Number & Operations—Fractions

Adding & Subtracting Unlike Denominators

Using equivalent fractions to add and subtract fractions with unlike denominators.

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

Why You Can't Just Add the Numbers You See

What is 1/2 + 1/3? Before reading on, try it — and if you got 2/5 by adding the numerators and denominators separately, draw a picture of half a pizza and a third of a different-size pizza combined. Does 2/5 of a pizza look right?

It doesn't, and here's why: halves and thirds are different-size pieces, cut from wholes divided in different ways. Adding 1 half-size piece and 1 third-size piece the same way you'd add two counts of the same object doesn't make sense — the pieces themselves aren't the same size yet.

The fix, using what you already know about equivalent fractions from Grade 4, is to rewrite both fractions with the same denominator first — so both are now built from equal-size pieces — and then the numerators really can be added directly.

Worked Example — Adding Fractions with a Common Denominator

Add 1/2 + 1/3. Rewrite both with a common denominator of 6: 1/2 = 3/6, and 1/3 = 2/6. Now both fractions are built from sixths, so the numerators can be added: 3/6 + 2/6 = 5/6.

Worked Example — Subtracting Fractions with a Common Denominator

Subtract 5/6 − 1/4. Rewrite both with a common denominator of 12: 5/6 = 10/12, and 1/4 = 3/12. Subtract the numerators: 10/12 − 3/12 = 7/12.

\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}

Tip

Multiplying the two denominators together always produces a common denominator that works, even if it isn't the smallest possible one — for 1/2 and 1/3, 2 × 3 = 6 happens to be the smallest, but this trick works for any pair of fractions.

Common Mistakes

Adding or subtracting the numerators and denominators straight across, without first finding a common denominator.
Fractions can only be added or subtracted once they share a common denominator — rewrite both fractions as equivalent fractions with the same denominator first.
Finding a common denominator but forgetting to also convert the numerators to match the new denominator.
Whatever the denominator is multiplied by to reach the common denominator, the numerator must be multiplied by that same number — both parts of the fraction scale together.

Key Takeaways

Fractions with different denominators represent different-size pieces and can't be added or subtracted directly.
Rewriting both fractions with a common denominator makes the pieces the same size, so the numerators can be combined.
Multiplying the two original denominators together always produces a valid common denominator.

Summary

Finding a common denominator makes adding and subtracting any two fractions possible, no matter how different they start out. The next lesson turns to a different fraction operation entirely — multiplication — and what it really means to multiply by a fraction.

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