Subtraction of Fractions with Different Denominators

This lesson shows how to subtract fractions whose denominators are different. The idea is the same as for addition: find the LCM of the denominators, rewrite each fraction over that common denominator, and then subtract.

The method

To subtract fractions with different denominators:

1. Find the LCM of the denominators.
2. Rewrite each fraction so its denominator is the LCM.
3. Subtract the numerators and keep the common denominator.

Example: $\tfrac{3}{5} - \tfrac{1}{2}$

The denominators are $5$ and $2$. Both are prime, so the LCM is just their product:

$\operatorname{lcm}(5,2) = 5 \times 2 = 10.$

Rewrite each fraction over $10$:

$\frac{3}{5} - \frac{1}{2} = \frac{3 \times 2}{5 \times 2} - \frac{1 \times 5}{2 \times 5} = \frac{6}{10} - \frac{5}{10} = \frac{6 - 5}{10} = \frac{1}{10}.$

Example: $\tfrac{12}{13} - \tfrac{1}{6}$

Here $13$ is prime and shares no factor with $6$, so the LCM is again the product:

$\operatorname{lcm}(13,6) = 13 \times 6 = 78.$

When the two denominators have no common factor, you can cross multiply directly:

$\frac{12}{13} - \frac{1}{6} = \frac{12 \times 6}{13 \times 6} - \frac{1 \times 13}{6 \times 13} = \frac{72}{78} - \frac{13}{78} = \frac{72 - 13}{78} = \frac{59}{78}.$

Example: $1\tfrac{1}{3} - 2\tfrac{1}{2}$

First turn the mixed numbers into improper fractions:

$1\tfrac{1}{3} = \frac{4}{3}, \qquad 2\tfrac{1}{2} = \frac{5}{2}.$

The LCM of $3$ and $2$ is $6$:

$\frac{4}{3} - \frac{5}{2} = \frac{4 \times 2}{6} - \frac{5 \times 3}{6} = \frac{8}{6} - \frac{15}{6} = \frac{8 - 15}{6} = \frac{-7}{6}.$

Subtracting a larger number. When the second numerator is bigger, the answer is negative. You can think of the subtraction as adding the opposite: $8 - 15 = 8 + (-15) = -7$, so the result is $-\tfrac{7}{6}$.

Example: combining several fractions

Consider $\dfrac{8}{5} + \dfrac{1}{5} - \dfrac{1}{6} - \dfrac{7}{6}$. First combine the terms that already share a denominator:

$\frac{8}{5} + \frac{1}{5} = \frac{9}{5}, \qquad \frac{1}{6} + \frac{7}{6} = \frac{8}{6},$

so the expression becomes $\dfrac{9}{5} - \dfrac{8}{6}$. The LCM of $5$ and $6$ is $30$:

$\frac{9}{5} - \frac{8}{6} = \frac{9 \times 6}{30} - \frac{8 \times 5}{30} = \frac{54}{30} - \frac{40}{30} = \frac{14}{30} = \frac{7}{15}.$

Dividing numerator and denominator by $2$ gives the simplified answer $\tfrac{7}{15}$.

Example: $1\tfrac{1}{3} + 2\tfrac{1}{5} - 1\tfrac{1}{2}$

Convert the mixed numbers to improper fractions:

$1\tfrac{1}{3} = \frac{4}{3}, \qquad 2\tfrac{1}{5} = \frac{11}{5}, \qquad 1\tfrac{1}{2} = \frac{3}{2}.$

The denominators $3$, $5$ and $2$ are all prime, so the LCM is $3 \times 5 \times 2 = 30$:

$\frac{4}{3} + \frac{11}{5} - \frac{3}{2} = \frac{40}{30} + \frac{66}{30} - \frac{45}{30} = \frac{40 + 66 - 45}{30} = \frac{61}{30}.$

Key takeaways

• To subtract fractions with different denominators, rewrite them over the LCM, then subtract the numerators.
• If the denominators share no common factor, you can cross multiply to combine them in one step.
• A subtraction can be read as adding the opposite, which makes a negative result easy to handle.
• Convert mixed numbers to improper fractions before finding the common denominator, and simplify the final fraction where possible.