What are equivalent fractions?
Equivalent fractions are fractions that name the same value, like 12, 24 and 36. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.
Equivalent Fractions in a picture
Formal definition and notation
Two fractions ab and cd (with b, d ≠ 0) are equivalent when they represent the same number. This happens exactly when their cross products are equal: a × d = b × c. The Equivalent Fractions Property says ab = n × an × b for any n ≠ 0. Every fraction has infinitely many equivalent fractions and exactly one in simplest form.
| Symbol | Read it as | Meaning |
|---|---|---|
| 12 = 24 | one half equals two fourths | an equation between equivalent fractions |
| nn = 1 | n over n equals one | why multiplying top and bottom by n keeps the value |
Key facts
- Multiplying makes an equivalent fraction with smaller pieces (more of them); dividing makes one with bigger pieces (fewer of them).
- Dividing by the greatest common factor gives the simplest form in one step: 1218 = 23.
- Adding the same number to the top and bottom does not give an equivalent fraction.
- Equivalent fractions let you add, subtract and compare fractions with different denominators.
- Whole numbers have equivalent fractions too: 3 = 31 = 62 = 124.
Equivalent Fractions Property
- a, b
- the numerator and denominator of the fraction, b ≠ 0
- n
- any nonzero number; for division, a common factor of a and b
To test two fractions, compare cross products: 68 and 1520 give 6 × 20 = 120 and 8 × 15 = 120, so they are equivalent.
Try it: find equivalent fractions
Change the numerator and denominator. The bars below cut the same amount into more pieces, so each one shows an equivalent fraction.
Equivalent Fractions examples, step by step
Example 1: Build equivalent fractions
Write two fractions equivalent to 35.
- Multiply top and bottom by 2: 3 × 25 × 2 = 610.
- Multiply top and bottom by 3: 3 × 35 × 3 = 915.
- Each step multiplies by 22 or 33, which equal 1.
Answer: 610 and 915 are both equivalent to 35.
Example 2: Find the missing number
Fill in the blank: 47 = ?35.
- Ask what turns 7 into 35: 35 ÷ 7 = 5.
- Do the same to the numerator: multiply 4 by 5.
- 4 × 5 = 20, so 47 = 2035.
Answer: The missing numerator is 20.
Example 3: Test with cross products
Are 68 and 1520 equivalent?
- Cross multiply: 6 × 20 = 120 and 8 × 15 = 120.
- The cross products are equal, so the fractions are equivalent.
- Check by simplifying: 68 = 34 and 1520 = 34.
Answer: Yes. Both equal 34.
Common mistakes
Multiply or divide the numerator and denominator by the same number. Adding changes the value: 23 ≈ 0.67 but 34 = 0.75.
Changing only one part multiplies the fraction by 2, not by 1. 43 is more than a whole.
The size of the numbers is not the size of the fraction. Compare by renaming with a common denominator or by cross products.
Not to be confused with
Simplest form is the one equivalent fraction whose numerator and denominator share no factor except 1, such as 23. 46 and 812 are equivalent to it but not simplified.
Equivalent ratios, like 2:3 and 4:6, follow the same multiply-both-parts rule, but a ratio compares two quantities and is not always a part of a whole.
Quick check: 3 questions
Which fraction is equivalent to 23?
Answer: C. 2 × 23 × 2 = 46. Choices A and B add the same number to the top and bottom, and choice D changes only the denominator.Show answer and explanation
What is 1218 in simplest form?
Answer: A. Divide by the greatest common factor, 6: 12 ÷ 618 ÷ 6 = 23. 69 and 46 are equivalent but can still be simplified, and 34 is a different value.Show answer and explanation
56 = 25?
Answer: D. 5 was multiplied by 5 to get 25, so multiply 6 by 5: 30. Choices A and B add instead of multiply, and 36 multiplies 6 by 6.Show answer and explanation
Equivalent Fractions: frequently asked questions
How do you find equivalent fractions?
Multiply the numerator and the denominator by the same whole number: 34 = 68 = 912. Or divide both by a common factor: 1015 = 23. Each step multiplies or divides by a form of 1, such as 22, so the value stays the same. You can make as many as you like.
How do you check if two fractions are equivalent?
Simplify both and see if they match, or compare cross products. For 410 and 615: 4 × 15 = 60 and 10 × 6 = 60, so they are equivalent (both are 25). You can also divide: equivalent fractions give the same decimal, here 0.4.
Why do we need equivalent fractions?
To add, subtract or compare fractions, the pieces must be the same size, so you rename the fractions with a common denominator. For example, 13 + 14 = 412 + 312 = 712. Equivalent fractions also let you write answers in simplest form and link fractions to decimals and percents, such as 34 = 75100 = 75%.
Practice and tools
Sources
- Common Core State Standards, Grade 3 Number and Operations, Fractions (3.NF.A.3.b)
- Common Core State Standards, Grade 4 Number and Operations, Fractions (4.NF.A.1)
- OpenStax, Prealgebra 2e, 4.1 Visualize Fractions
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 3.NF.A.3.b, CCSS 4.NF.A.1
Cite this entry
Effortless Math Team. “Equivalent Fractions.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/equivalent-fractions/
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