Math Core

You can add $\dfrac{2}{5} + \dfrac{1}{5}$ , and you can add $3 + 1$ . Adding mixed numbers like $3\dfrac{2}{5} + 1\dfrac{1}{5}$ just puts those two skills together: add the wholes, add the fractions.

Add the wholes, add the fractions

A mixed number is a whole number plus a fraction. So you can work with the two parts separately.

\begin{aligned} 3\frac{2}{5} + 1\frac{1}{5} &= (3 + 1) + \left(\frac{2}{5} + \frac{1}{5}\right) \\ &= 4 + \frac{3}{5} \\ &= 4\frac{3}{5} \end{aligned}

Subtracting works the same way when the fraction you take away is not too big:

5\frac{7}{8} - 2\frac{3}{8} = (5 - 2) + \left(\frac{7}{8} - \frac{3}{8}\right) = 3\frac{4}{8}.

Adding and subtracting mixed numbers

When the denominators are the same:

Add or subtract the fraction parts.
Add or subtract the whole number parts.
If the fraction part ends up being 1 or more, trade it for a whole. If you can't subtract the fractions, trade a whole for fraction pieces first.

When the fractions make a whole

Sometimes the fraction parts add up to more than 1.

Worked example: Regrouping when adding

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

Wholes: $2 + 1 = 3$ . Fractions: $\dfrac{3}{4} + \dfrac{2}{4} = \dfrac{5}{4}$ .

So far that's $3\dfrac{5}{4}$ . But $\dfrac{5}{4}$ is more than 1: $\dfrac{5}{4} = 1\dfrac{1}{4}$ . Trade 4 fourths for 1 whole:

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

When you need to borrow a whole

What about $4\dfrac{1}{6} - 1\dfrac{5}{6}$ ? You can't take 5 sixths away from 1 sixth. So break one of the 4 wholes into sixths first:

4\frac{1}{6} = 3 + 1 + \frac{1}{6} = 3 + \frac{6}{6} + \frac{1}{6} = 3\frac{7}{6}.

Now subtract: $3\dfrac{7}{6} - 1\dfrac{5}{6} = 2\dfrac{2}{6}$ .

Worked example: Borrowing when subtracting

Find $5\dfrac{2}{8} - 2\dfrac{7}{8}$ .

$\dfrac{2}{8}$ is smaller than $\dfrac{7}{8}$ , so borrow a whole. One whole is $\dfrac{8}{8}$ :

5\frac{2}{8} = 4\frac{10}{8}.

Now subtract the wholes and the fractions:

4\frac{10}{8} - 2\frac{7}{8} = 2\frac{3}{8}.

Worked example: Another way: use fractions

Find $3\dfrac{1}{3} - 1\dfrac{2}{3}$ by changing both mixed numbers into fractions.

$3\dfrac{1}{3} = \dfrac{10}{3}$ and $1\dfrac{2}{3} = \dfrac{5}{3}$ . Subtract the numerators:

\frac{10}{3} - \frac{5}{3} = \frac{5}{3} = 1\frac{2}{3}.

Worked example: A word problem

A plant was $6\dfrac{3}{10}$ inches tall. It grew $2\dfrac{9}{10}$ inches. How tall is it now?

Wholes: $6 + 2 = 8$ . Fractions: $\dfrac{3}{10} + \dfrac{9}{10} = \dfrac{12}{10} = 1\dfrac{2}{10}$ .

8 + 1\frac{2}{10} = 9\frac{2}{10}

The plant is $9\dfrac{2}{10}$ inches tall.

Common mistake

When the top fraction is too small to subtract, don't flip the problem around. In $4\dfrac{1}{6} - 1\dfrac{5}{6}$ , it's wrong to do $\dfrac{5}{6} - \dfrac{1}{6}$ and get $3\dfrac{4}{6}$ . Borrow a whole instead. The real answer is $2\dfrac{2}{6}$ .

Tip

Estimate first. $5\dfrac{2}{8} - 2\dfrac{7}{8}$ is a bit more than $5 - 3 = 2$ , so an answer of $2\dfrac{3}{8}$ makes sense.

Practice

Practice 1

Find $2\dfrac{1}{5} + 3\dfrac{2}{5}$ .

Type your answer like 1 2/3.

Type a number.

Practice 2

Find $7\dfrac{5}{6} - 3\dfrac{3}{6}$ .

Type your answer like 1 2/3.

Type a number.

Practice 3

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

Type your answer like 1 2/3.

Type a number.

Practice 4

To find $6\dfrac{1}{4} - 2\dfrac{3}{4}$ , Sara first rewrites $6\dfrac{1}{4}$ . Which is the correct way to rewrite it?

Practice 5

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

Type your answer like 1 2/3.

Type a number.

Practice 6

Mo has a board that is 4 feet long. He cuts off $1\dfrac{3}{10}$ feet. How long is the board now?

Type your answer like 1 2/3.

Type a number.

Practice 7

A hiker walks $2\dfrac{5}{8}$ miles in the morning, $3\dfrac{1}{8}$ miles at lunch time, and $1\dfrac{3}{8}$ miles in the afternoon. How far does she walk in all?

Type your answer like 1 2/3.

Type a number.

Practice 8

Ella's cat weighs $9\dfrac{2}{12}$ pounds. Her dog weighs $10\dfrac{7}{12}$ pounds. How much more does the dog weigh than the cat?

Type your answer like 1 2/3.

Type a number.