Adding Fractions — Like & Unlike Denominators

Adding fractions means combining parts of the same whole. The whole method turns on a single question you ask first: do the two fractions already count equal-size pieces, or not? If the denominators match, you add directly. If they differ, you rewrite first.

When to use this method

Reach for the common-denominator approach whenever you are combining fractions that may be measured in different units. The basic rule is

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. That single check, same unit or not, decides every step that follows.

The steps

Step 1 — Check the denominators. If they already match, the pieces are the same size and you can add right away:

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

Step 2 — Find a common denominator. When the denominators differ, rewrite each fraction so they share a denominator. The least common denominator is often easiest because it keeps the numbers smaller.

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

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

You are not changing the amount. You are changing the unit so both fractions describe equal-size parts.

Step 3 — Add and simplify. Add the numerators, keep the common denominator, and reduce the result if it has a common factor.

A full example from start to finish: $\frac{3}{8} + \frac{1}{6}$

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:

Now add the numerators:

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

Where students get stuck, and how to self-check

The most common stumble is at Step 1, adding both numerators and denominators: writing $\frac{1}{3} + \frac{1}{4} = \frac{2}{7}$. That is not valid because thirds and fourths are different-sized parts. Self-check: confirm the two denominators are equal before writing anything in the answer.

A second stumble is at Step 2, changing the denominator without changing the numerator. If you rewrite $\frac{1}{3}$ in twelfths, it becomes $\frac{4}{12}$, not $\frac{1}{12}$. Self-check: multiply top and bottom by the same number, every time.

A third is at Step 3, forgetting to simplify when the result reduces, as in $\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$. Self-check: scan the final fraction for any shared factor before calling it done.

Adding fractions shows up whenever you combine parts of one whole: recipes, measurement, probability, and rational expressions in algebra. The same common-denominator idea drives fraction subtraction too.

Test the method yourself

Combine $\frac{5}{12} + \frac{1}{8}$. Run the three steps in order: check the denominators, build a common denominator, then add and reduce. If your common denominator is $24$, you are on the right track.