Chapter 7 · Add and subtract fractions

Subtract Mixed Numbers: Regrouping and Fixing Mistakes

Learn to subtract mixed numbers when the first fraction part is too small, by renaming a whole or using improper fractions. Then find and fix common mistakes.

12 min read

A mixed number, such as $3\frac{1}{4}$ , has a whole-number part and a fraction part. You already know how to subtract mixed numbers: subtract the fraction parts, then subtract the whole numbers. But what if the first fraction part is too small?

Picture pies that are all the same size. One slice is one fourth of a pie. You have $3\frac{1}{4}$ pies: 3 whole pies and 1 loose slice. You want to give away $1\frac{3}{4}$ pies: 1 whole pie and 3 slices.

You have only 1 loose slice, so you cannot hand over 3 slices yet. But you have plenty of pie. Cut one whole pie into 4 slices. Now you have 2 whole pies and 5 loose slices. It is still the same amount of pie.

Writing an amount in a new way, without changing how much it is, is called renaming or regrouping.

The same amount of pie, before and after you cut one whole pie into fourths

When	In words	As parts added together	As a mixed number
Before the cut	3 whole pies and 1 fourth	$3 + \frac{1}{4}$	$3\frac{1}{4}$
After the cut	2 whole pies and 5 fourths	$2 + \frac{4}{4} + \frac{1}{4}$	$2\frac{5}{4}$

Key concept

Rename one whole to get more fraction pieces

When the first fraction part is smaller than the second, take one whole out of the whole-number part. Write that whole as a fraction with the same denominator (the bottom number, which tells how many equal pieces make one whole). Add it to the fraction part.

The whole number goes down by 1. The fraction part goes up by one whole. The amount stays the same, so the answer to the subtraction does not change.

Use this when both mixed numbers refer to the same whole, or to wholes of equal size, and when the number you start with is not smaller than the number you take away.

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

Now finish the pie problem. Both amounts count pies of the same size, so you can subtract them.

Rename the first number: $3\frac{1}{4} = 2\frac{5}{4}$ .
Hand over 3 of your 5 slices. That leaves 2 slices: $\frac{5}{4} - \frac{3}{4} = \frac{2}{4}$ .
Hand over 1 of your 2 whole pies. That leaves 1 pie: $2 - 1 = 1$ .

So $3\frac{1}{4} - 1\frac{3}{4} = 1\frac{2}{4}$ . Since $\frac{2}{4}$ is the same amount as $\frac{1}{2}$ , you have $1\frac{1}{2}$ pies left.

The fraction part $\frac{5}{4}$ is more than one whole. That is fine for a working step. It does not appear in the final answer.

Here is the rule for two mixed numbers that refer to wholes of the same size: rename only when the first fraction part is smaller than the second. If the two fraction parts are equal, their difference (the result of the subtraction) is $0$ , and you do not rename.

Worked example

Rename a whole when the denominators are different

Find $5\frac{1}{3} - 2\frac{3}{4}$ . Both mixed numbers refer to wholes of the same size.

Write the fraction parts with a common denominator, which is the same denominator for both, as you did in the last lesson. Thirds and fourths can both be written as twelfths. Now all the pieces are the same size.
$5\frac{1}{3} - 2\frac{3}{4} = 5\frac{4}{12} - 2\frac{9}{12}$
Compare the fraction parts. Both are twelfths of same-size wholes, so you can compare them. 4 twelfths is smaller than 9 twelfths, so you cannot take 9 twelfths away from 4 twelfths. You need to rename a whole.
$\frac{4}{12} < \frac{9}{12}$
Take one whole out of the 5, so the whole number goes down to 4. Write that whole as 12 twelfths and add it to the 4 twelfths. The amount is still the same.
$5\frac{4}{12} = 4 + \frac{12}{12} + \frac{4}{12} = 4\frac{16}{12}$
Subtract the twelfths: 16 twelfths take away 9 twelfths leaves 7 twelfths. Subtract the wholes: 4 take away 2 leaves 2. Put the two parts together.
$4\frac{16}{12} - 2\frac{9}{12} = 2\frac{7}{12}$

Answer

$5\frac{1}{3} - 2\frac{3}{4} = 2\frac{7}{12}$ , so the difference is $2\frac{7}{12}$ .

There is a second way to do the same subtraction.

An improper fraction is a fraction whose numerator (the top number) is greater than or equal to its denominator, such as $\frac{5}{4}$ or $\frac{4}{4}$ . You can write any mixed number as an improper fraction that names the same amount.

Take $5\frac{1}{3}$ . It is 5 wholes and 1 third. Each whole is 3 thirds, so 5 wholes are $5 \times 3 = 15$ thirds. One more third makes 16 thirds. So $5\frac{1}{3} = \frac{16}{3}$ .

When both numbers are improper fractions, there is no whole-number part to rename. You subtract two fractions. In the next example, you then write the answer as a mixed number again.

Worked example

The same subtraction with improper fractions

Find $5\frac{1}{3} - 2\frac{3}{4}$ again. This time, write both mixed numbers as improper fractions first. Both numbers refer to wholes of the same size.

