Find common denominators

A common denominator is a shared multiple of the denominators that allows fractions to be rewritten as equivalent fractions with equal-sized parts; the numerator and denominator must be multiplied by the same nonzero whole number, preserving the fraction’s value. The least common denominator is typically preferred because it avoids unnecessary enlargement and supports accurate addition and subtraction of fractions, while simply adding denominators is not valid; this scope concerns numerical fractions with positive whole-number denominators, not algebraic rational expressions.

Detailed Explanation: Find common denominators

A common denominator is a denominator that two or more fractions can share. To find one, choose a number that is a multiple of each denominator. The least common denominator (LCD) is usually the best choice.

Example: Rewrite $\frac{2}{3}$ and $\frac{3}{4}$ with a common denominator

List multiples of each denominator.

Multiples of $3$ : $3, 6, 9, 12, \ldots$

Multiples of $4$ : $4, 8, 12, \ldots$

The smallest number in both lists is $12$ , so the LCD is $12$ .
Rewrite $\frac{2}{3}$ with denominator $12$ .

Since $3 \times 4 = 12$ , multiply both the numerator and denominator by $4$ :

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

Rewrite $\frac{3}{4}$ with denominator $12$ .

Since $4 \times 3 = 12$ , multiply both the numerator and denominator by $3$ :

\frac{3}{4}\times\frac{3}{3}=\frac{9}{12}

Now the fractions have the common denominator $12$ :

\frac{2}{3}=\frac{8}{12} \qquad\text{and}\qquad \frac{3}{4}=\frac{9}{12}

Always multiply the numerator and denominator by the same number so the fraction keeps its value. Do not add the denominators; $3+4=7$ is not a valid common denominator.

Learn by doing: Find common denominators

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Fraction Addition - Mixed (No Simplifying Answers) - Two Changed Denominators

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