Add and Subtract Mixed Numbers with Like Denominators | Grade 5 Math | LevelKite

You already know how to add and subtract fractions with like denominators. Like denominators means the bottom numbers are the same, so each whole has the same number of equal parts. In this lesson you add and subtract mixed numbers too.

A mixed number is a whole number and a fraction written together. For example, $2\frac{3}{5}$ means 2 wholes and 3 fifths of one more whole. In symbols, $2\frac{3}{5} = 2 + \frac{3}{5}$ . The 2 is the whole-number part. The $\frac{3}{5}$ is the fraction part.

One thing must be true before you add or subtract fractions or mixed numbers as amounts. They must refer to the same whole, or to wholes of equal size. Fifths of two same-size pizzas are the same size, so you can count them together. Fifths of a small pizza and fifths of a large pizza are different sizes, so you cannot.

Think of a mixed number as two counts: a count of wholes and a count of parts. $1\frac{1}{5}$ is 1 whole and 1 fifth. $2\frac{3}{5}$ is 2 wholes and 3 fifths.

When both numbers refer to the same whole, or to wholes of equal size, and have like denominators, you can add count by count. Wholes join wholes: $1 + 2 = 3$ . Fifths join fifths: $\frac{1}{5} + \frac{3}{5} = \frac{4}{5}$ . So $1\frac{1}{5} + 2\frac{3}{5} = 3\frac{4}{5}$ . This works because joining amounts in a different order does not change the total. That total is the sum.

Sometimes the fraction parts add up to one whole or more. Then you regroup. To regroup means to trade a full set of parts for 1 whole. With fifths, a full set is 5 fifths, because $\frac{5}{5} = 1$ .

Worked example

Add and regroup

Add $2\frac{3}{5} + 1\frac{4}{5}$ . Both mixed numbers refer to the same whole.

Add the wholes. The wholes are the same size, so you can count them together. 2 wholes and 1 whole make 3 wholes.
$2 + 1 = 3$
Add the fraction parts. The denominators are alike and the whole is the same, so the fifths are the same size. 3 fifths and 4 fifths make 7 fifths.
$\frac{3}{5} + \frac{4}{5} = \frac{7}{5}$
Regroup, because 7 fifths is more than one whole. One whole is 5 fifths. Trade 5 of the fifths for 1 whole. 2 fifths are left over.
$\frac{7}{5} = \frac{5}{5} + \frac{2}{5} = 1\frac{2}{5}$
Put everything together. You had 3 wholes. Regrouping gave you 1 more whole and 2 fifths.
$3 + 1\frac{2}{5} = 4\frac{2}{5}$

Answer

The sum of $2\frac{3}{5}$ and $1\frac{4}{5}$ is $4\frac{2}{5}$ .

When do you regroup? Three sums in sevenths of the same whole

Add	Wholes	Fraction parts	A whole or more?	Sum
$2\frac{2}{7} + 1\frac{3}{7}$	$2 + 1 = 3$	$\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$	No. 5 sevenths is less than 7 sevenths, so you do not regroup.	$3\frac{5}{7}$
$2\frac{3}{7} + 1\frac{4}{7}$	$2 + 1 = 3$	$\frac{3}{7} + \frac{4}{7} = \frac{7}{7}$	Yes, exactly one whole: $\frac{7}{7} = 1$ .	$3 + 1 = 4$
$2\frac{5}{7} + 1\frac{4}{7}$	$2 + 1 = 3$	$\frac{5}{7} + \frac{4}{7} = \frac{9}{7}$	Yes, with 2 sevenths left over: $\frac{9}{7} = 1\frac{2}{7}$ .	$3 + 1\frac{2}{7} = 4\frac{2}{7}$

Subtraction uses the same two counts. The answer to a subtraction is the difference. When both mixed numbers refer to the same whole, or to wholes of equal size, and have like denominators, take wholes from wholes and parts from parts. Try $4\frac{2}{3} - 1\frac{1}{3}$ . Wholes: $4 - 1 = 3$ . Thirds: $\frac{2}{3} - \frac{1}{3} = \frac{1}{3}$ . The difference is $3\frac{1}{3}$ . That worked because you had enough thirds to take from.

Now swap the fraction parts: $4\frac{1}{3} - 1\frac{2}{3}$ . You have only 1 third, and you must take away 2 thirds. The fraction part is too small. You can still subtract, because $4\frac{1}{3}$ is more than $1\frac{2}{3}$ . The thirds you need are inside the wholes.

The fix is to rename. To rename a number means to write the same amount in a different way. Here you open one whole into 3 thirds, because $1 = \frac{3}{3}$ . In this lesson the number you start with is never smaller than the number you take away, so there is always a whole to open. The table shows three mixed numbers renamed this way, in words and in symbols.

One whole opened into parts: the amount stays the same

Mixed number	In words	Open one whole	Renamed
$4\frac{1}{3}$	4 wholes and 1 third	$3 + \frac{3}{3} + \frac{1}{3}$	$3\frac{4}{3}$ , which is 3 wholes and 4 thirds
$2\frac{1}{4}$	2 wholes and 1 fourth	$1 + \frac{4}{4} + \frac{1}{4}$	$1\frac{5}{4}$ , which is 1 whole and 5 fourths
$5\frac{3}{8}$	5 wholes and 3 eighths	$4 + \frac{8}{8} + \frac{3}{8}$	$4\frac{11}{8}$ , which is 4 wholes and 11 eighths

Worked example

Subtract by renaming a whole

Subtract $4\frac{1}{3} - 1\frac{2}{3}$ . Both mixed numbers refer to the same whole.

