How to Add Fractions with Different Denominators
A fraction names part of a whole. The denominator (bottom number) tells you how many equal pieces make that whole. The numerator (top number) tells you how many of those pieces you have.
The Big Idea
To add fractions of the same whole, first make the pieces the same size. Then count the pieces. You change how the fraction is written, not how much it is worth.
Work Through an Example
Add + .
Choose a common denominator: 6.
Why: A whole can be split into 6 equal pieces. Both halves and thirds can be described using sixths. Six is the least common multiple of 2 and 3.
Rewrite as and as .
Why: Multiply the top and bottom of by 3. Multiply the top and bottom of by 2. Multiplying by or means multiplying by 1, so each value stays the same.
Add the numerators: + = .
Why: You have 3 sixths and 2 more sixths: 5 sixths altogether. Keep the denominator 6 because the pieces are still sixths.
Check whether can be simplified.
Why: The only positive factor shared by 5 and 6 is 1, so is already in simplest form. It is bigger than and smaller than 1, which fits the original sum.
A Common Mix-Up
Do not add both tops and bottoms: + is not . Two fifths is even smaller than one half, so it cannot be the result of adding a positive third to one half.
Try Another Way to Picture It
Imagine one same-sized chocolate bar marked into 6 equal squares. Half the bar is 3 squares, and one third is 2 squares. Together those amounts are 5 of the 6 squares.
Your Turn
Try these on paper, then open each answer to check your reasoning. These examples do not record a score or change your learning progress.
Find + .
Show answer and explanation
Rewrite as . Then + = .
Find + . Give the answer as a mixed number.
Show answer and explanation
1
= , so the sum is = = 1 . A sum of fractions can be greater than one.
Keep This in Mind
Same-sized pieces first. Add the numerators, keep the common denominator, then simplify.