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

When the parts make more than a whole

Add $1\frac{3}{4}+2\frac{3}{4}$1 3/4 + 2 3/4. The wholes are 1 + 2 = 3. The parts are $\frac{3}{4}+\frac{3}{4}=\frac{6}{4}$3/4 + 3/4 = 6/4, and $\frac{6}{4}$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}$6/4 = 1 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}$1 3/4 + 2 3/4 = 3 + 1 2/4 = 4 2/4. In simplest form that is $4\frac{1}{2}$4 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}$2/4 left over.

When the parts will not subtract

Subtract $3\frac{1}{4}-1\frac{3}{4}$3 1/4 − 1 3/4. The wholes would be 3 − 1, but the parts would be $\frac{1}{4}-\frac{3}{4}$1/4 − 3/4, and $\frac{1}{4}$1/4 is too small to take $\frac{3}{4}$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}$4/4, which joins the $\frac{1}{4}$1/4 to make $\frac{5}{4}$5/4. So $3\frac{1}{4}=2\frac{5}{4}$3 1/4 = 2 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}$1/4 below, and $1\frac{1}{4}=\frac{4}{4}+\frac{1}{4}=\frac{5}{4}$1 1/4 = 4/4 + 1/4 = 5/4. The other 2 wholes stay as they are, so $3\frac{1}{4}=2\frac{5}{4}$3 1/4 = 2 5/4.

Then subtract as before

Now the parts subtract: $2\frac{5}{4}-1\frac{3}{4}$2 5/4 − 1 3/4 has wholes 2 − 1 = 1 and parts $\frac{5}{4}-\frac{3}{4}=\frac{2}{4}$5/4 − 3/4 = 2/4. The answer is $1\frac{2}{4}$1 2/4, which is $1\frac{1}{2}$1 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}$3 1/4 = 13/4 and $1\frac{3}{4}=\frac{7}{4}$1 3/4 = 7/4, so $\frac{13}{4}-\frac{7}{4}=\frac{6}{4}$13/4 − 7/4 = 6/4, and $\frac{6}{4}=1\frac{2}{4}$6/4 = 1 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}$2 1/2 + 1 3/4, write $\frac{1}{2}=\frac{2}{4}$1/2 = 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}$2/4 + 3/4 = 5/4 = 1 1/4, so the answer is $3+1\frac{1}{4}=4\frac{1}{4}$3 + 1 1/4 = 4 1/4.

The usual mistakes

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

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

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