Quick answer
Equivalent fractions represent the same value with different numerators and denominators. To find one: multiply both numerator and denominator by the same integer k. To verify: cross-multiply - if a/b = c/d, then a × d = b × c.
- Multiply, never add. Only multiplying (or dividing) both parts by the same k preserves the proportion. Adding the same number to both changes the value: 3/4 + 1 = 4/5, not 3/4.
- Cross-multiplication verifies equivalence without reducing either fraction first. If a × d = b × c, the fractions are equal. If the products differ, they are not.
- All equivalents share one simplest form. 6/8, 9/12, and 12/16 all reduce to 3/4 - meaning they are all equivalent to each other.
- LCD work is equivalent fraction work. Converting 3/4 to 9/12 to add it to 5/6 is just finding an equivalent fraction with a useful denominator.
- k can go in either direction. Multiplying by k scales up; dividing by the GCD scales down to lowest terms. Both produce valid equivalents.
How to use our equivalent fractions calculator
Enter any fraction. The calculator simplifies it to lowest terms (showing the GCD), then generates a table of 12 equivalent fractions by multiplying the numerator and denominator by 1 through 12 - both from your original input and from the simplified form.
What are equivalent fractions?
Two fractions are equivalent when they represent the same point on the number line - the same proportion of a whole. The fundamental property is:
$$\frac{a}{b} = \frac{a \times k}{b \times k} \quad \text{for any integer } k \neq 0$$
Multiplying (or dividing) both numerator and denominator by the same non-zero integer k does not change the value of the fraction. It changes only how the fraction is written.
$$\frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{12}{16} = \frac{15}{20} \ldots$$
All of these fractions equal 0.75. In lowest terms, they all reduce to 3/4.
Equivalent fractions explained to a beginner
Think of a fraction as a slice count. If you cut a cake into 4 equal pieces and take 3, you have 3/4 of the cake. Now take the same cake and cut it into 8 pieces instead - you would need to take 6 pieces to get the exact same amount. That is 6/8. The two fractions look different but describe the same portion of the whole.
All you did was cut each of the original 4 slices in two - doubling both the total count and your share. Formally, you multiplied by 2/2. Because the numerator and denominator scale by the same factor, the ratio stays locked. This is the entire logic behind equivalent fractions: same proportion, different notation.
How to find equivalent fractions
Choose any non-zero integer k and multiply both parts of the fraction by k. To generate 5 equivalent fractions for 2/5:
| k | Numerator (2 × k) | Denominator (5 × k) | Fraction |
|---|---|---|---|
| 2 | 4 | 10 | 4/10 |
| 3 | 6 | 15 | 6/15 |
| 4 | 8 | 20 | 8/20 |
| 5 | 10 | 25 | 10/25 |
| 10 | 20 | 50 | 20/50 |
All of these are equivalent to 2/5 = 0.4.
How to verify equivalence on a fraction
Use cross-multiplication to test whether two fractions are equivalent:
$$\frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c$$
To check 3/4 = 9/12: 3 × 12 = 36 and 4 × 9 = 36. Equal products confirm they are equivalent. To check 2/3 vs 3/4: 2 × 4 = 8 and 3 × 3 = 9. Different products - not equivalent.
The cross-multiplication check is the one I reach for whenever I need a fast yes-or-no answer. You do not need to reduce either fraction to lowest terms first - just multiply diagonally and compare.
It works even when the two fractions look nothing alike, such as 5/8 and 15/24, where reducing first takes more steps than a quick 5 × 24 = 120 vs 8 × 15 = 120.
Using equivalent fractions to add and subtract
Equivalent fractions are essential when adding or subtracting fractions with unlike denominators. The goal is to express both fractions with the same denominator (LCD) so numerators can be compared or combined:
$$\frac{3}{4} \text{ and } \frac{5}{6}: \quad \text{LCD}(4,6)=12 \implies \frac{9}{12} = \frac{10}{12}$$
3/4 and 5/6 share no common denominator. Find the LCD (12), then convert: 3/4 → 9/12 and 5/6 → 10/12. Now they can be added: 9/12 + 10/12 = 19/12.
Also see our dedicated add fractions calculator and subtract fractions calculator.
Worked examples for equivalent fractions
Example 1: find three fractions equivalent to 1/3
Multiply by 2, 3, 4: 2/6, 3/9, 4/12. Cross-check: 1 × 6 = 6 = 3 × 2 ✓, 1 × 9 = 9 = 3 × 3 ✓, 1 × 12 = 12 = 3 × 4 ✓.
Example 2: are 5/8 and 15/24 equivalent?
Cross-multiply: 5 × 24 = 120 and 8 × 15 = 120. Equal products - yes, they are equivalent. (15/24 = 5/8 with k = 3.)
Example 3: find the fraction equivalent to 3/7 with denominator 35
35 ÷ 7 = 5, so k = 5. Multiply numerator by 5: 3 × 5 = 15. The equivalent fraction is 15/35.
How to reduce fraction to simplest form
Dividing both numerator and denominator by their GCD is the reverse of generating equivalents. It produces the simplest (lowest-terms) fraction:
$$\frac{12}{16} \div \gcd(12,16) = \frac{12 \div 4}{16 \div 4} = \frac{3}{4}$$
12/16 - GCD(12,16) = 4. Divide both: 12 ÷ 4 = 3, 16 ÷ 4 = 4. Simplest form: 3/4.
Common mistakes when calculating equivalent fractions
Adding instead of multiplying
3/4 → add 1 to both → 4/5. This is NOT equivalent to 3/4 (0.75 ≠ 0.8). Only multiplying (or dividing) both parts by the same factor preserves the value.
Multiplying by zero
k must be non-zero. Multiplying by 0 gives 0/0, which is undefined.
Confusing "equivalent" with "equal"
3/4 and 9/12 are the same number written differently. All equivalent fractions reduce to the same simplest form and have the same decimal value.
The "add instead of multiply" error - turning 3/4 into 4/5 - is more persistent than it looks. What makes it easy to catch in practice: convert both to decimals.
3/4 = 0.75, but 4/5 = 0.80. The new fraction is actually larger, which is an immediate signal that the value changed. Comparing decimal expansions is a fast sanity check whenever an equivalent fraction result looks suspect.
FAQs about equivalent fractions
What are equivalent fractions?
Fractions that represent the same value. Produced by multiplying or dividing both numerator and denominator by the same non-zero integer.
How do you find equivalent fractions?
Multiply both numerator and denominator by any integer k ≠ 0. The result is an equivalent fraction.
How do you check if two fractions are equivalent?
Cross-multiply. If a/b = c/d, then a × d = b × c. Equal products confirm equivalence.
Are 2/3 and 4/6 equivalent?
Yes. 2 × 6 = 12 and 3 × 4 = 12. They are equivalent - 4/6 is 2/3 multiplied by 2/2.
Quiz: how well do you know equivalent fractions?
1. What is the fundamental property that makes two fractions equivalent?
2. Worked example 3 on the page: find the fraction equivalent to 3/7 with denominator 35.
3. How does worked example 2 on the page verify that 5/8 and 15/24 are equivalent?
4. The page adds 3/4 and 5/6 by converting to equivalent fractions. What is the LCD and what are the converted fractions?
5. The page warns that converting 3/4 to 4/5 by adding 1 to both parts is a common mistake. Why is 4/5 not equivalent to 3/4?