Number & Operations—Fractions

Equivalent Fractions

Generating and recognizing equivalent fractions.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Different Fractions, Same Value

1/2 and 2/4 look like different fractions, but they represent the exact same amount — half of a whole, whether it's cut into 2 pieces or 4. Fractions like these, which name the same value in different ways, are called equivalent fractions.

Definition — Equivalent Fractions

Fractions that represent the same value, even though they have different numerators and denominators. Multiplying (or dividing) both the numerator and denominator of a fraction by the same number always produces an equivalent fraction.

Multiplying the top and bottom of a fraction by the same number doesn't change its value, because it's the same as multiplying the fraction by a version of 1, like 2/2 or 3/3 — and multiplying anything by 1 never changes its value.

Worked Example — Generating an Equivalent Fraction

Find a fraction equivalent to 2/3 with a denominator of 12. The denominator needs to be multiplied by 4 to go from 3 to 12, so multiply the numerator by 4 too: 2 × 4 = 8. The equivalent fraction is 8/12.

Worked Example — Checking Whether Two Fractions Are Equivalent

Are 3/5 and 9/15 equivalent? Check whether the same multiplier connects both the numerators and denominators: 3 × 3 = 9, and 5 × 3 = 15. Since the same multiplier, 3, works for both, 3/5 and 9/15 are equivalent.

\dfrac{2}{3} = \dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}

Tip

Equivalent fractions can also be found in reverse, by dividing the numerator and denominator by the same number — this is called simplifying a fraction, and it produces the same equivalence relationship.

Common Mistakes

Multiplying only the numerator or only the denominator by a number, instead of multiplying both by the same amount.
Both the numerator and denominator must be multiplied (or divided) by the exact same number for the result to stay equivalent to the original fraction.
Adding the same number to the numerator and denominator instead of multiplying, assuming it would also create an equivalent fraction.
Only multiplying or dividing both parts by the same number preserves the fraction's value — adding the same number to both parts changes the value.

Key Takeaways

Equivalent fractions represent the same value with different numerators and denominators.
Multiplying (or dividing) both the numerator and denominator by the same number produces an equivalent fraction.
Two fractions are equivalent exactly when the same multiplier connects both their numerators and their denominators.

Summary

Equivalent fractions show that the same value can be written in more than one way. That flexibility is exactly what's needed next, to compare fractions that don't already share a numerator or denominator.

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