How to Add, Subtract, Multiply, and Divide Fractions

The Anatomy of a Fraction

A fraction represents a part of a whole, expressed as one integer divided by another: Numerator / Denominator. The numerator (top number) counts how many parts you have, while the denominator (bottom number) defines how many equal parts make up the complete unit.

Because the denominator defines the unit size, arithmetic rules differ fundamentally between addition/subtraction (which require equal units) and multiplication/division (which scale units directly).

1. Adding and Subtracting Fractions (Finding Common Denominators)

You cannot directly add fractions with different denominators (e.g. 1/2 + 1/3 is not 2/5). You must first find a Least Common Denominator (LCD) to convert them into like parts:

1. Find the least common multiple of the two denominators.
2. Multiply numerator and denominator of each fraction by the factor needed to reach the LCD.
3. Add or subtract the numerators while keeping the common denominator unchanged.
4. Simplify the resulting fraction to lowest terms.

Worked Example: 2/3 + 3/4

1. The least common multiple of 3 and 4 is 12.
2. Convert 2/3: (2 × 4) / (3 × 4) = 8/12.
3. Convert 3/4: (3 × 3) / (4 × 3) = 9/12.
4. Add numerators: 8/12 + 9/12 = 17/12 (or 1 5/12 as a mixed number).

2. Multiplying Fractions (The Direct Method)

Multiplication is the simplest fraction operation because no common denominator is required. Multiply straight across:

Worked Example: 2/5 × 3/7

Multiply numerators: 2 × 3 = 6. Multiply denominators: 5 × 7 = 35. Result: 6/35 (already in simplest form).

3. Dividing Fractions (Keep, Change, Flip)

Dividing by a fraction is mathematically identical to multiplying by its reciprocal (inverting the numerator and denominator). Remember the 3-word rule: Keep, Change, Flip:

• Keep the first fraction unchanged.
• Change the division sign (÷) into multiplication (×).
• Flip the second fraction upside down (c/d becomes d/c).

Worked Example: 1/2 ÷ 1/4

1. Keep 1/2.
2. Change ÷ to ×.
3. Flip 1/4 to 4/1: 1/2 × 4/1 = (1 × 4) / (2 × 1) = 4/2 = 2.

There are exactly two quarters in one half.

4. Simplifying Fractions & Mixed Numbers

Always reduce fractions by dividing both numerator and denominator by their Greatest Common Divisor (GCD):

• Simplifying 18/24: The GCD of 18 and 24 is 6. Dividing both by 6 yields 3/4.
• Improper to Mixed: For 17/12, divide 17 by 12 = 1 with a remainder of 5 → 1 5/12.

Solve Fraction Problems Instantly

Compute fraction arithmetic, reduce improper fractions, and convert mixed numbers with our free math tools:

• Get instant step-by-step solutions with our Fraction Calculator.
• Solve proportional comparisons with our Ratio Calculator.