Adding and Subtracting Mixed Numbers

Wholes with wholes, parts with parts

A mixed number is a whole number and a fraction added together: $2\frac{1}{5}$ means $2 + \frac{1}{5}$, and $1\frac{3}{5}$ means $1 + \frac{3}{5}$. To add two mixed numbers, add the wholes together and add the parts together.

For $2\frac{1}{5} + 1\frac{3}{5}$, the wholes are 2 + 1 = 3 and the parts are $\frac{1}{5} + \frac{3}{5} = \frac{4}{5}$. Put them back together: $2\frac{1}{5} + 1\frac{3}{5} = 3\frac{4}{5}$.

The 2 wholes and the 1 whole make 3 whole bars. The 1 fifth and the 3 fifths make 4 fifths of a fourth bar: $3\frac{4}{5}$.

When the parts make more than a whole

Add $1\frac{3}{4} + 2\frac{3}{4}$. The wholes are 1 + 2 = 3. The parts are $\frac{3}{4} + \frac{3}{4} = \frac{6}{4}$, and $\frac{6}{4}$ is more than a whole.

Trade the full whole into the whole-number part. 4 of the 6 quarters make 1 whole, and 2 quarters are left, so $\frac{6}{4} = 1\frac{2}{4}$. The 3 wholes gain that 1 and become 4: $1\frac{3}{4} + 2\frac{3}{4} = 3 + 1\frac{2}{4} = 4\frac{2}{4}$. In simplest form that is $4\frac{1}{2}$.

The 6 quarters from the two parts: quarters 1 to 4 fill a whole bar, and quarters 5 and 6 are $\frac{2}{4}$ left over.

When the parts will not subtract

Subtract $3\frac{1}{4} - 1\frac{3}{4}$. The wholes would be 3 − 1, but the parts would be $\frac{1}{4} - \frac{3}{4}$, and $\frac{1}{4}$ is too small to take $\frac{3}{4}$ away from.

Break one of the 3 wholes into quarters, just as a ten is broken into ones in column subtraction. That leaves 2 wholes, and the broken whole is $\frac{4}{4}$, which joins the $\frac{1}{4}$ to make $\frac{5}{4}$. So $3\frac{1}{4} = 2\frac{5}{4}$, the same amount written another way.

Drag the handle to break the whole into quarters. At 4 pieces it matches the $\frac{1}{4}$ below, and $1\frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}$. The other 2 wholes stay as they are, so $3\frac{1}{4} = 2\frac{5}{4}$.

Then subtract as before

Now the parts subtract: $2\frac{5}{4} - 1\frac{3}{4}$ has wholes 2 − 1 = 1 and parts $\frac{5}{4} - \frac{3}{4} = \frac{2}{4}$. The answer is $1\frac{2}{4}$, which is $1\frac{1}{2}$ in simplest form.

A second route avoids breaking a whole: write both numbers as improper fractions first. $3\frac{1}{4} = \frac{13}{4}$ and $1\frac{3}{4} = \frac{7}{4}$, so $\frac{13}{4} - \frac{7}{4} = \frac{6}{4}$, and $\frac{6}{4} = 1\frac{2}{4}$. It is the same answer. This route is safer when the numbers are awkward.

Different denominators

When the fractions have different denominators, give them a common denominator first, exactly as for fractions on their own. For $2\frac{1}{2} + 1\frac{3}{4}$, write $\frac{1}{2} = \frac{2}{4}$. The wholes are 2 + 1 = 3 and the parts are $\frac{2}{4} + \frac{3}{4} = \frac{5}{4} = 1\frac{1}{4}$, so the answer is $3 + 1\frac{1}{4} = 4\frac{1}{4}$.

The usual mistakes

Swapping the parts round. In $3\frac{1}{4} - 1\frac{3}{4}$, it is tempting to work out $\frac{3}{4} - \frac{1}{4} = \frac{2}{4}$ and answer $2\frac{2}{4}$. But $\frac{3}{4}$ is being taken away from $\frac{1}{4}$, not the other way round, and $\frac{1}{4}$ is too small. Break a whole instead.

Forgetting the traded whole. In $1\frac{3}{4} + 2\frac{3}{4}$ the parts make $1\frac{2}{4}$, and the 1 whole from them must join the 3 in front: the answer is $4\frac{2}{4}$, not $3\frac{2}{4}$. When subtracting, the broken whole leaves the front, so 3 becomes 2.

Adding the denominators. $\frac{1}{5} + \frac{3}{5}$ is $\frac{4}{5}$, not $\frac{4}{10}$. The pieces are fifths before and after.