Addition of Fractions

A fraction counts equal parts of a whole: $\frac{3}{8}$ means 3 parts, each one eighth of the whole. Adding fractions is easy when the parts are the same size – you just count them. When the parts are different sizes, you first cut them into equal pieces. This page shows both cases, then mixed numbers, negative fractions and fractions with letters.

Fractions with the same denominator

Add the numerators and keep the denominator. $$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$$

Two sevenths plus three sevenths is five sevenths, just as 2 apples plus 3 apples is 5 apples. The denominator only names the size of the parts, so it doesn't change.

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

If the answer can be simplified, simplify it:

$$\frac{5}{12}+\frac{1}{12}=\frac{6}{12}=\frac{1}{2}$$

Never add the denominators. $\frac{1}{2}+\frac{1}{2}$ is one whole, not $\frac{2}{4}$. Half a pizza plus half a pizza is a whole pizza, while $\frac{2}{4}$ is still only half.

Fractions with different denominators

You can't count thirds and quarters together, because they are different sizes. First rewrite both fractions so that they have the same denominator. This doesn't change their value, because multiplying the top and the bottom of a fraction by the same number gives an equal fraction.

Find a common denominator: a number that both denominators divide into. The smallest one, the least common multiple of the denominators, keeps the numbers small.
Rewrite each fraction with that denominator.
Add the numerators and keep the common denominator.
Simplify the result, and turn an improper fraction into a mixed number if you need one.

Example 1: $\frac{1}{4}+\frac{1}{6}$

Multiples of 4: 4, 8, 12, 16, … Multiples of 6: 6, 12, 18, … The least common denominator is 12. Since $12 = 4\cdot 3 = 6\cdot 2$:

$$\begin{aligned}\frac{1}{4}+\frac{1}{6}&=\frac{1\cdot 3}{4\cdot 3}+\frac{1\cdot 2}{6\cdot 2}\\&=\frac{3}{12}+\frac{2}{12}=\frac{5}{12}\end{aligned}$$

Example 2: $\frac{2}{3}+\frac{3}{4}$

3 and 4 have no common factor, so the least common denominator is their product, $3\cdot 4=12$.

$$\frac{2}{3}+\frac{3}{4}=\frac{8}{12}+\frac{9}{12}=\frac{17}{12}=1\frac{5}{12}$$

Shortcut for two fractions

The product of the denominators is always a common denominator:

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$

It always works, but when the denominators share a factor the answer needs simplifying at the end. With Example 1:

$$\frac{1}{4}+\frac{1}{6}=\frac{1\cdot 6+1\cdot 4}{4\cdot 6}=\frac{10}{24}=\frac{5}{12}$$

Three or more fractions

Use one common denominator for all of them. For $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$ it is 6:

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

Whole numbers and mixed numbers

A whole number plus a fraction

Write the whole number as a fraction with the same denominator: $3=\frac{15}{5}$.

$$3+\frac{2}{5}=\frac{15}{5}+\frac{2}{5}=\frac{17}{5}=3\frac{2}{5}$$

Adding mixed numbers

Method 1: add the whole parts and the fractions separately.

$$\begin{aligned}2\frac{1}{3}+1\frac{3}{4}&=(2+1)+\left(\frac{4}{12}+\frac{9}{12}\right)\\&=3+\frac{13}{12}\end{aligned}$$

$\frac{13}{12}$ is more than 1: $\frac{13}{12}=1\frac{1}{12}$, so the sum is $3+1\frac{1}{12}=4\frac{1}{12}$.

Method 2: change to improper fractions first.

$$\begin{aligned}2\frac{1}{3}+1\frac{3}{4}&=\frac{7}{3}+\frac{7}{4}=\frac{28}{12}+\frac{21}{12}\\&=\frac{49}{12}=4\frac{1}{12}\end{aligned}$$

Method 1 keeps the numbers smaller. Method 2 is easier to use when some of the numbers are negative.

