A mixed number is a whole number and a proper fraction written together, such as 3 1/4. Adding and subtracting them is ordinary fraction work with one extra step at the end, and one case that needs regrouping.
Adding
- Rewrite the fractions with a common denominator.
- Add the whole numbers. Add the fractions.
- If the fraction part is now improper, convert the extra whole out of it.
2 3/4 plus 1 5/6
common denominator is 12 2 9/12 1 10/12 -------- 3 19/12 19/12 is one whole and 7/12 so the answer is 4 7/12
Subtracting, the easy case
5 7/8 minus 2 3/8
5 7/8 2 3/8 -------- 3 4/8 which simplifies to 3 1/2
Subtracting when the top fraction is smaller
This is the case that produces most of the errors. When the fraction being subtracted is larger, borrow one whole from the whole number part and add it to the fraction as a full denominator.
4 1/5 minus 1 3/5
1/5 is smaller than 3/5, so regroup 4 1/5 = 3 + 1 + 1/5 = 3 6/5 3 6/5 1 3/5 -------- 2 3/5
The one that was borrowed became 5/5, which added to 1/5 makes 6/5.
The improper fraction route
The regrouping step can be avoided completely by converting both mixed numbers to improper fractions first. It gives the same answer, and which route is better depends on the numbers.
4 1/5 minus 1 3/5 again, without regrouping
4 1/5 = 21/5 1 3/5 = 8/5 21/5 - 8/5 = 13/5 = 2 3/5
Improper fractions are cleaner when the whole numbers are small. Regrouping is easier when they are large, because 18 1/4 becomes the unwieldy 73/4 and every later step carries those big numbers along.
Estimate before you start
Round each mixed number to the nearest whole and add or subtract those. 2 3/4 + 1 5/6 is roughly 3 + 2 = 5, so an answer of 4 7/12 is plausible and an answer of 3 7/12 is not.
Common mistakes
Adding the denominators
1/4 + 1/6 is not 2/10. Denominators name the size of the pieces, and only pieces of the same size can be counted together.
Borrowing one instead of a whole denominator
When borrowing from the whole number in 4 1/5, the fraction becomes 6/5, not 2/5. One whole is 5/5 here.
Check yourself
1 2/3 plus 2 1/2
Common denominator 6: 1 4/6 + 2 3/6 = 3 7/6 = 4 1/6.
6 1/4 minus 2 3/4
Regroup: 6 1/4 = 5 5/4, then 5 5/4 - 2 3/4 = 3 2/4 = 3 1/2.
3 5/6 minus 1 1/3
Common denominator 6: 3 5/6 - 1 2/6 = 2 3/6 = 2 1/2.
Why does borrowing turn 4 1/5 into 3 6/5?
Because the whole that was taken from the 4 is worth 5/5 in fifths. Adding it to the existing 1/5 gives 6/5, and the whole number drops to 3.
See also
Sources
- Mixed number operations as taught in United States middle grades mathematics. Examples on this page are our own.