Math Core

A hike is $2\dfrac{1}{2}$ miles to the waterfall and $1\dfrac{3}{4}$ miles more to the lake. Real measurements often come as mixed numbers like these. To add or subtract them, you work with the whole numbers and the fractions, using a common denominator for the fraction parts.

Adding mixed numbers

Add the whole numbers, add the fractions, then put them together. If the fractions have different denominators, rename them first.

For $2\dfrac{1}{2} + 1\dfrac{3}{4}$ , rename $\dfrac{1}{2}$ as $\dfrac{2}{4}$ :

2\frac{2}{4} + 1\frac{3}{4} = 3\frac{5}{4}

But $\dfrac{5}{4}$ is more than a whole! It is $1\dfrac{1}{4}$ . Move that extra whole over to the whole numbers:

3\frac{5}{4} = 3 + 1\frac{1}{4} = 4\frac{1}{4}

The hike to the lake is $4\dfrac{1}{4}$ miles.

Adding and subtracting mixed numbers

Rename the fraction parts with a common denominator.
Add or subtract the fractions, then the whole numbers.
When adding, if the fraction part is 1 or more, regroup it into a whole.
When subtracting, if the top fraction is too small, borrow 1 whole first.

Subtracting mixed numbers

Subtracting works the same way, as long as the first fraction is big enough. For $5\dfrac{2}{3} - 2\dfrac{1}{6}$ , rename $\dfrac{2}{3}$ as $\dfrac{4}{6}$ :

5\frac{4}{6} - 2\frac{1}{6} = 3\frac{3}{6} = 3\frac{1}{2}

Borrowing a whole

What about $4\dfrac{1}{5} - 1\dfrac{1}{2}$ ? Rename with tenths: $4\dfrac{2}{10} - 1\dfrac{5}{10}$ . You can't take 5 tenths from 2 tenths.

So break one of the 4 wholes into tenths. One whole is $\dfrac{10}{10}$ :

4\frac{2}{10} = 3 + \frac{10}{10} + \frac{2}{10} = 3\frac{12}{10}

Now subtract:

3\frac{12}{10} - 1\frac{5}{10} = 2\frac{7}{10}

Common mistake

When you borrow 1 whole, it becomes as many pieces as the denominator, not 10 every time. In fourths, 1 whole is $\dfrac{4}{4}$ . In sixths, it is $\dfrac{6}{6}$ . So $3\dfrac{1}{4}$ becomes $2\dfrac{5}{4}$ , not $2\dfrac{11}{4}$ .

Worked example: Adding with regrouping

Find $3\dfrac{2}{3} + 2\dfrac{1}{2}$ .

Use sixths: $\dfrac{2}{3} = \dfrac{4}{6}$ and $\dfrac{1}{2} = \dfrac{3}{6}$ .

\begin{aligned} 3\frac{4}{6} + 2\frac{3}{6} &= 5\frac{7}{6} \\ &= 5 + 1\frac{1}{6} \\ &= 6\frac{1}{6} \end{aligned}

Worked example: Subtracting without borrowing

Find $6\dfrac{3}{4} - 2\dfrac{1}{3}$ .

Use twelfths: $\dfrac{3}{4} = \dfrac{9}{12}$ and $\dfrac{1}{3} = \dfrac{4}{12}$ .

6\frac{9}{12} - 2\frac{4}{12} = 4\frac{5}{12}

Worked example: Subtracting with borrowing

Find $5\dfrac{1}{4} - 2\dfrac{5}{6}$ .

Use twelfths: $5\dfrac{3}{12} - 2\dfrac{10}{12}$ . Since 3 twelfths is less than 10 twelfths, borrow a whole. One whole is $\dfrac{12}{12}$ :

\begin{aligned} 5\frac{3}{12} &= 4\frac{15}{12} \\ 4\frac{15}{12} - 2\frac{10}{12} &= 2\frac{5}{12} \end{aligned}

Check by estimating: about 5 minus about 3 is about 2. The answer $2\dfrac{5}{12}$ is close.

Tip

Another way: change both mixed numbers to improper fractions, then subtract. $5\dfrac{1}{4} - 2\dfrac{5}{6} = \dfrac{63}{12} - \dfrac{34}{12} = \dfrac{29}{12} = 2\dfrac{5}{12}$ . This avoids borrowing, but the numbers get bigger.

Practice

Practice 1

Find $2\dfrac{1}{4} + 3\dfrac{1}{2}$ .

Type a number.

Practice 2

Find $4\dfrac{1}{2} - 1\dfrac{1}{3}$ .

Type a number.

Practice 3

Rename $3\dfrac{3}{4}$ by borrowing 1 whole: $3\dfrac{3}{4} = 2\dfrac{?}{4}$ . What is the missing numerator?

Type a number.

Practice 4

Find $1\dfrac{3}{4} + 2\dfrac{3}{8}$ .

Type a number.

Practice 5

Which shows a correct first step for $6\dfrac{1}{3} - 4\dfrac{2}{3}$ ?

Practice 6

Find $3\dfrac{1}{10} - 1\dfrac{4}{5}$ .

Type a number.

Practice 7

Find $4\dfrac{5}{6} + 2\dfrac{1}{4}$ .

Type a number.

Practice 8

Find $5 - 2\dfrac{5}{12}$ .

Type a number.