Compare the fraction parts. Both are thirds of the same whole. You have 1 third and you must take away 2 thirds. 1 third is too small, so you need to rename.
$\frac{1}{3} < \frac{2}{3}$
Rename one of the 4 wholes as 3 thirds, because one whole cut into thirds is 3 thirds. Join them with the 1 third you already have. The amount stays the same. Only the way you write it changes.
$4\frac{1}{3} = 3 + \frac{3}{3} + \frac{1}{3} = 3\frac{4}{3}$
Subtract the wholes. After renaming you have 3 wholes. Take away 1 whole and 2 wholes are left.
$3 - 1 = 2$
Subtract the fraction parts. Now you have enough thirds. 4 thirds take away 2 thirds leaves 2 thirds.
$\frac{4}{3} - \frac{2}{3} = \frac{2}{3}$
Put the two counts together. You have 2 wholes and 2 thirds.
$2 + \frac{2}{3} = 2\frac{2}{3}$

Answer

The difference $4\frac{1}{3} - 1\frac{2}{3}$ is $2\frac{2}{3}$ .

Key concept

Wholes with wholes, parts with parts

A mixed number is a count of wholes plus a count of equal parts. When two mixed numbers have like denominators and refer to the same whole, or to wholes of equal size, you can add or subtract the wholes and the parts separately.

One whole and a full set of parts are the same amount, such as $1 = \frac{5}{5}$ or $1 = \frac{3}{3}$ . So you can trade one for the other:

Regroup when the parts add up to a whole or more: $\frac{7}{5} = 1\frac{2}{5}$ .
Rename a whole when you have too few parts to subtract from: $4\frac{1}{3} = 3\frac{4}{3}$ .

A trade changes how the number is written. It does not change the amount.

A word problem puts the amounts inside a story. Ask two questions before you compute.

Do the amounts refer to the same whole, or to wholes of equal size? Amounts in the same unit do. Every mile is the same length, and every measuring cup holds the same amount. Parts of different-size things do not. $\frac{1}{4}$ of a small pizza and $\frac{1}{4}$ of a large pizza do not make $\frac{2}{4}$ of either pizza, because the fourths are different sizes. Add or subtract the numbers only when the answer is yes.
What does the story do? If the story joins amounts, add. If the story takes an amount away, or compares two amounts to find how much more one is, subtract.

Then compute, and answer in a sentence with the unit. The table shows four short stories. Look at the last row: an answer can be 0.

Four short stories about amounts of the same whole

Story	Same whole?	What the story does	Number sentence
You pour $\frac{1}{3}$ cup of milk into an empty bowl, then $\frac{1}{3}$ cup more. How much milk is in the bowl?	Yes. Both are parts of one cup.	It joins two amounts, so add.	$\frac{1}{3} + \frac{1}{3} = \frac{2}{3}$ cup
A ribbon is $\frac{7}{8}$ yard long. You cut off $\frac{2}{8}$ yard. How much ribbon is left?	Yes. Both are parts of one yard.	It takes an amount away, so subtract.	$\frac{7}{8} - \frac{2}{8} = \frac{5}{8}$ yard
Kai walks $\frac{7}{10}$ mile. Zoe walks $\frac{4}{10}$ mile. How much farther does Kai walk?	Yes. Both are parts of one mile.	It compares two amounts, so subtract.	$\frac{7}{10} - \frac{4}{10} = \frac{3}{10}$ mile
You have $\frac{3}{4}$ cup of juice. You drink $\frac{3}{4}$ cup. How much juice is left?	Yes. Both are parts of one cup.	It takes an amount away, so subtract.	$\frac{3}{4} - \frac{3}{4} = 0$ cups

Worked example

A two-step trail problem

A trail is $6\frac{2}{5}$ miles long. Ana hikes $1\frac{4}{5}$ miles of the trail before lunch and $2\frac{4}{5}$ miles more after lunch. How many miles of the trail does she have left?

Check the whole and make a plan. All three amounts are measured in miles, so they refer to the same whole, one mile. All are in fifths, so the denominators are alike. You can add and subtract them. First add the two hikes to find how far Ana has gone. Then subtract that from the length of the trail.
Add the two hikes. 1 whole and 2 wholes make 3 wholes. 4 fifths and 4 fifths make 8 fifths.
$1\frac{4}{5} + 2\frac{4}{5} = 3\frac{8}{5}$
Regroup, because 8 fifths is more than one whole. Trade 5 fifths for 1 whole. 3 fifths are left over. Now you have 4 wholes and 3 fifths, so Ana has hiked 4 and 3 fifths miles.
$3\frac{8}{5} = 3 + \frac{5}{5} + \frac{3}{5} = 4\frac{3}{5}$
Next, subtract what Ana has hiked from the length of the trail. The trail has only 2 fifths in its fraction part, and you must take away 3 fifths. So rename one of the 6 wholes as 5 fifths.
$6\frac{2}{5} = 5 + \frac{5}{5} + \frac{2}{5} = 5\frac{7}{5}$
Subtract. 5 wholes take away 4 wholes leaves 1 whole. 7 fifths take away 3 fifths leaves 4 fifths.
$5\frac{7}{5} - 4\frac{3}{5} = 1\frac{4}{5}$

Answer

Ana has $1\frac{4}{5}$ miles of the trail left.

Here is the idea to keep. A mixed number is wholes plus parts. When two mixed numbers have like denominators and refer to the same whole, or to wholes of equal size, you add or subtract the wholes and the parts separately. If the parts add up to a whole or more, regroup them into a whole. If the fraction part you have is too small to subtract from, rename one whole as parts. In a word problem, first check that the amounts refer to the same whole. Then decide whether the story joins amounts, takes an amount away or compares two amounts.