Chapter 9 · Divide fractions

Add, Subtract, Multiply and Divide Fractions

Mix all four fraction operations. Decide when you need a common denominator, choose a way to work with mixed numbers, and estimate first to catch wrong-size answers.

16 min read

You already know how to add, subtract, multiply and divide fractions. So far, each practice set used just one of these four operations. In this lesson they are mixed together, so the method can change from one problem to the next.

That means you have to stop and look at each problem before you start. You will learn to:

say which operations need pieces of the same size, and why
choose a good way to work with numbers that have a whole part and a fraction part, such as $2\frac{1}{2}$
guess the size of an answer before you compute, so that a wrong-size answer stands out

Start with what a fraction tells you. The denominator is the bottom number. It tells you the size of the pieces: fourths are bigger pieces than twelfths. The numerator is the top number. It counts how many pieces there are.

When you add or subtract, you count pieces together or take pieces away. You can only do that when the pieces are the same size. First you rename both fractions to give them the same denominator, called a common denominator. Renaming means writing the same amount in a different way, such as $\frac{2}{3} = \frac{8}{12}$ . The pieces are smaller, but there are more of them.

Multiplying and dividing do a different job. When you multiply, you take a part of a part. When you divide a fraction by a whole number, you split it into equal shares. Both of these cut the pieces into new, smaller pieces. You are not counting two sets of pieces together, so the pieces do not have to match. You do not need a common denominator.

The table shows all four operations at work. The first three rows use the same two fractions, so you can see how much the operation matters.

The four operations side by side

Operation	What happens, in words	Common denominator needed?	Number sentence
Add	Count the pieces together: 8 twelfths and 3 twelfths make 11 twelfths.	Yes	$\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$
Subtract	Take pieces away: 3 twelfths taken from 8 twelfths leaves 5 twelfths.	Yes	$\frac{2}{3} - \frac{1}{4} = \frac{8}{12} - \frac{3}{12} = \frac{5}{12}$
Multiply	Take two thirds of one fourth: cut the fourth into 3 equal pieces and keep 2. That is 2 twelfths, or 1 sixth.	No	$\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}$
Divide	Share one fourth 3 ways: each share is one twelfth.	No	$\frac{1}{4} \div 3 = \frac{1}{12}$ , because $3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}$

Key concept

The operation tells you the method

Look at the operation sign before you do anything else. Adding and subtracting count pieces, so the pieces must be the same size: rename with a common denominator first. Multiplying and dividing make new pieces, so you do not need a common denominator.

Worked example

A subtraction from a mixed practice set

In a mixed practice set you meet $\frac{5}{6} - \frac{3}{4}$ . Do you need a common denominator? Find the answer.

Look at the sign first. This is subtraction, and subtraction takes pieces away. Sixths and fourths are pieces of different sizes, so you do need a common denominator.
Pick a denominator that works for both fractions. 12 is in the 6 times table and in the 4 times table, so sixths and fourths can both be renamed as twelfths.
$6 \times 2 = 12 \qquad 4 \times 3 = 12$
Rename each fraction as twelfths. Multiply the top and bottom of five sixths by 2, and the top and bottom of three fourths by 3. Multiplying top and bottom by the same number keeps the amount the same. Only the size of the pieces changes.
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \qquad \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
Now the pieces match, so subtract the numerators. Taking 9 twelfths from 10 twelfths leaves 1 twelfth. The denominator stays 12 because the pieces are still twelfths.
$\frac{10}{12} - \frac{9}{12} = \frac{1}{12}$

Answer

Yes, subtraction needs a common denominator, and $\frac{5}{6} - \frac{3}{4}$ is $\frac{1}{12}$ .

The same thinking carries over to mixed numbers. A mixed number is a whole number and a fraction together, such as $2\frac{1}{2}$ . There are two ways to work with mixed numbers.

Work with the parts. Handle the whole numbers and the fraction parts separately, then put the results together.

Rename as improper fractions. An improper fraction has a numerator that is equal to or greater than its denominator, such as $\frac{5}{2}$ . To rename $2\frac{1}{2}$ , think of the 2 wholes as 4 halves, then count 1 more half. That makes 5 halves: $2\frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$ .

You already know both ways. The new skill is choosing between them. The best choice depends on the operation and on the numbers.

Worked example

Adding mixed numbers by working with the parts

Find $2\frac{1}{2} + 1\frac{1}{3}$ . Choose a way to work before you start.

Choose the way. This is addition, and adding counts things together. Wholes can be counted with wholes, and fraction pieces with fraction pieces, so work with the parts. Start with the whole numbers.
$2 + 1 = 3$
Now look at the fraction parts. Halves and thirds are different sizes, and adding needs same-size pieces. 6 is in the 2 times table and in the 3 times table, so rename both as sixths. One half is 3 sixths and one third is 2 sixths.
$\frac{1}{2} = \frac{3}{6} \qquad \frac{1}{3} = \frac{2}{6}$
The pieces match now, so add the numerators. 3 sixths and 2 sixths make 5 sixths.
$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$
Put the two parts back together. 3 wholes and 5 sixths make three and five sixths.
$3 + \frac{5}{6} = 3\frac{5}{6}$

Answer

Adding $2\frac{1}{2}$ and $1\frac{1}{3}$ gives $3\frac{5}{6}$ . Working with the parts fits addition well.

