How to Add, Subtract, Multiply and Divide Fractions

Fractions are one of the most important topics in school maths, and knowing how to add, subtract, multiply and divide them is essential. Each operation has its own method, but none of them are as complicated as they might first seem.

This guide covers all four operations with clear steps and worked examples. Grab a pen and paper and work through them yourself. That is how these methods stick.

Before you start: the basics

A quick reminder of fraction terminology. In the fraction 3/4, the top number (3) is the numerator and the bottom number (4) is the denominator. The denominator tells you how many equal parts the whole has been divided into. The numerator tells you how many of those parts you have.

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Adding fractions

Same denominator

If the denominators are already the same, just add the numerators and keep the denominator.

Example: 2/7 + 3/7 = 5/7

You have 2 sevenths plus 3 sevenths, which gives you 5 sevenths. Straightforward.

Different denominators

When the denominators are different, you need to rewrite the fractions so they share a common denominator before you can add them.

Example: 1/3 + 1/4

1. Find the lowest common denominator (LCD). For 3 and 4, it is 12.
2. Convert each fraction: 1/3 = 4/12 and 1/4 = 3/12.
3. Add the numerators: 4/12 + 3/12 = 7/12.

How to find the LCD: List the multiples of each denominator until you find the smallest one they share. Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. The LCD is 12.

Another example: 2/5 + 1/3

1. LCD of 5 and 3 is 15.
2. 2/5 = 6/15 and 1/3 = 5/15.
3. 6/15 + 5/15 = 11/15.

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Subtracting fractions

Subtraction works exactly like addition, except you subtract the numerators instead.

Same denominator

Example: 5/8 - 2/8 = 3/8

Different denominators

Example: 3/4 - 1/6

1. LCD of 4 and 6 is 12.
2. 3/4 = 9/12 and 1/6 = 2/12.
3. 9/12 - 2/12 = 7/12.

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Multiplying fractions

Multiplying fractions is actually simpler than adding them. You do not need a common denominator at all.

Method: Multiply the numerators together, then multiply the denominators together.

Example 1: 2/3 × 4/5
Numerators: 2 × 4 = 8
Denominators: 3 × 5 = 15
Answer: 8/15

Example 2: 3/7 × 2/9
Numerators: 3 × 2 = 6
Denominators: 7 × 9 = 63
Simplify: 6/63 = 2/21

Simplify before you multiply

You can make your life easier by cancelling common factors before multiplying. In the example above, 3 and 9 share a factor of 3. Cancel first: (1/7) × (2/3) = 2/21. Same answer, smaller numbers to work with.

Multiplying a fraction by a whole number

Write the whole number as a fraction over 1, then multiply as normal.

Example: 3/4 × 6 = 3/4 × 6/1 = 18/4 = 9/2 = 4 and 1/2

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Dividing fractions

This is the one that confuses students most, but the method is simple once you know it: flip the second fraction and multiply.

The technical name for this is "multiply by the reciprocal." The reciprocal of 2/3 is 3/2. You just swap the numerator and denominator.

Example 1: 3/4 divided by 2/5

1. Flip the second fraction: 2/5 becomes 5/2.
2. Multiply: 3/4 × 5/2 = 15/8.
3. Convert if needed: 15/8 = 1 and 7/8.

Example 2: 5/6 divided by 1/3

1. Flip: 1/3 becomes 3/1.
2. Multiply: 5/6 × 3/1 = 15/6.
3. Simplify: 15/6 = 5/2 = 2 and 1/2.

Why does this work?

Dividing by a fraction asks "how many of these fit into that?" How many thirds fit into 5/6? Well, each third is smaller than 5/6, so more than one fits. The answer is 2 and 1/2, which makes sense when you think about it on a number line.

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Dealing with mixed numbers

A mixed number like 2 and 1/3 needs to be converted to an improper fraction before you can use any of these methods.

To convert: Multiply the whole number by the denominator, add the numerator, and put the result over the original denominator.

2 and 1/3 = (2 × 3 + 1) / 3 = 7/3

Example: 1 and 1/2 + 2 and 2/3

1. Convert: 1 and 1/2 = 3/2. And 2 and 2/3 = 8/3.
2. LCD of 2 and 3 is 6.
3. 3/2 = 9/6 and 8/3 = 16/6.
4. 9/6 + 16/6 = 25/6 = 4 and 1/6.

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Common mistakes

• Adding the denominators: 1/3 + 1/4 does not equal 2/7. You need a common denominator first.
• Forgetting to flip when dividing: You flip the second fraction, not the first.
• Not simplifying: Always check if your answer can be reduced. Divide the numerator and denominator by their highest common factor.
• Mixed number errors: Convert to improper fractions first, then convert back at the end.

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Summary of methods

Operation	Method
Addition	Common denominator, add numerators
Subtraction	Common denominator, subtract numerators
Multiplication	Multiply numerators, multiply denominators
Division	Flip the second fraction, then multiply

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Keep practising

Fraction operations become second nature with practice. The key is to understand why each method works, not just memorise the steps. Once you have that understanding, you will rarely make mistakes.

If you want guided practice that builds up your fraction skills step by step, imSteyn covers all four operations as part of the Year 7 and Year 8 Australian Curriculum. It walks you through problems without giving the answer away, so you learn by doing. Sign up for free to give it a try.