Negative fractions

A minus sign can stand in front of the fraction, in the numerator, or in the denominator – all three mean the same number: $-\frac{3}{4}=\frac{-3}{4}=\frac{3}{-4}$. Move the sign into the numerator, then add the numerators as signed numbers:

$$\begin{aligned}-\frac{3}{4}+\frac{1}{6}&=\frac{-9}{12}+\frac{2}{12}\\&=\frac{-9+2}{12}=-\frac{7}{12}\end{aligned}$$

Adding a negative fraction is the same as subtracting a positive one, see subtraction of fractions.

Algebraic fractions

Fractions with letters are added by exactly the same rules. The common denominator is built from the factors of the denominators.

Example 3: $\frac{a}{b}+\frac{c}{d}$

$$\frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{bc}{bd}=\frac{ad+bc}{bd}$$

Example 4: $a+\frac{1}{b}$

Write $a$ as $\frac{ab}{b}$:

$$a+\frac{1}{b}=\frac{ab}{b}+\frac{1}{b}=\frac{ab+1}{b}$$

Example 5: $\frac{a}{a+b}+\frac{b}{a-b}$

The denominators have no common factor, so the common denominator is their product $(a+b)(a-b)=a^2-b^2$:

$$\begin{aligned}&\frac{a}{a+b}+\frac{b}{a-b}\\&=\frac{a(a-b)+b(a+b)}{(a+b)(a-b)}\\&=\frac{a^2-ab+ab+b^2}{a^2-b^2}=\frac{a^2+b^2}{a^2-b^2}\end{aligned}$$

Example 6: $\frac{1}{x-1}+\frac{2}{x^2-1}$

Factor first: $x^2-1=(x-1)(x+1)$. It already contains $x-1$, so it is the least common denominator. Multiply the first fraction by $\frac{x+1}{x+1}$:

$$\begin{aligned}&\frac{1}{x-1}+\frac{2}{x^2-1}\\&=\frac{x+1}{x^2-1}+\frac{2}{x^2-1}=\frac{x+3}{x^2-1}\end{aligned}$$

This holds for every $x$ except $x=1$ and $x=-1$, where a denominator is zero. More on this in rational expressions.

Common mistakes

Adding the denominators: $\frac{1}{3}+\frac{1}{4}\ne\frac{2}{7}$. The correct sum is $\frac{7}{12}$.
Changing only the denominator: $\frac{1}{4}$ is $\frac{3}{12}$, not $\frac{1}{12}$. Multiply the numerator by the same number.
Forgetting to simplify: $\frac{10}{24}$ is correct but not finished, the answer is $\frac{5}{12}$.
Cancelling terms in a sum: in $\frac{x+3}{x^2-1}$ nothing cancels. You may cancel only a factor of the whole numerator and the whole denominator.

Practice

Try each problem, then open it to check your answer.

1. $\frac{3}{8}+\frac{1}{8}$

$\frac{3+1}{8}=\frac{4}{8}=\frac{1}{2}$

2. $\frac{1}{3}+\frac{1}{5}$

$\frac{5}{15}+\frac{3}{15}=\frac{8}{15}$

3. $\frac{5}{6}+\frac{3}{4}$

The least common denominator is 12: $\frac{10}{12}+\frac{9}{12}=\frac{19}{12}=1\frac{7}{12}$

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

$3+\frac{3}{6}+\frac{4}{6}=3+\frac{7}{6}$, and $\frac{7}{6}=1\frac{1}{6}$, so the sum is $4\frac{1}{6}$.

5. $-\frac{2}{5}+\frac{1}{2}$

$\frac{-4}{10}+\frac{5}{10}=\frac{1}{10}$

6. $\frac{2}{x}+\frac{3}{y}$

$\frac{2y}{xy}+\frac{3x}{xy}=\frac{2y+3x}{xy}$

7. $\frac{1}{x+1}+\frac{1}{x-1}$

$\frac{(x-1)+(x+1)}{(x+1)(x-1)}=\frac{2x}{x^2-1}$

You can check any sum of numeric fractions, step by step, with the fraction calculator below.