Write each mixed number as an improper fraction. 5 wholes are 15 thirds, and 1 more third makes 16 thirds. 2 wholes are 8 fourths, and 3 more fourths make 11 fourths.
$5\frac{1}{3} = \frac{16}{3} \qquad 2\frac{3}{4} = \frac{11}{4}$
Write both fractions as twelfths, so the pieces are the same size. Multiply the numerator and the denominator of the first fraction by 4, and those of the second fraction by 3. Each fraction gets a new name but keeps its amount.
$\frac{16}{3} = \frac{64}{12} \qquad \frac{11}{4} = \frac{33}{12}$
Subtract the numerators: 64 take away 33 leaves 31. The denominator stays 12, because the pieces are still twelfths.
$\frac{64}{12} - \frac{33}{12} = \frac{31}{12}$
Write the answer as a mixed number. 12 twelfths make one whole, so 24 twelfths make 2 wholes. That leaves 7 twelfths.
$\frac{31}{12} = \frac{24}{12} + \frac{7}{12} = 2\frac{7}{12}$

Answer

With improper fractions, $5\frac{1}{3} - 2\frac{3}{4} = 2\frac{7}{12}$ . This is the same answer that renaming a whole gave.

Two methods for the same subtraction

What to compare	Rename a whole	Improper fractions
How you start	Write twelfths, then rename $5\frac{4}{12}$ as $4\frac{16}{12}$	Write $\frac{16}{3}$ and $\frac{11}{4}$ , then write them as $\frac{64}{12}$ and $\frac{33}{12}$
What you subtract	$\frac{16}{12} - \frac{9}{12}$ and $4 - 2$	$\frac{64}{12} - \frac{33}{12}$
How you finish	Put the two parts together	Change $\frac{31}{12}$ back to a mixed number
Where to take care	Lower the whole number by 1 when you rename	Work carefully with the larger numerators
Answer	$2\frac{7}{12}$	$2\frac{7}{12}$

Both methods are correct when the two mixed numbers refer to the same whole, or to wholes of equal size. They give the same answer because each one only gives the numbers new names. Neither one changes an amount.

In this example, renaming a whole kept the numbers small. The improper fractions needed no renaming, but the numerators were larger, and you had to change the answer back to a mixed number.

You can choose either method. You can also use one method to check the other. If your two answers are different amounts, at least one of them has a mistake.

A common mistake: flipping the fraction parts. Take $6\frac{1}{5} - 2\frac{4}{5}$ , where both numbers refer to wholes of the same size. Some learners see that $\frac{1}{5}$ is too small, so they subtract the smaller fraction from the larger one: $\frac{4}{5} - \frac{1}{5} = \frac{3}{5}$ . With $6 - 2 = 4$ , they write $4\frac{3}{5}$ .

This fails because the $\frac{4}{5}$ is part of the amount you take away. All of it must come out of the first number. The first fraction part is 3 fifths short, so those 3 fifths must come out of a whole. The mistaken answer adds them on instead.

To fix it, rename a whole: $6\frac{1}{5} = 5\frac{6}{5}$ . Then $5\frac{6}{5} - 2\frac{4}{5} = 3\frac{2}{5}$ .

Worked example

Find and fix a renaming mistake

Sam works out $7\frac{1}{6} - 3\frac{5}{6}$ . Both numbers refer to wholes of the same size. Sam writes $7\frac{1}{6} = 7\frac{7}{6}$ , and then $7\frac{7}{6} - 3\frac{5}{6} = 4\frac{2}{6} = 4\frac{1}{3}$ . Find Sam's mistake and correct it.

Read Sam's work one line at a time. Start with the renaming, which must keep the amount the same. 7 and 7 sixths is 7 wholes, plus one more whole, plus 1 sixth. That is 8 and 1 sixth, not 7 and 1 sixth. So Sam's first line is wrong.
$7\frac{7}{6} = 7 + \frac{6}{6} + \frac{1}{6} = 8\frac{1}{6}$
Say what went wrong. Sam added 6 sixths to the fraction part but left the whole number at 7. The whole that became 6 sixths is counted twice, so Sam's first number is one whole too big.
Rename correctly. Take one whole out of the 7, so the whole number goes down to 6. Write that whole as 6 sixths and add it to the 1 sixth.
$7\frac{1}{6} = 6 + \frac{6}{6} + \frac{1}{6} = 6\frac{7}{6}$
Subtract the sixths: 7 sixths take away 5 sixths leaves 2 sixths. Subtract the wholes: 6 take away 3 leaves 3. Then write 2 sixths as 1 third, which is the same amount.
$6\frac{7}{6} - 3\frac{5}{6} = 3\frac{2}{6} = 3\frac{1}{3}$

Answer

Sam renamed a whole but did not lower the whole number, so Sam's answer was one whole too big. The correct difference is $3\frac{1}{3}$ , not $4\frac{1}{3}$ .

Takeaway. When the first fraction part is too small, do not flip the fractions. Rename one whole: the whole number goes down by 1, the fraction part goes up by one whole, and the amount stays the same. Or write both mixed numbers as improper fractions and subtract. Both methods give the same answer when the numbers refer to the same whole, or to wholes of equal size.

To find a mistake, ask two questions. Did all of the second number come out of the first number? Does each renamed number name the same amount as before?

Next, you will find a missing mixed number in an addition or subtraction sentence, and you will solve word problems with mixed numbers.

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Missing Mixed Numbers and Word Problems

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