So which way should you choose? Both can lead to the right answer. Pick the one with fewer chances to make a mistake, and let the operation and the numbers guide you.

Adding: work with the parts. Wholes and fraction pieces can each be counted on their own.
Subtracting: compare the two fraction parts first. If the first one is the same size or bigger, work with the parts. If it is smaller, the fraction part alone does not have enough to take away from, so renaming is easier. The table below shows both cases.
Multiplying: rename as improper fractions. Every part of one number has to multiply every part of the other, and parts are easy to lose. The last example in this lesson shows how.

Two subtractions, two different choices

Problem	Compare the fraction parts	Way that fits	Number sentence
$2\frac{1}{2} - 1\frac{1}{3}$	$\frac{1}{2}$ is more than $\frac{1}{3}$ , so there is enough to take away from.	Work with the parts	$2 - 1 = 1$ and $\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}$ , so $1\frac{1}{6}$
$2\frac{1}{3} - 1\frac{1}{2}$	$\frac{1}{3}$ is less than $\frac{1}{2}$ , so the fraction part alone does not have enough.	Rename as improper fractions: $2\frac{1}{3} = \frac{7}{3}$ and $1\frac{1}{2} = \frac{3}{2}$	$\frac{7}{3} - \frac{3}{2} = \frac{14}{6} - \frac{9}{6} = \frac{5}{6}$

One more habit keeps mixed practice safe: estimate first. An estimate is a quick, sensible guess at the size of an answer. Make it before you compute. Then if your answer is far from your estimate, you know to go back and look for a mistake.

To estimate, notice roughly how big each number is. Then think about what the operation does to the size:

Adding gives more than either number you add.
Subtracting gives less than the number you started with.
Multiplying by a fraction less than $1$ takes only a part, so you get less than you started with. Multiplying by a number greater than $1$ gives more.
Dividing by a whole number greater than $1$ shares the amount out, so each share is less than the amount you started with.

Test this on answers you already have. In $\frac{5}{6} - \frac{3}{4}$ , both fractions are close to $1$ , so the answer should be small. The answer $\frac{1}{12}$ fits. In $2\frac{1}{2} + 1\frac{1}{3}$ , the wholes make $3$ and the fraction parts together make less than $1$ , so the answer should be between $3$ and $4$ . The answer $3\frac{5}{6}$ fits.

Watch out. Working with the parts is fine for adding, so it is tempting to multiply the same way: $2\frac{1}{2} \times 1\frac{1}{3}$ becomes $2 \times 1 = 2$ and $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ , giving $2\frac{1}{6}$ .

An estimate shows the size is wrong. Multiplying $2\frac{1}{2}$ by a number greater than $1$ must give more than $2\frac{1}{2}$ , but $2\frac{1}{6}$ is less.

Here is why it fails. $2\frac{1}{2} \times 1\frac{1}{3}$ means 2 groups of $1\frac{1}{3}$ and half a group more. The 2 groups need $2 \times 1$ and $2 \times \frac{1}{3}$ . The half group needs $\frac{1}{2} \times 1$ and $\frac{1}{2} \times \frac{1}{3}$ . The shortcut does only two of these four multiplications. Renaming as improper fractions leaves nothing out.

Worked example

Multiplying mixed numbers: estimate, rename, check

Find $2\frac{1}{2} \times 1\frac{1}{3}$ . Estimate first.

Estimate first. Two and a half times 1 is two and a half. Two and a half times 2 is 5. One and one third is between 1 and 2, so the answer should be between two and a half and 5.
$2\frac{1}{2} \times 1 = 2\frac{1}{2} \qquad 2\frac{1}{2} \times 2 = 5$
Choose the way. This is multiplication, so rename both numbers as improper fractions. 2 wholes are 4 halves, and 1 more half makes 5 halves. 1 whole is 3 thirds, and 1 more third makes 4 thirds.
$2\frac{1}{2} = \frac{5}{2} \qquad 1\frac{1}{3} = \frac{4}{3}$
Multiplying makes new pieces, so you do not need a common denominator. Multiply the denominators to find the new piece size: 2 times 3 is 6, so the pieces are sixths. Multiply the numerators to count the pieces: 5 times 4 is 20.
$\frac{5}{2} \times \frac{4}{3} = \frac{5 \times 4}{2 \times 3} = \frac{20}{6}$
Rename 20 sixths as a mixed number. 18 sixths make 3 wholes, and 2 sixths are left over. 2 sixths is the same amount as 1 third.
$\frac{20}{6} = 3\frac{2}{6} = 3\frac{1}{3}$
Check the size. Three and one third is between two and a half and 5, so it fits the estimate.

Answer

Multiplying $2\frac{1}{2}$ by $1\frac{1}{3}$ gives $3\frac{1}{3}$ , which fits the estimate of between $2\frac{1}{2}$ and $5$ .

Here is what to take away. The operation tells you the method. Adding and subtracting count pieces, so they need a common denominator. Multiplying and dividing make new pieces, so they do not. With mixed numbers, work with the parts when you add, compare the fraction parts when you subtract, and rename as improper fractions when you multiply. Estimate first, every time, so that a wrong-size answer stands out.

Next, you will use these skills in word problems, where you first have to decide which operation the situation calls for.

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