Equivalent Fractions: Definition, Examples and How to Find Them
Part of the Math Concepts guide for Grades 3 to 8.
What Are Equivalent Fractions?
Equivalent fractions are fractions that represent the same value or proportion, even though they have different numerators and denominators. For example, 1/2, 2/4, 3/6, and 4/8 are all equivalent; they all represent exactly half of a whole. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. The value of the fraction does not change because you are multiplying by a form of 1.
Key Formulas
| Method | Rule | Example |
|---|---|---|
| Multiply to find an equivalent fraction | a/b = (a×n)/(b×n) | 1/3 = 2/6 = 3/9 = 4/12 |
| Divide to simplify (find lowest terms) | a/b = (a÷n)/(b÷n) | 8/12 = 4/6 = 2/3 |
| Test if two fractions are equivalent | Cross multiply: a×d = b×c | 1/2 = 3/6 because 1×6 = 2×3 = 6 |
How to Find Equivalent Fractions
Method 1: Multiply to create equivalent fractions
Problem: Find three fractions equivalent to 2/5
- Multiply numerator and denominator by 2: (2×2)/(5×2) = 4/10
- Multiply by 3: (2×3)/(5×3) = 6/15
- Multiply by 10: (2×10)/(5×10) = 20/50
Answer: 2/5 = 4/10 = 6/15 = 20/50
Method 2: Simplify to lowest terms
Problem: Simplify 18/24
- Find the greatest common factor of 18 and 24. GCF = 6.
- Divide both numerator and denominator by 6: (18÷6)/(24÷6) = 3/4
Answer: 18/24 = 3/4 (simplest form). A fraction is in simplest form when the numerator and denominator share no common factors other than 1.
Method 3: Check if two fractions are equivalent
Problem: Are 3/4 and 9/12 equivalent?
- Cross multiply: 3 × 12 = 36 and 4 × 9 = 36
- 36 = 36, so the fractions are equivalent.
Answer: Yes, 3/4 and 9/12 are equivalent.
Why Equivalent Fractions Matter
Students encounter equivalent fractions constantly once they start working with fractions. Here are the three main situations where this skill comes up.
- Adding fractions with different denominators: To add 1/3 + 1/4, you first convert to a common denominator: 4/12 + 3/12 = 7/12.
- Comparing fractions: To compare 3/4 and 2/3, convert to the same denominator: 9/12 vs 8/12. Since 9 > 8, we know 3/4 > 2/3.
- Simplifying answers: A final answer of 6/8 should always be simplified. Divide by GCF = 2: 6/8 = 3/4.
Common Mistakes to Avoid
Mistake 1: Only multiplying the numerator
To create an equivalent fraction, you must multiply both the numerator and the denominator by the same number. Multiplying only the numerator changes the value of the fraction. For example, 1/3 × 2 does not give 2/3; it gives 2/3 only if you also multiply the denominator: (1×2)/(3×2) = 2/6.
Mistake 2: Adding instead of multiplying to find equivalents
Adding the same number to both the numerator and denominator does not produce an equivalent fraction. 1/3 + 2/2 ≠ 3/5. Equivalents only work with multiplication and division, not addition and subtraction.
Mistake 3: Not fully simplifying
When simplifying, divide by the greatest common factor, not just any common factor. If you divide 12/18 by 2, you get 6/9, which can be simplified further by 3 to get 2/3. Dividing by the GCF (6) in one step gives the simplest form directly.
Common Questions About Equivalent Fractions
What are equivalent fractions?
Equivalent fractions are fractions that represent the same value despite having different numbers. For example, 1/2, 2/4, 3/6, and 50/100 are all equivalent; each represents exactly half of a whole.
How do you find equivalent fractions?
Multiply or divide both the numerator and the denominator by the same non-zero number. For example: 2/3 multiplied by 4/4 gives 8/12. Or simplify: 10/15 divided by 5/5 gives 2/3.
How do you check if two fractions are equivalent?
Use cross multiplication. If a/b and c/d are equivalent, then a times d equals b times c. For 3/4 and 9/12: 3 × 12 = 36 and 4 × 9 = 36. They are equal, so the fractions are equivalent.
What does simplest form mean for fractions?
A fraction is in simplest form (also called lowest terms) when the numerator and denominator share no common factors other than 1. To simplify, divide both by their GCF. For 8/12: GCF = 4, so 8/12 = 2/3.
Are equivalent fractions the same as equal fractions?
Yes. Equivalent fractions have the same value, even with different numerators and denominators. 1/2 and 4/8 are equivalent because they both equal exactly 0.5.
What grade do students learn equivalent fractions?
Equivalent fractions are introduced in 3rd grade and developed through 5th grade in the US Common Core standards. In 3rd grade, students identify visual equivalents. By 5th grade, they use them to add, subtract, and compare fractions.
