Equivalent fractions are fractions that look different but describe the same amount: 1/2, 2/4 and 4/8 all name one half. To make an equivalent fraction, multiply (or divide) the numerator and denominator by the same number — 1/2 × 2/2 = 2/4. To check whether two fractions are equivalent, cross-multiply: if the products match, they're equal.
1/2 × 3/3 = 3/6. Multiplying by 3/3 is multiplying by 1, so the value can't change — you've cut each half into three smaller pieces and kept all of them. Any whole number works, so every fraction has infinitely many equivalents.
Going the other way is simplifying: 4/8 ÷ 4 = 1/2. Dividing works only when the number divides both exactly.
For 3/4 = ?/20, ask what the denominator was multiplied by: 4 × 5 = 20. Apply the same factor on top: 3 × 5 = 15, so 3/4 = 15/20.
For a/b and c/d, compare a × d with b × c. For 4/6 and 6/9: 4 × 9 = 36 and 6 × 6 = 36 — equal products, so the fractions are equivalent (both simplify to 2/3). Unequal products mean they're not.
Building a family of equivalents
Write three fractions equivalent to 1/2
Answer: 2/4, 3/6 and 4/8
Finding a missing numerator
3/4 = ?/20
Answer: 15/20
Checking with cross-multiplication
Are 4/6 and 6/9 equivalent?
Answer: Yes — both equal 2/3
Adding the same number to top and bottom does not make an equivalent fraction: 1/2 with 1 added to each is 2/3, which is more than a half. Only multiplying or dividing both by the same number preserves the value.
3/4 = ?/20 needs ×5 on both. Multiplying the top by 4 because "the bottom is 4" gives 12/20 = 3/5 — a different fraction. Find the one scale factor and apply it twice.
15/20 has bigger numbers than 3/4 but is exactly the same amount. The size of a fraction depends on the relationship between top and bottom, not on how large the numbers are.
Fractions that name the same amount with different numbers: 1/2, 2/4, 3/6 and 4/8 are all one half. Cut a pizza in half, or into quarters and take two — you get the same amount of pizza.
Multiply the numerator and denominator by the same whole number: 2/3 × 4/4 = 8/12. Or divide both by a common factor to go the other way: 8/12 ÷ 4 = 2/3.
Cross-multiply: for a/b and c/d, compare a × d with b × c. For 4/6 and 6/9, both products are 36, so they're equivalent. You can also simplify both fully — equivalent fractions always share the same simplest form.
Because multiplying by 3/3 (or any n/n) is multiplying by 1, and multiplying by 1 changes nothing. Visually, you're slicing each existing piece into 3 smaller ones and keeping them all — more pieces, same amount.
Infinitely many — one for every whole number you could multiply by. But each family has exactly one member in simplest form (1/2 for the halves family), which is why answers are usually given simplified.
They're the machinery behind almost everything else with fractions: finding common denominators to add and subtract, simplifying answers, comparing fractions and converting to decimals and percentages all rely on swapping a fraction for an equivalent one.
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