Common Denominators: How to Compare, Add and Subtract Fractions

Why you cannot add fractions by adding the tops and the bottoms, how to rewrite fractions over a common denominator using the LCM, and the cross-multiplication shortcut for comparing.

3 min read Supports 3 lessons

Adding $\frac{1}{5} + \frac{2}{5}$ is easy: one fifth plus two fifths is three fifths. The pieces are the same size, so you just count them.

Adding $\frac{1}{3} + \frac{1}{4}$ is harder. Thirds and quarters are different sizes, so counting pieces makes no sense. The fix is to cut both into pieces of a size they share.

\tfrac{1}{3}

\tfrac{1}{4}

\tfrac{4}{12}

\tfrac{3}{12}

\tfrac{7}{12}

Thirds and quarters both cut evenly into twelfths: $\frac13 + \frac14 = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$ .

Finding a common denominator

A common denominator must be a multiple of every denominator involved. Multiplying the denominators always works, but the least common multiple gives the smallest numbers and least simplifying afterwards.

$\frac{5}{6} - \frac{1}{4}$
Sixths and quarters: different sized pieces.

1/5

Adding fractions

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

This always works. Using the LCM instead of $bd$ keeps the numbers smaller.

Watch out

Never add the denominators. $\frac{1}{2} + \frac{1}{3}$ is not $\frac{2}{5}$ . Two fifths is less than a half, and you started with more than a half.

Comparing fractions

Which is bigger, $\frac{2}{3}$ or $\frac{3}{5}$ ? Over a common denominator of $15$ the answer is clear:

\frac{2}{3} = \frac{10}{15} \qquad \frac{3}{5} = \frac{9}{15} \qquad\Rightarrow\qquad \frac{2}{3} > \frac{3}{5}

Tip

Cross-multiplication shortcut. To compare $\frac{a}{b}$ and $\frac{c}{d}$ , compare $a \times d$ with $c \times b$ . Here $2 \times 5 = 10$ beats $3 \times 3 = 9$ , so $\frac{2}{3}$ is bigger. These are exactly the numerators you would get over the denominator $bd$ .

Mixed numbers

To add or subtract mixed numbers, either handle the wholes and the fractions separately, or convert everything to improper fractions first. The second way is slower but never goes wrong when you need to borrow.

Worked example

Compute $2\frac{1}{4} - 1\frac{2}{3}$ .

1
Convert to improper fractions.
$\frac{9}{4} - \frac{5}{3}$

The same idea in algebra

Algebraic fractions need a common denominator too, and the reasoning is identical:

\frac{1}{x} - \frac{1}{x + h} = \frac{(x + h) - x}{x(x + h)} = \frac{h}{x(x + h)}

This exact calculation appears when you first meet derivatives.

Common questions

Do I have to use the lowest common denominator?: No. Any common denominator gives the right answer. The lowest one just keeps the numbers small, so there is less to simplify at the end.
Do I need a common denominator to multiply fractions?: No. Multiplying is simpler: multiply the numerators together and the denominators together. Common denominators are only needed to add, subtract or compare.

Practise common denominators

The Adding & Subtracting Fractions lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Do I have to use the lowest common denominator?

No. Any common denominator gives the right answer. The lowest one just keeps the numbers small, so there is less to simplify at the end.

Do I need a common denominator to multiply fractions?

No. Multiplying is simpler: multiply the numerators together and the denominators together. Common denominators are only needed to add, subtract or compare.