Duplicate Ratio: Formula, Rules and Solved Examples

Duplicate Ratio means the ratio obtained by squaring both terms of a given ratio. For a:b, the duplicate ratio is a²:b². This page explains the duplicate ratio formula, simplification rules, methods for finding missing ratios and numerical examples commonly used in quantitative aptitude.

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What is Duplicate Ratio?

The duplicate ratio of a:b is a²:b², obtained by multiplying each term of the ratio by itself.

If x:y is a ratio of positive quantities, its duplicate ratio is x²:y². For example, the duplicate ratio of 3:5 is 9:25. The ratio should be simplified after squaring if the resulting terms have a common factor.

Duplicate Ratio Formula & Tricks

Important Formulas

Duplicate ratio formula
Duplicate ratio of a:b = a²:b²

Square the first term and the second term separately.

Fraction form
If a:b = a/b, then its duplicate ratio = (a/b)² = a²/b²

The ratio is squared as a complete fraction.

Finding the original ratio
If the duplicate ratio is a²:b², the original ratio is a:b

For positive ratio terms, take the positive square root of both terms or of their quotient.

Quick Tricks

Square after simplifying

First reduce the given ratio to its simplest form, then square both terms. This keeps the calculation smaller and avoids unnecessary simplification.

Example: For 12:18, simplify to 2:3. Its duplicate ratio is 2²:3² = 4:9.
Use the ratio value for unknown terms

If the duplicate ratio is given as p:q, compare the squared ratio with p/q and take the positive square root to obtain the original ratio.

Example: If the duplicate ratio is 25:49, the original ratio is √25:√49 = 5:7.

Duplicate Ratio Concepts

Calculating the duplicate ratio

To calculate a duplicate ratio, square both terms of the simplified original ratio.

For a:b, multiply a by a and b by b. The result is a²:b². If the original ratio is not in simplest form, simplifying it first gives the same final ratio.

Example: The duplicate ratio of 4:7 is 4²:7² = 16:49.

Finding the original ratio from a duplicate ratio

Take the positive square root of both terms when the duplicate ratio is given in square form.

If a duplicate ratio is 36:121, the corresponding positive original ratio is √36:√121 = 6:11. In aptitude questions, ratio terms represent positive quantities, so the positive roots are used.

Example: For a duplicate ratio of 64:81, the original ratio is 8:9.

Duplicate ratio of an equivalent ratio

Equivalent forms of the same ratio have the same duplicate ratio after simplification.

Multiplying both terms by the same non-zero number does not change a ratio. Thus, 2:3 and 10:15 both produce the duplicate ratio 4:9 after simplification.

Example: For 10:15, first reduce to 2:3; therefore, its duplicate ratio is 4:9.

Comparing a ratio with its duplicate ratio

For positive unequal terms, duplicating a ratio changes its value by squaring the original ratio value.

If a:b has value a/b, its duplicate ratio has value (a/b)². If a>b, the duplicate ratio is greater than the original ratio value; if a<b, it is smaller. When a=b, both ratios equal 1.

Example: The ratio 2:5 has value 2/5, while its duplicate ratio 4:25 has value 4/25.

Duplicate Ratio Video Lessons

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Duplicate, Triplicate and Sub-Duplicate Ratios

Learn the definitions and application of duplicate, triplicate, and sub-duplicate ratios, with clear methods for understanding how these special ratios are formed and used.

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Practice Duplicate Ratio Questions

Practise published questions related to this topic.

1A person sells two items: one at a profit of 20% and another at a loss of 10%. If both items cost ₹100 each, what is the overall gain or loss percentage?→ 2The HCF of 1/2, 3/4, 27/16 and 81/64 is:→ 3Which of the following numbers is divisible by 3?→ 4At what percentage above the cost price of an article is to be marked to gain 20%, after allowing a discount of 25%?→ 539P6 is divisible by 3. Which of the following cannot be the value of P?→ 6What is the minimum value that must be assigned to A so that the 8-digit number 130A1568 is divisible by 8?→ 7A shopkeeper marks a washing machine at ₹24,000. During a festival, he offers a 20% discount and sells it for an additional ₹800 less. What is the final selling price of the washing machine?→ 8The marked price of an article is 160 percent of its cost price. Its selling price is 70 percent of the marked price. What is the profit percentage?→ 9An item is marked at ₹2000 . To sell it for ₹1500 , what discount percentage must be given?→ 10Which of the following numbers is divisible by both 6 and 9, but not divisible by 12? 126, 252, 72, 108→

Duplicate Ratio Quick Quiz

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Quick Revision Notes

Duplicate Ratio Revision Points

Use these rules to solve duplicate ratio questions quickly and accurately.

  • Duplicate ratio of a:b is a²:b².
  • Simplify the original ratio before squaring when the terms are large.
  • The duplicate ratio of 3:4 is 9:16.
  • To recover a positive original ratio from a²:b², take the positive square roots.
  • Equivalent original ratios produce equivalent duplicate ratios.
  • Duplicate ratio is different from triplicate ratio, which is a³:b³.

Duplicate Ratio FAQs

What is the duplicate ratio formula for a:b?

The duplicate ratio formula is a²:b². Each term of the ratio is multiplied by itself.

What is the duplicate ratio of 5:8?

Square both terms: 5²:8² = 25:64.

If the duplicate ratio is 49:100, what is the original ratio?

Take positive square roots: √49:√100 = 7:10.

What is the duplicate ratio of 18:24?

Simplify 18:24 to 3:4. Therefore, the duplicate ratio is 3²:4² = 9:16.

Is the duplicate ratio of a:b equal to 2a:2b?

No. The duplicate ratio is a²:b², whereas 2a:2b is equivalent to the original ratio a:b.

What is the difference between duplicate ratio and triplicate ratio?

The duplicate ratio of a:b is a²:b², while the triplicate ratio is a³:b³.

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