Equivalent Ratios: Formula, Rules and Examples

Equivalent Ratios represent the same comparison even when their terms are different. They are formed by multiplying or dividing both terms by the same non-zero number. This page explains the equivalent ratios formula, methods to test equivalence, missing-term calculations, solved examples and useful shortcuts for ratio and proportion questions.

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What Are Equivalent Ratios?

Equivalent ratios are ratios that have the same value or represent the same comparison. Two ratios a:b and c:d are equivalent when a/b = c/d, or when ad = bc.

Multiplying or dividing both terms of a ratio by the same non-zero number produces an equivalent ratio. For example, multiplying 3:5 by 4 gives 12:20. Both ratios represent the same comparison because 3/5 = 12/20.

Equivalent Ratios Video Lessons

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Ratio Basics: Simplify, Compare and Scale

Learn the fundamentals of ratios by simplifying ratio expressions, comparing quantities, and scaling ratios while preserving their relationships.

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Equivalent Ratios Formula & Tricks

Important Formulas

Multiplication rule
a:b = (a × k):(b × k), where k ≠ 0

Multiplying both terms by the same non-zero number creates an equivalent ratio.

Division rule
a:b = (a ÷ k):(b ÷ k), where k divides both a and b

Dividing both terms by their common factor gives an equivalent, often simpler, ratio.

Cross-multiplication test
a:b = c:d if and only if a × d = b × c

The ratios are equivalent when the products of opposite terms are equal.

Missing fourth term
If a:b = c:x, then x = (b × c) ÷ a

Cross-multiplication gives a × x = b × c, so x can be calculated directly.

Quick Tricks

Reduce both ratios before comparing

Divide each ratio by the greatest common divisor of its terms. If the reduced forms match, the original ratios are equivalent.

Example: 18:24 reduces to 3:4, and 15:20 also reduces to 3:4. Therefore, 18:24 and 15:20 are equivalent.
Use cross-products for a quick check

For ratios a:b and c:d, calculate a × d and b × c. Equal products confirm equivalence without converting to decimals.

Example: Check 7:9 and 21:27: 7 × 27 = 189 and 9 × 21 = 189. Hence, the ratios are equivalent.

Equivalent Ratios Concepts

Generating Equivalent Ratios

An equivalent ratio is obtained by multiplying or dividing both terms by the same non-zero number.

The multiplier must be applied to both terms. Applying it to only one term changes the comparison and does not produce an equivalent ratio. For example, 4:7 multiplied by 5 becomes 20:35.

Example: 2:9 multiplied by 3 gives 6:27. Thus, 2:9 = 6:27.

Testing Whether Two Ratios Are Equivalent

Two ratios are equivalent when their cross-products are equal.

For a:b and c:d, compare a × d with b × c. Alternatively, convert both ratios to their simplest forms. If both tests give the same result, the ratios are equivalent.

Example: For 8:12 and 10:15, 8 × 15 = 120 and 12 × 10 = 120. Therefore, the ratios are equivalent.

Simplest Form of a Ratio

The simplest form is obtained by dividing both terms by their greatest common divisor.

The greatest common divisor must divide both terms exactly. The resulting ratio has no common factor other than 1 and remains equivalent to the original ratio.

Example: For 36:48, the greatest common divisor is 12. Dividing both terms by 12 gives 3:4.

Finding a Missing Term

A missing term in an equivalent-ratio equation is found by cross-multiplication.

If a:b = c:x, then a × x = b × c, so x = bc/a. For a:x = c:d, cross-multiplication gives a × d = c × x, so x = ad/c.

Example: If 5:8 = 20:x, then x = (8 × 20) ÷ 5 = 32. Hence, 5:8 = 20:32.

Equivalent Ratios Data Table

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Examples of Equivalent Ratios

Each row shows an original ratio and an equivalent ratio formed using the stated multiplier.

Original ratio Multiplier Equivalent ratio
2:3 4 8:12
5:7 3 15:21
6:11 2 12:22
9:14 5 45:70

Note: The same multiplier is applied to both terms in every row.

Practice Equivalent Ratios Questions

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10.4 × 0.4 + 0.6 - 0.34 = ?→ 2What is the percentage equivalent to 3/4 ?→ 3A student bought a calculator for ₹400 and sold it for ₹440. What is the profit percentage?→ 4In an election, there were only two candidates. One of the candidates secured 30% of the votes and was defeated by the other candidate by 3,65,104 votes. What is the total number of votes polled?→ 5The LCM of the numbers 2.7 and 0.064 is:→ 6A fashion app lists a smartwatch at a marked price of ₹2,750. During a seasonal sale, the app offers a flat 12% discount. What is the selling price of the smartwatch (in ₹)?→ 7The price of an item was increased by 12%. This reduced the monthly total sales by 16%. The overall effect on the value of monthly sales is __________.→ 8The sum of the first 9 prime numbers is:→ 9A manufacturer sells a product to a wholesaler at a profit of 33%. The wholesaler, in turn, sells it to a retailer at a profit of 28%, and the retailer sells it to a customer at a profit of 25%. If the customer pays ₹532, find the cost price (in ₹) for the manufacturer.→ 10Two numbers are in the ratio of 2 : 5. If the Highest Common Factor of the numbers is 11, then the sum of the numbers will be _____.→

Equivalent Ratios Quick Quiz

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Quick Revision Notes

Equivalent Ratios Revision Points

Use these rules to solve equivalent ratios questions and missing-term problems.

  • Equivalent ratios have the same value and represent the same comparison.
  • Multiply or divide both terms by the same non-zero number to preserve a ratio.
  • For a:b and c:d, equivalence requires a × d = b × c.
  • Reduce ratios by dividing both terms by their greatest common divisor.
  • If a:b = c:x, then x = bc/a.
  • Never multiply or divide only one term when forming an equivalent ratio.

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Equivalent Ratios FAQs

What is the formula for equivalent ratios?

For ratios a:b and c:d, the condition is a/b = c/d or a × d = b × c. An equivalent ratio can also be formed as a:b = ak:bk, where k is non-zero.

Are 12:18 and 20:30 equivalent ratios?

Yes. Both reduce to 2:3. Cross-products also match: 12 × 30 = 360 and 18 × 20 = 360.

How do you find x if 7:9 = 21:x?

Using x = (9 × 21) ÷ 7, x = 27. Therefore, 7:9 = 21:27.

Can both terms of a ratio be divided by different numbers?

No. Both terms must be multiplied or divided by the same non-zero number. For example, 6:10 can become 3:5 by dividing both terms by 2, but dividing them by different numbers changes the ratio.

How can 4:5 be written with the second term as 35?

The multiplier is 35 ÷ 5 = 7. Multiply the first term by 7: 4 × 7 = 28. Therefore, 4:5 = 28:35.

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