Adding and Subtracting Fractions

To add or subtract fractions, first rewrite them with a common denominator (LCD), then add or subtract the numerators.

Same Denominator

When denominators are equal, simply add (or subtract) the numerators:

$\frac{3}{8} + \frac{2}{8} = \frac{3+2}{8} = \frac{5}{8}$

Different Denominators

Find the Lowest Common Denominator (LCD) first.

1
Find the LCD
Find the smallest number divisible by both denominators. LCD(4, 6) = 12.
2
Rewrite both fractions
$\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$
3
Add the numerators
$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$
4
Simplify if possible
5/12 is already in simplest form.

Addition of Fractions

\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + c \cdot b}{b \cdot d}

Multiply cross-wise, then add — but always simplify afterwards.

Tip Always simplify your answer. If GCD of numerator and denominator is greater than 1, divide both by it.

Subtracting Fractions

The process is the same — subtract the numerators after finding the LCD.

$\frac{7}{10} - \frac{2}{5} = \frac{7}{10} - \frac{4}{10} = \frac{3}{10}$

Add 2/3 + 3/4

LCD(3, 4) = 12

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

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

Real-World Applications

Real world: cooking a recipe

A recipe calls for $\dfrac{1}{4}$ teaspoon of salt and $\dfrac{1}{3}$ teaspoon of baking soda. To find the total, add the fractions.

LCD(4, 3) = 12, so $\dfrac{1}{4} = \dfrac{3}{12}$ and $\dfrac{1}{3} = \dfrac{4}{12}$ .

Total: $\dfrac{3}{12} + \dfrac{4}{12} = \dfrac{7}{12}$ teaspoon.

Real world: fuel in a tank

A motorbike tank is $\dfrac{2}{5}$ full. The rider adds $\dfrac{1}{3}$ of a tank during the week. How full is the tank now?

LCD(5, 3) = 15: $\dfrac{2}{5} = \dfrac{6}{15}$ , $\dfrac{1}{3} = \dfrac{5}{15}$ .

$\dfrac{6}{15} + \dfrac{5}{15} = \dfrac{11}{15}$ of a full tank.

Real world: splitting household chores

One housemate cleans $\dfrac{3}{8}$ of the flat each week; another cleans $\dfrac{1}{4}$ . What fraction do they cover together?

LCD(8, 4) = 8: $\dfrac{1}{4} = \dfrac{2}{8}$ .

$\dfrac{3}{8} + \dfrac{2}{8} = \dfrac{5}{8}$ — leaving $\dfrac{3}{8}$ still to be done.

Tip

Where do you see this? Recipes using cups and teaspoons, filling a fuel tank over several trips, budgeting time between tasks, and reading rainfall totals split across days all use fraction addition.

Practice Problems

Calculate 5/6 − 1/4.

Show answer

LCD = 12. 5/6 = 10/12, 1/4 = 3/12. 10/12 − 3/12 = 7/12.

Calculate 3/5 + 2/3.

Show answer

LCD = 15. 3/5 = 9/15, 2/3 = 10/15. 9/15 + 10/15 = 19/15 = 1 4/15.