Why do fractions need a common denominator?
To add fractions, count pieces of the same size. For 1/2 + 1/3, sixths give a shared unit: 3/6 + 2/6 = 5/6.
Start with equal wholes
We are adding 1/2 and 1/3 of the same-sized whole. Halves and thirds are different-sized pieces, so their counts cannot simply be combined.
Read the complete explanation +
Start with equal wholes
We are adding 1/2 and 1/3 of the same-sized whole. Halves and thirds are different-sized pieces, so their counts cannot simply be combined.
Choose a shared piece size
Divide each whole into six equal parts. Six is the least common multiple of two and three. Each new piece represents 1/6.
Rename without changing value
Multiply 1/2 by 3/3 to get 3/6. Multiply 1/3 by 2/2 to get 2/6. Multiplying by one keeps each value unchanged.
Add the matching pieces
Add the numerators while keeping the common denominator: 3/6 + 2/6 = 5/6. The answer is less than one whole.
About this visual model +
Three equally sized bars are each divided into six equal pieces. Filled pieces show three sixths, two sixths and five sixths. The separate ruler marks one sixth of a whole.
The bars use the same whole and exact equal partitions. Playback highlights the reasoning without changing the quantities.
- One half: One of two equal parts of a whole. Three sixths cover the same amount.
- One third: One of three equal parts of the same-sized whole. Two sixths cover the same amount.
- Shared unit: Six is a common multiple of two and three, so sixths can represent both fractions.
- Five sixths: Three sixths plus two sixths equals five sixths.
Further reading: Adding fractions with different denominators ↗