Sub-Duplicate Ratio: Formula, Rules and Examples

Sub-Duplicate Ratio is the ratio formed by taking the square roots of the terms of a given ratio. This page explains the sub duplicate ratio formula, its relation with duplicate ratio, methods for simplifying square root ratios, and numerical examples commonly used in quantitative aptitude.

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What is Sub-Duplicate Ratio?

The sub-duplicate ratio of a:b is √a:√b. It is obtained by taking the square root of both terms of the original ratio.

If two positive quantities are in the ratio a:b, their sub-duplicate ratio is √a:√b. For example, the sub-duplicate ratio of 25:49 is √25:√49 = 5:7. The terms must be non-negative when real square roots are used.

Sub-Duplicate Ratio Formula & Tricks

Important Formulas

Sub-duplicate ratio formula
Sub-duplicate ratio of a:b = √a:√b

Take the positive square root of each term in the given ratio.

Duplicate and sub-duplicate relation
Duplicate ratio of a:b = a²:b²; sub-duplicate ratio of a:b = √a:√b

A duplicate ratio squares both terms, while a sub-duplicate ratio takes their square roots.

Ratio of squared quantities
If x²:y² = a:b, then x:y = √a:√b

For positive x and y, taking square roots gives the ratio of the original quantities.

Quick Tricks

Extract perfect-square factors

When a term is not a perfect square, factor it into a perfect square and another factor before taking its square root.

Example: The sub-duplicate ratio of 72:98 is √72:√98 = 6√2:7√2 = 6:7.
Compare square roots by squaring

For non-negative numbers, √a < √b exactly when a < b. This avoids decimal approximation while comparing square-root terms.

Example: Since 18 < 32, √18 < √32.
Simplify the ratio after taking roots

Cancel common factors only after expressing both square roots in simplified form.

Example: √45:√80 = 3√5:4√5 = 3:4.

Sub-Duplicate Ratio Concepts

Finding the sub-duplicate ratio of a given ratio

To find the sub-duplicate ratio, take the square root of both terms and simplify the resulting ratio.

For a ratio a:b, write √a:√b. If both terms are perfect squares, the calculation is direct. For non-perfect squares, factor out common square factors and cancel any common radical factor.

Example: For 144:225, the sub-duplicate ratio is √144:√225 = 12:15 = 4:5.

Sub-duplicate ratio with non-perfect squares

A ratio containing non-perfect squares can still be simplified by separating perfect-square factors from the radicals.

Use √(mn) = √m × √n when m and n are non-negative. Then cancel common numerical or radical factors from the two terms.

Example: For 50:128, √50:√128 = 5√2:8√2 = 5:8.

Recovering the original ratio from a sub-duplicate ratio

If p:q is a sub-duplicate ratio, the corresponding original ratio is p²:q².

Squaring both terms reverses the square-root operation. Thus, if the sub-duplicate ratio is 3:5, the original ratio is 9:25.

Example: If √a:√b = 4:7, then a:b = 4²:7² = 16:49.

Difference between duplicate and sub-duplicate ratios

A duplicate ratio squares both terms, whereas a sub-duplicate ratio takes the square root of both terms.

For a:b, the duplicate ratio is a²:b² and the sub-duplicate ratio is √a:√b. These operations are inverse only when they are applied to the same positive ratio in the appropriate order.

Example: For 4:9, the duplicate ratio is 16:81, while the sub-duplicate ratio is 2:3.

Sub-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 Sub-Duplicate Ratio Questions

Practise published questions related to this topic.

1Simplify: ((2x²y⁻¹)³)/((4xy)⁴) × ((4x⁻³y⁴)³)/((2x²y)³)→ 2Find the greatest number that divides 75, 105 and 150 exactly.→ 3A shopkeeper is offering a 20% discount on a Mixer that originally costs Rs 400. What is the sale price of the Mixer?→ 4An officer computes (√50 + √8).Simplify.→ 5What is the HCF of 32, 28, and 24?→ 6The sum of a number and its positive square root is 42. What is the value of the number?→ 728P3 is divisible by 11 . What is the value of P ?→ 8What is the Highest Common Factor of 57 x 113 and 55 x 116?→ 9A store selling mobile accessories has a special offer on screen guards: when a customer buys two screen guards priced at ₹ 400 each, they receive a 5% discount on the total bill. How much does the customer have to pay after the discount is applied?→ 10If 3^x - 3^(x-1) = 18 , then the value of (2x-1)/(2x+3) is:→

Sub-Duplicate Ratio Quick Quiz

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Quick Revision Notes

Sub-Duplicate Ratio Revision Points

Use these rules to revise the definition, calculation and inverse relationship of sub-duplicate ratios.

  • The sub-duplicate ratio of a:b is √a:√b.
  • Take the positive square root of each non-negative term.
  • Simplify radicals before cancelling common factors.
  • The sub-duplicate ratio of 25:49 is 5:7.
  • If the sub-duplicate ratio is p:q, the original ratio is p²:q².
  • Duplicate ratio means squaring both terms; sub-duplicate ratio means taking their square roots.
  • For non-negative numbers, √a:√b can be simplified using common square factors.

Sub-Duplicate Ratio FAQs

What is the sub-duplicate ratio of 36:64?

√36:√64 = 6:8 = 3:4.

What is the sub-duplicate ratio formula?

The sub-duplicate ratio of a:b is √a:√b, where a and b are non-negative.

What is the sub-duplicate ratio of 72:98?

√72:√98 = 6√2:7√2 = 6:7.

If the sub-duplicate ratio is 5:8, what is the original ratio?

Square both terms: 5²:8² = 25:64.

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

The duplicate ratio of a:b is a²:b², while the sub-duplicate ratio is √a:√b.

How is √45:√80 simplified?

√45:√80 = 3√5:4√5 = 3:4.

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