Addition of Fractions with Different Denominators

This lesson shows how to add fractions whose denominators are different. The key idea is that you can only add fractions once they share a denominator, so the first job is always to find a common denominator using the LCM.

The method

To add fractions with different denominators:

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

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

The denominators are $3$ and $5$. Both are prime, so the LCM is just their product:

$\operatorname{lcm}(3,5) = 3 \times 5 = 15.$

Rewrite each fraction over $15$:

$\frac{1}{3} + \frac{1}{5} = \frac{1 \times 5}{3 \times 5} + \frac{1 \times 3}{5 \times 3} = \frac{5}{15} + \frac{3}{15} = \frac{5 + 3}{15} = \frac{8}{15}.$

Example: $\tfrac{15}{12} + \tfrac{1}{3}$

To find the LCM of $12$ and $3$, divide by shared factors until you reach $1$:

$\operatorname{lcm}(12,3) = 3 \times 2 \times 2 = 12.$

The first denominator is already $12$, so it stays as it is. Rewrite the second fraction over $12$:

$\frac{15}{12} + \frac{1}{3} = \frac{15}{12} + \frac{1 \times 4}{3 \times 4} = \frac{15}{12} + \frac{4}{12} = \frac{15 + 4}{12} = \frac{19}{12}.$

As a mixed number this is $1\tfrac{7}{12}$.

Example: $1\tfrac{1}{3} + 2\tfrac{1}{6}$

First turn the mixed numbers into improper fractions:

$1\tfrac{1}{3} = \frac{4}{3}, \qquad 2\tfrac{1}{6} = \frac{13}{6}.$

The LCM of $3$ and $6$ is $6$, since $6$ is a multiple of $3$:

$\frac{4}{3} + \frac{13}{6} = \frac{4 \times 2}{6} + \frac{13}{6} = \frac{8}{6} + \frac{13}{6} = \frac{8 + 13}{6} = \frac{21}{6}.$

Example: $\tfrac{2}{5} + \tfrac{3}{4} + \tfrac{1}{6}$

With three fractions, find the LCM of all three denominators $5$, $4$ and $6$ by repeated division:

$\operatorname{lcm}(5,4,6) = 2 \times 2 \times 3 \times 5 = 60.$

Rewrite each fraction over $60$:

$\frac{2}{5} = \frac{24}{60}, \qquad \frac{3}{4} = \frac{45}{60}, \qquad \frac{1}{6} = \frac{10}{60}.$

Now add the numerators:

$\frac{24}{60} + \frac{45}{60} + \frac{10}{60} = \frac{24 + 45 + 10}{60} = \frac{79}{60}.$

Example: $\tfrac{8}{11} + \tfrac{3}{22}$

The denominators are $11$ and $22$, and $22 = 11 \times 2$, so the LCM is $22$:

$\frac{8}{11} + \frac{3}{22} = \frac{8 \times 2}{22} + \frac{3}{22} = \frac{16}{22} + \frac{3}{22} = \frac{16 + 3}{22} = \frac{19}{22}.$

Key takeaways

• To add fractions with different denominators, rewrite them over the LCM, then add the numerators.
• Find the LCM by dividing the denominators by shared prime factors until you reach $1$.
• The same method works for three or more fractions: put them all over one common denominator first.
• Convert mixed numbers to improper fractions before finding the common denominator.