Adding Fractions

What You’ll Learn

In this lesson you’ll learn how to add fractions, both when the denominators match and when they don’t. If you need a refresher on what fractions are, check out the Introduction to Fractions lesson first.

The Concept

Adding fractions with the same denominator is straightforward. Just add the numerators:

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

For example: $\frac{2}{5} + \frac{1}{5} = \frac{3}{5}$

Adding fractions with different denominators requires a common denominator. The least common denominator (LCD) is the smallest number that both denominators divide into evenly.

The general formula is:

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

After adding, always check whether the result can be simplified by dividing the numerator and denominator by their greatest common factor.

Worked Examples

Let’s add $\frac{1}{2} + \frac{1}{3}$ step by step.

1. Find the LCD. The denominators are 2 and 3. The smallest number both divide into is 6.

2. Convert each fraction.

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

3. Add the numerators.

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

4. Simplify if possible. 5 and 6 share no common factors, so $\frac{5}{6}$ is already in simplest form.

Real-World Applications

Fractions show up constantly in everyday life. Imagine you’re following a recipe that calls for $\frac{1}{2}$ cup of flour and you want to add an extra $\frac{1}{3}$ cup for a thicker batter. Knowing that $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ means you can measure out exactly the right amount instead of guessing.

Fractions are also essential in construction (measuring lumber), music (time signatures), and finance (splitting costs).

You’ve Got This

If fractions feel tricky at first, that’s completely normal. The key insight is that you can only add fractions when they speak the same “language,” that is, when they have the same denominator. Once you find that common ground, the rest is just addition. Take your time and practice with small numbers until it clicks.

Quiz

What is $\frac{1}{2} + \frac{1}{3}$?

• A.$\frac{2}{5}$
• B.$\frac{5}{6}$
• C.$\frac{1}{6}$
• D.$\frac{2}{6}$

What is $\frac{2}{5} + \frac{1}{5}$?

• A.$\frac{3}{10}$
• B.$\frac{3}{5}$
• C.$\frac{1}{5}$
• D.$\frac{2}{10}$

What is the least common denominator of 4 and 6?

• A.24
• B.10
• C.12
• D.6

What is $\frac{3}{4} + \frac{1}{6}$?

• A.$\frac{4}{10}$
• B.$\frac{11}{12}$
• C.$\frac{5}{6}$
• D.$\frac{7}{12}$

What is $\frac{5}{8} - \frac{1}{4}$?

• A.$\frac{3}{8}$
• B.$\frac{4}{8}$
• C.$\frac{1}{2}$
• D.$\frac{1}{8}$