Adding Fractions — Like & Unlike Denominators

Adding fractions means combining parts of the same whole. If the denominators already match, add the numerators and keep the denominator. If the denominators are different, rewrite the fractions with a common denominator first.

The basic rule is

$$\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}$$

but it works only when both fractions are counting equal-size parts. You can add $\frac{2}{7}$ and $\frac{3}{7}$ right away because both are sevenths. You cannot add $\frac{1}{3}$ and $\frac{1}{4}$ until you rewrite them in the same unit.

How to add fractions with like denominators

Fractions with like denominators are already measured in the same unit, so the addition is direct.

For example,

$$\frac{2}{7} + \frac{3}{7} = \frac{5}{7}.$$

The denominator stays $7$ because the size of each piece has not changed. You are only counting how many sevenths there are in total.

How to add fractions with unlike denominators

When the denominators are different, first rewrite the fractions so they use the same denominator. The least common denominator is often the easiest choice because it keeps the numbers smaller.

For $\frac{1}{3} + \frac{1}{4}$, a common denominator is $12$:

$$\frac{1}{3} = \frac{4}{12}, \qquad \frac{1}{4} = \frac{3}{12}.$$

Now both fractions are written in twelfths, so the addition is valid:

$$\frac{4}{12} + \frac{3}{12} = \frac{7}{12}.$$

This is the key idea: you are not changing the amount. You are changing the unit so both fractions describe equal-size parts.

Worked Example: $\frac{3}{8} + \frac{1}{6}$

The denominators are different, so do not add $3+1$ and $8+6$. First find a common denominator.

The least common multiple of $8$ and $6$ is $24$, so rewrite both fractions in twenty-fourths:

$$\frac{3}{8} = \frac{9}{24}, \qquad \frac{1}{6} = \frac{4}{24}.$$

Now add the numerators:

$$\frac{9}{24} + \frac{4}{24} = \frac{13}{24}.$$

Since $13$ and $24$ have no common factor greater than $1$, $\frac{13}{24}$ is already simplified. So

$$\frac{3}{8} + \frac{1}{6} = \frac{13}{24}.$$

Common mistakes when adding fractions

One common mistake is adding both the numerators and denominators, as in

$$\frac{1}{3} + \frac{1}{4} = \frac{2}{7}.$$

That is not valid because thirds and fourths are different-sized parts.

Another mistake is changing the denominator without changing the numerator to keep the fraction equivalent. If you rewrite $\frac{1}{3}$ in twelfths, it becomes $\frac{4}{12}$, not $\frac{1}{12}$.

A third mistake is forgetting to simplify when the result can be reduced. For example,

$$\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}.$$

Where adding fractions is used

Adding fractions shows up whenever you combine parts of one whole. Common examples include recipes, measurement, probability, and algebra problems with rational expressions.

The same common-denominator idea also drives fraction subtraction. Once that idea is clear, both operations become much easier to check.

Try a similar problem

Try $\frac{5}{12} + \frac{1}{8}$ on your own. Find a common denominator, rewrite both fractions, and simplify the result if possible.