Free Math Guide

Adding and Subtracting Fractions Explained

Learn how to add and subtract fractions by finding a common denominator, rewriting equivalent fractions, combining the numerators, and simplifying the answer.

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Adding and subtracting fractions is easiest when you treat the denominator as the size of the pieces and the numerator as the number of pieces. Before combining fractions, the pieces must be the same size.

On This Page

The Basic Idea

Understanding the Parts of a Fraction

A fraction describes equal parts of a whole. The numerator tells how many parts you have, and the denominator tells how many equal parts make one whole.

Numerator

The numerator is the top number. It tells how many equal parts are being counted.

Denominator

The denominator is the bottom number. It tells the size of the equal parts.

3 8 means 3 pieces when one whole is divided into 8 equal pieces

Why Denominators Matter

Fractions can only be combined directly when their denominators match, because the pieces must be the same size.

The Main Process

How to Add or Subtract Fractions

Use these steps whenever you add or subtract fractions. Some problems may skip the rewriting or simplifying step.

1

Find a Common Denominator

Check whether the denominators already match. If not, find the least common denominator.

2

Rewrite the Fractions When Needed

Create equivalent fractions so both denominators equal the common denominator.

3

Combine the Numerators

Add or subtract the numerators while keeping the common denominator.

4

Simplify When Possible

Reduce the answer only if the numerator and denominator share a common factor greater than 1.

Easier Case

Fractions With the Same Denominator

When the denominators already match, you do not need to rewrite the fractions. Combine the numerators and keep the denominator.

Addition Example

2 7 + 3 7 = 5 7
  1. The denominators are already both 7.
  2. Add the numerators: 2 + 3 = 5.
  3. Keep the denominator 7.
  4. The answer is already simplified.

Subtraction Example

6 9 − 2 9 = 4 9
  1. The denominators are already both 9.
  2. Subtract the numerators: 6 − 2 = 4.
  3. Keep the denominator 9.
  4. The answer is already simplified.

Same-Denominator Shortcut

If the denominators match, combine the top numbers and leave the bottom number unchanged.

Finding Equal-Sized Pieces

Fractions With Different Denominators

When the denominators are different, first find a common denominator. The least common denominator is usually the best choice because it keeps the numbers smaller.

Find the Least Common Denominator

Multiples of 3: 3, 6, 9, 12...
Multiples of 4: 4, 8, 12...

The first shared multiple is 12, so the least common denominator is 12.

Rewrite Equivalent Fractions

1 3 = 4 12 and 1 4 = 3 12

Multiply the numerator and denominator by the same number so the fraction keeps the same value.

Equivalent Fractions Keep the Same Value

Multiplying both the numerator and denominator by the same nonzero number changes the name of the fraction, but not its value.

Worked Example

Adding Fractions With Different Denominators

Find a common denominator, rewrite both fractions, and then add the numerators.

1 3 + 1 4 = 4 12 + 3 12 = 7 12
  1. The denominators are 3 and 4.
  2. The least common denominator is 12.
  3. Rewrite 1/3 as 4/12.
  4. Rewrite 1/4 as 3/12.
  5. Add the numerators: 4 + 3 = 7.
  6. Keep the denominator 12.
  7. 7/12 is already simplified.

Worked Example

Subtracting Fractions With Different Denominators

Subtraction follows the same denominator process as addition. The only difference is that you subtract the rewritten numerators.

5 6 − 1 4 = 10 12 − 3 12 = 7 12
  1. The denominators are 6 and 4.
  2. The least common denominator is 12.
  3. Rewrite 5/6 as 10/12.
  4. Rewrite 1/4 as 3/12.
  5. Subtract the numerators: 10 − 3 = 7.
  6. Keep the denominator 12.
  7. 7/12 is already simplified.

Keep the Order

In subtraction, subtract the second rewritten numerator from the first rewritten numerator.

Final Step

Simplifying a Fraction

A fraction is in simplest form when its numerator and denominator do not share any common factor greater than 1.

Fraction That Needs Simplifying

6 8 = 3 4
  1. 6 and 8 are both divisible by 2.
  2. Divide the numerator and denominator by 2.
  3. The simplified fraction is 3/4.

Fraction Already Simplified

6 7

The only common factor of 6 and 7 is 1, so 6/7 is already in simplest form.

Do Not Simplify When Nothing Reduces

If the numerator and denominator share no factor greater than 1, the fraction is already simplified.

Important Exceptions

Special Fraction Cases

A Zero Numerator

0 6 = 0

Any fraction with a numerator of zero and a nonzero denominator equals zero.

A Denominator of 1

5 1 = 5

A fraction with a denominator of 1 is equal to its numerator as a whole number.

Equal Fractions Subtract to Zero

4 9 − 4 9 = 0

A quantity minus itself always equals zero.

The Denominator Cannot Be Zero

Division by zero is undefined, so zero can never be used as the denominator of a valid fraction.

Verify Your Work

How to Check a Fraction Answer

Check the Denominator

After combining, the denominator should remain the common denominator.

Check Equivalent Fractions

Make sure each rewritten fraction has the same value as its original fraction.

Estimate the Size

Decide whether the answer should be less than 1, equal to 1, or greater than 1.

Check Simplest Form

Confirm that the numerator and denominator share no common factor greater than 1.

What to Watch For

Common Fraction Addition and Subtraction Mistakes

Adding the Denominators

Once the denominators match, keep the common denominator. Only combine the numerators.

Combining Fractions Before the Denominators Match

Fractions with different denominators represent different-sized pieces and cannot be combined directly.

Changing Only the Denominator

When rewriting a fraction, multiply both the numerator and denominator by the same number.

Forgetting to Simplify

After combining, check whether the numerator and denominator share a common factor greater than 1.

Simplifying a Fraction That Is Already Reduced

If the only common factor is 1, the fraction is already in simplest form.

Writing Zero as a Fraction

A result such as 0/6 should be written simply as 0.

Ready to Practice?

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Work through each fraction problem one step at a time. Find the common denominator, rewrite when needed, combine the numerators, and simplify with immediate feedback.

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