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How to find a common denominator for fractions and mixed numbers

The rule: use the least common multiple of the denominators

To give fractions a common denominator, find a number divisible by every denominator. The smallest useful choice is the least common denominator, which is the least common multiple (LCM) of the denominators. For each fraction, divide the common denominator by its old denominator. Multiply both the numerator and denominator by that result. Multiplying by the same number on top and bottom does not change the fraction. For mixed numbers, keep the whole-number part and apply the same method to the fractional part.

Problem

Write $\frac{5}{18}$ and $\frac{7}{15}$ as equivalent fractions with the least common denominator.

Answer

\frac{5}{18} = \frac{25}{90}

and

\frac{7}{15} = \frac{42}{90}

Checked by reducing both converted fractions back to their original forms.

Step-by-step solution

1. Find the least common denominator

Factor the two denominators:

18 = 2 \cdot 3^{2}, 15 = 3 \cdot 5

The LCM must contain every required prime factor:

LCM (18, 15) = 2 \cdot 3^{2} \cdot 5 = 90

So $90$ is the least common denominator.

2. Convert the first fraction

Because $90 \div 18 = 5$ , multiply the numerator and denominator by $5$ :

\frac{5}{18} = \frac{5 \cdot 5}{18 \cdot 5} = \frac{25}{90}

3. Convert the second fraction

Because $90 \div 15 = 6$ , multiply the numerator and denominator by $6$ :

\frac{7}{15} = \frac{7 \cdot 6}{15 \cdot 6} = \frac{42}{90}

The two equivalent fractions now have the same denominator.

Mixed-number example

For $3 \frac{2}{5}$ and $3 \frac{4}{7}$ , the least common denominator of $5$ and $7$ is $35$ . Keep each whole-number part and convert only the fractional parts:

3 \frac{2}{5} = 3 \frac{14}{35}, 3 \frac{4}{7} = 3 \frac{20}{35}

How to check

Reduce the answers. Dividing $25/90$ by $5$ gives $5/18$ , and dividing $42/90$ by $6$ gives $7/15$ .

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