How to Add Fractions

Adding fractions is one of those things that seems tricky at first, but once you see the pattern, it clicks pretty quickly. The key is making sure the fractions are ready to be added before you actually add them.

When the Bottoms Are the Same

The bottom number of a fraction is called the denominator. When two fractions already have the same denominator, adding them is easy — just add the top numbers and keep the bottom the same.

Example: \(\frac{2}{7} + \frac{3}{7}\)

Both fractions are in sevenths, so:

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

Think of it like slices of pizza. If a pizza is cut into 7 slices and you take 2 pieces, then someone hands you 3 more pieces, you now have 5 pieces — all out of 7. The denominator (7 slices total) never changed.

One more: \(\frac{4}{9} + \frac{3}{9}\)

$$\frac{4}{9} + \frac{3}{9} = \frac{7}{9}$$

Sometimes the answer can be simplified. If you got \(\frac{6}{9}\), you'd simplify it to \(\frac{2}{3}\) by dividing both numbers by 3.

When the Bottoms Are Different

This is where most students get stuck. You can't just add fractions when the denominators are different — you have to make them match first.

Imagine trying to add 2 quarters and 3 dimes. You can't just say "5 coins" because quarters and dimes aren't the same size. You'd first convert everything to cents: 50 cents + 30 cents = 80 cents.

Fractions work the same way. Before you add, convert both fractions so they have the same denominator.

Quick check — what's wrong with this "solution": \(\frac{1}{2} + \frac{1}{3} = \frac{2}{5}\)? Show answerThe student just added the tops and added the bottoms, which doesn't work. You need a common denominator first. The correct answer is \(\frac{5}{6}\).

Example: \(\frac{1}{2} + \frac{1}{3}\)

The denominators are 2 and 3. The simplest common denominator to use is \(2 \times 3 = 6\).

Now rewrite each fraction as sixths:

$$\frac{1}{2} = \frac{3}{6} \qquad \frac{1}{3} = \frac{2}{6}$$

Now add:

$$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$$

How do you rewrite a fraction with a new denominator?

Ask: what do I multiply the old denominator by to get the new one? Then multiply the top by that same number.

For \(\frac{1}{2}\) → sixths: multiply bottom by 3, so multiply top by 3 too. \(\frac{1 \times 3}{2 \times 3} = \frac{3}{6}\).

For \(\frac{1}{3}\) → sixths: multiply bottom by 2, so multiply top by 2. \(\frac{1 \times 2}{3 \times 2} = \frac{2}{6}\).

Another example: \(\frac{3}{4} + \frac{1}{3}\)

The denominators are 4 and 3. Use \(4 \times 3 = 12\) as the common denominator.

$$\frac{3}{4} = \frac{9}{12} \qquad \frac{1}{3} = \frac{4}{12}$$

$$\frac{9}{12} + \frac{4}{12} = \frac{13}{12}$$

That's an improper fraction (the top is bigger than the bottom). You can leave it that way, or convert it to a mixed number: \(\frac{13}{12} = 1\frac{1}{12}\).

Finding a Smaller Common Denominator

Multiplying the two denominators together always works, but sometimes it gives you bigger numbers than you need. If you can spot a smaller common denominator, use it — you'll get the same answer with easier math.

Example: \(\frac{1}{4} + \frac{1}{6}\)

You could use \(4 \times 6 = 24\), but 12 also works since both 4 and 6 divide evenly into 12.

$$\frac{1}{4} = \frac{3}{12} \qquad \frac{1}{6} = \frac{2}{12}$$

$$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$$

The smallest number that two denominators both divide into is called the least common denominator, or LCD.

A quick trick: if one denominator is a multiple of the other, the bigger one is your LCD. For \(\frac{1}{3} + \frac{1}{6}\), since 6 is already a multiple of 3, just use 6. No multiplication needed.

If multiplying the denominators feels easier, go ahead — just simplify your answer at the end if needed. Either method works.

Adding Mixed Numbers

A mixed number has a whole number part and a fraction part, like \(2\frac{1}{3}\).

The easiest way to add mixed numbers is to handle the whole numbers and fractions separately.

Example: \(1\frac{1}{2} + 2\frac{1}{4}\)

Add the whole numbers: \(1 + 2 = 3\)

Add the fractions: \(\frac{1}{2} + \frac{1}{4}\). Use 4 as the common denominator.

$$\frac{1}{2} = \frac{2}{4} \qquad \text{so} \qquad \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$$

Combine: \(3 + \frac{3}{4} = 3\frac{3}{4}\)

What if the fractions add up to more than 1?

Example: \(1\frac{2}{3} + 2\frac{3}{4}\)

Add the whole numbers: \(1 + 2 = 3\)

Add the fractions: \(\frac{2}{3} + \frac{3}{4}\). Use 12 as the common denominator.

$$\frac{2}{3} = \frac{8}{12} \qquad \frac{3}{4} = \frac{9}{12}$$

$$\frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\frac{5}{12}$$

The fractions added up to more than 1, so carry that extra 1 over to the whole number:

$$3 + 1\frac{5}{12} = 4\frac{5}{12}$$

Try These

1. \(\frac{3}{8} + \frac{1}{8}\) Show answer\(\frac{1}{2}\) — same denominator: \(\frac{4}{8}\), simplified to \(\frac{1}{2}\)

2. \(\frac{1}{2} + \frac{1}{4}\) Show answer\(\frac{3}{4}\) — rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\): \(\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\)

3. \(\frac{2}{3} + \frac{1}{5}\) Show answer\(\frac{13}{15}\) — common denominator 15: \(\frac{10}{15} + \frac{3}{15} = \frac{13}{15}\)

4. \(\frac{5}{6} + \frac{1}{4}\) Show answer\(1\frac{1}{12}\) — common denominator 12: \(\frac{10}{12} + \frac{3}{12} = \frac{13}{12} = 1\frac{1}{12}\)

5. \(2\frac{1}{3} + 1\frac{1}{2}\) Show answer\(3\frac{5}{6}\) — whole numbers: \(3\); fractions: \(\frac{2}{6} + \frac{3}{6} = \frac{5}{6}\)

6. \(3\frac{3}{4} + 2\frac{2}{3}\) Show answer\(6\frac{5}{12}\) — whole numbers: \(5\); fractions: \(\frac{9}{12} + \frac{8}{12} = \frac{17}{12} = 1\frac{5}{12}\); total: \(6\frac{5}{12}\)