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.
How to use this 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.
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
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.
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.
Worked examples
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.
Reducing 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
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.
Frequently asked questions
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.