Mastering Fractions: How to Add, Subtract, Multiply, and Divide

Fractions are one of the most fundamental concepts in mathematics, yet they remain a source of confusion for many students. Whether you are adding recipe measurements, solving algebraic equations, or working with ratios, a solid understanding of fraction operations is essential.

This guide covers the four arithmetic operations on fractions, how to simplify results, and how to work with mixed numbers.

What Is a Fraction?

A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number):

\frac{\text{numerator}}{\text{denominator}} = \frac{a}{b}

The denominator tells you how many equal parts the whole is divided into. The numerator tells you how many of those parts you have.

Adding and Subtracting Fractions

To add or subtract fractions, they must have the same denominator. If they do, simply add or subtract the numerators:

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

If the denominators are different, find the least common denominator (LCD) first.

Example 1: Same Denominator

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

Example 2: Different Denominators

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

The LCD of 3 and 4 is 12. Convert each fraction:

\frac{2}{3} = \frac{8}{12}, \quad \frac{1}{4} = \frac{3}{12}

\frac{8}{12} + \frac{3}{12} = \frac{11}{12}

Example 3: Subtraction

Subtract $\frac{5}{6} - \frac{1}{4}$ .

The LCD of 6 and 4 is 12:

\frac{5}{6} = \frac{10}{12}, \quad \frac{1}{4} = \frac{3}{12}

\frac{10}{12} - \frac{3}{12} = \frac{7}{12}

Multiplying Fractions

Multiplication is the simplest operation. Multiply the numerators together and the denominators together:

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

Example 4: Multiplication

\frac{3}{5} \times \frac{2}{7} = \frac{3 \times 2}{5 \times 7} = \frac{6}{35}

Dividing Fractions

To divide by a fraction, multiply by its reciprocal (flip the second fraction):

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}

Example 5: Division

\frac{4}{9} \div \frac{2}{3} = \frac{4}{9} \times \frac{3}{2} = \frac{12}{18} = \frac{2}{3}

Simplifying Fractions

A fraction is in its simplest form when the numerator and denominator share no common factors other than 1. To simplify, divide both by their greatest common divisor (GCD).

Example 6: Simplification

Simplify $\frac{18}{24}$ .

The GCD of 18 and 24 is 6:

\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}

Mixed Numbers and Improper Fractions

A mixed number combines a whole number and a fraction, such as $2\frac{3}{4}$ . An improper fraction has a numerator larger than its denominator, such as $\frac{11}{4}$ .

To convert a mixed number to an improper fraction:

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

Example 7: Conversion

Convert $3\frac{2}{5}$ to an improper fraction:

3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}

To convert back, divide the numerator by the denominator: $17 \div 5 = 3$ remainder $2$ , giving $3\frac{2}{5}$ .

Common Mistakes

Adding denominators. $\frac{1}{3} + \frac{1}{4} \neq \frac{2}{7}$ . You must find a common denominator first.
Forgetting to simplify. Always check whether the result can be reduced.
Not converting mixed numbers. When multiplying or dividing, convert mixed numbers to improper fractions first.

Try It Yourself

Practise fraction calculations with our free Fraction Calculator. It handles all four operations, simplification, and mixed number conversions with step-by-step solutions. For finding common denominators, our GCD & LCM Calculator can determine the least common denominator quickly.