antecedents and all consequents.
Comparing ratios
a : b > c : d if ad > bc
For positive terms. Cross-multiply the fractions a/b and c/d.

How to solve Ratio: Meaning and Types questions

Use this method for any question on types of ratio or comparison of ratios.

  1. 1Read the question and note which type of ratio is asked: duplicate, triplicate, sub-duplicate, inverse or compounded.
  2. 2Write each given ratio as a fraction in the same unit.
  3. 3Apply the rule for that type: square, cube, root, flip, or multiply across.
  4. 4If the question gives a duplicate or sub-duplicate ratio and asks for the original, reverse the operation: take roots for duplicate, square for sub-duplicate.
  5. 5Simplify the result by dividing both terms by their highest common factor.
  6. 6For comparison, cross-multiply (ad against bc) or convert to decimals, then arrange in the order asked.
  7. 7Check the answer against the options. Confirm the order of terms has not flipped.

Quickest way: Operate on terms, then reduce

When to use it: Use for most MCQs where options are clean ratios like 4 : 9 or 8 : 27.

  1. Do the operation on each term separately. Do not touch the colon.
  2. Reduce only at the end, if needed.
  3. For compounded ratio, cancel common factors across the numerators and denominators before multiplying.
  4. To compare ratios quickly, cross-multiply and see which product is bigger.
  5. Eliminate options that have the terms in the wrong order. This often removes two options at once.

Common mistakes in Ratio: Meaning and Types

  • Confusing duplicate ratio with double the ratio.

    The word 'duplicate' sounds like 'twice'.

    Fix: Duplicate means squared. The duplicate ratio of 2 : 3 is 4 : 9, not 4 : 6.

  • Mixing up duplicate and sub-duplicate ratios.

    Both words start with 'dup' and students rush.

    Fix: Duplicate squares. Sub-duplicate is 'under' the square, so it takes the square root.

  • Adding ratios to get the compounded ratio.

    Students think 'compound' means combine by adding.

    Fix: Multiply antecedent by antecedent and consequent by consequent. 2 : 3 and 4 : 5 compound to 8 : 15.

  • Ignoring units when forming a ratio.

    Quantities like ₹ and paise, or kg and g, are mixed in the question.

    Fix: Convert both to the same unit first. 50 paise : ₹2 is 50 : 200 = 1 : 4.

  • Reversing the order of terms.

    Students write the ratio in the order they read the data, not the order asked.

    Fix: Check the order: 'ratio of A to B' means A is the antecedent.

  • Adding the same number to both terms and expecting the ratio to stay the same.

    Students confuse this with multiplying both terms.

    Fix: Only multiplication or division by the same number keeps a ratio unchanged. Adding the same number changes it.

Worked examples

Example 1

Find the compounded ratio of the duplicate ratio of 2 : 3, the triplicate ratio of 1 : 2 and the sub-duplicate ratio of 9 : 16.

Show the solution
  1. Duplicate ratio of 2 : 3 = 2² : 3² = 4 : 9.
  2. Triplicate ratio of 1 : 2 = 1³ : 2³ = 1 : 8.
  3. Sub-duplicate ratio of 9 : 16 = √9 : √16 = 3 : 4.
  4. Compounded ratio = (4 × 1 × 3) : (9 × 8 × 4) = 12 : 288.
  5. Divide both by 12: 1 : 24.

Answer: 1 : 24

Example 2

Which is the greatest of the ratios 3 : 4, 5 : 7 and 7 : 9?

Show the solution
  1. Write them as fractions: 3/4, 5/7, 7/9.
  2. Use the common denominator 252 (4 × 7 × 9 = 252).
  3. 3/4 = 189/252. 5/7 = 180/252. 7/9 = 196/252.
  4. Compare numerators: 196 > 189 > 180.
  5. So the greatest is 7/9.

Answer: 7 : 9 is the greatest.

Exam tips

  • Learn the five rules by heart: square, cube, square root, flip, multiply. Most MCQs are direct applications.
  • Watch for questions that ask for the original ratio when the duplicate or sub-duplicate is given. Reverse the operation.
  • When asked to arrange ratios in order, cross-multiply pairs. It is faster than finding a common denominator for two ratios.
  • With no negative marking, always mark an answer. Eliminate options with terms in the wrong order first.
  • Reduce your final answer to the lowest terms. Options are usually given that way.

Practice questions from Ratios, Variations and Proportions

Ratio: Meaning and Types in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Ratio: Meaning and Types: frequently asked questions

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

Duplicate ratio squares both terms, so a : b becomes a² : b². Sub-duplicate ratio takes the square root of both terms, so a : b becomes √a : √b. They are opposite operations.

How do you find a compounded ratio?

Multiply the antecedents together to get the new first term. Multiply the consequents together to get the new second term. Then reduce to lowest terms. For 2 : 3 and 4 : 5, the answer is 8 : 15.

Does a ratio have a unit?

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

How do you compare two ratios?

Write them as fractions and cross-multiply. For a : b and c : d with positive terms, a : b is greater if ad > bc. You can also convert both to decimals.