Chapter 9 · Divide fractions

Multi-Step Fraction Division: Does the Answer Make Sense?

Solve story problems with more than one step that include dividing with a unit fraction, test whether a quotient makes sense, and find and fix common mistakes.

14 min read

You already know how to solve a story problem that has one fraction division in it. Now you will use that skill inside bigger problems. A multi-step problem is a problem that needs more than one calculation to reach the answer.

You will also learn a quick test that tells you whether an answer makes sense. Then you will use the test to find and fix mistakes.

Here are the words you need:

A whole number is one of the numbers 0, 1, 2, 3 and so on.
A unit fraction is a fraction whose top number, the numerator, is 1. Examples are $\frac{1}{2}$ , $\frac{1}{3}$ and $\frac{1}{8}$ . Every unit fraction in this lesson is less than 1.
In $6 \div \frac{1}{2} = 12$ , the dividend is $6$ . It is the amount that gets divided.
The divisor is $\frac{1}{2}$ . It is the number you divide by.
The quotient is $12$ . It is the answer to the division.

Every division with a fraction in this lesson has one of two forms.

A whole number divided by a unit fraction, like $3 \div \frac{1}{4}$ . In this lesson it asks: how many pieces of that size fit in the whole amount?
A unit fraction divided by a whole number that is not 0, like $\frac{1}{3} \div 4$ . In this lesson it asks: how big is each share when you split that piece into that many equal shares?

In a multi-step problem, the division is only one of the steps. So plan before you calculate:

First, read the whole story. What does the question ask for?
Next, find the amount that gets cut up or shared. That amount is the dividend.
Then find what you divide by. It is the size of each piece or the number of equal shares. That is the divisor.
Last, see whether another step, such as adding or subtracting, must come first to find the dividend or the divisor.

Worked example

Subtract first, then divide

Maya has $5$ yards of ribbon. She uses $2$ yards to wrap a gift. She cuts the rest into pieces that are each $\frac{1}{4}$ yard long. How many pieces does she cut?

Find how much ribbon is left, because Maya cuts only the ribbon she did not use. Take the 2 yards she used away from the 5 yards she had. That leaves 3 yards.
$5 - 2 = 3$
Pick the division. The 3 yards get cut up, so 3 is the dividend. Each piece is one fourth of a yard long, so one fourth is the divisor. You are asking how many of these pieces fit in 3 yards.
$3 \div \frac{1}{4}$
Divide. One yard holds 4 of these pieces, because 4 fourths make 1 whole. So 3 yards hold 3 groups of 4 pieces, which is 12 pieces.
$3 \div \frac{1}{4} = 3 \times 4 = 12$
Check with multiplication. The 12 pieces are each one fourth of a yard long, so together they should use all 3 yards. They do, because 12 fourths make 3 wholes.
$12 \times \frac{1}{4} = \frac{12}{4} = 3$

Answer

Maya cuts $12$ pieces that are each $\frac{1}{4}$ yard long.

An answer is reasonable when it makes sense for the story. You can test a quotient quickly: compare it with the dividend.

Think about Maya's ribbon. She cut $3$ yards into pieces that are each shorter than $1$ yard. Each yard gave her more than one piece. So the number of pieces had to be greater than $3$ . Her quotient, $12$ , fits.

Now think about sharing. Suppose $\frac{1}{2}$ of a sandwich is split equally between $2$ people. Each person gets only part of that half. So each share has to be less than $\frac{1}{2}$ . Each share is $\frac{1}{2} \div 2 = \frac{1}{4}$ of the sandwich, because $2$ fourths make one half. And $\frac{1}{4}$ is less than $\frac{1}{2}$ , so this quotient fits too.

In this lesson, a quotient can be greater than its dividend or less than it. The kind of division tells you which.

Key concept

Compare the quotient with the dividend

This test works for the two kinds of fraction division in this lesson.

A whole number greater than 0 divided by a unit fraction less than 1: the quotient is greater than the dividend. Each whole holds more than one piece of that size.
A unit fraction divided by a whole number greater than 1: the quotient is less than the dividend. Splitting a piece into 2 or more equal shares makes each share smaller than the piece.

If a quotient does not fit its pattern, look for a mistake. A quotient that fits can still be wrong, so also check with multiplication: the quotient times the divisor must equal the dividend.

Testing four quotients against their dividends

Division	What it asks in this lesson	Quotient should be	Quotient	Does it fit?
$5 \div \frac{1}{2}$	How many halves fit in $5$ ?	greater than $5$	$10$	Yes, $10$ is greater than $5$
$\frac{1}{2} \div 5$	How big is each share when $\frac{1}{2}$ is split into $5$ equal shares?	less than $\frac{1}{2}$	$\frac{1}{10}$	Yes, $\frac{1}{10}$ is less than $\frac{1}{2}$
$8 \div \frac{1}{3}$	How many thirds fit in $8$ ?	greater than $8$	$24$	Yes, $24$ is greater than $8$
$\frac{1}{4} \div 2$	How big is each share when $\frac{1}{4}$ is split into $2$ equal shares?	less than $\frac{1}{4}$	$\frac{1}{8}$	Yes, $\frac{1}{8}$ is less than $\frac{1}{4}$

Worked example

Count the shares, then divide

After a party, $\frac{1}{3}$ of a pan of lasagna is left. Ana shares it equally among herself and her $3$ brothers. What fraction of the whole pan does each person get?

Count the shares first. Ana gets a share and each of her 3 brothers gets a share, so there are 4 equal shares, not 3.
$1 + 3 = 4$
Pick the division. One third of the pan gets shared, so one third is the dividend. It is split into 4 equal shares, so 4 is the divisor.
$\frac{1}{3} \div 4$
Divide. Think of the whole pan as 3 equal thirds. Cutting each third into 4 equal parts makes 3 groups of 4 parts, which is 12 equal parts in the whole pan. One share is 1 of those 12 parts.
$\frac{1}{3} \div 4 = \frac{1}{12}$
Test the quotient. A unit fraction is divided by a whole number greater than 1, so the quotient should be less than the dividend. One twelfth is less than one third, because cutting a pan into 12 equal parts makes smaller parts than cutting it into 3. It fits.
$\frac{1}{12} < \frac{1}{3}$
Check with multiplication. The 4 shares of one twelfth should make the one third that was left. They do, because each third was cut into 4 twelfths.
$4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$

Answer

Each person gets $\frac{1}{12}$ of the whole pan of lasagna.

Now use the test to hunt for mistakes. Look at the first two rows of the table. Both use $5$ and $\frac{1}{2}$ , but $5 \div \frac{1}{2} = 10$ and $\frac{1}{2} \div 5 = \frac{1}{10}$ . When you divide with a whole number greater than 0 and a unit fraction less than 1, switching the order changes the answer.

To check a solved problem, yours or someone else's:

Read the story again. Decide which amount gets cut up or shared. That is the dividend.
Use the test to decide whether the quotient should be greater than or less than the dividend.
Compare. If the answer does not fit, look for two common mistakes. The numbers may be divided in the wrong order. Or they may be multiplied when the story divides.

Watch out: multiplying when the story divides. A string is $6$ feet long. You cut it into pieces that are each $\frac{1}{3}$ foot long. How many pieces do you get?

A common wrong answer is $6 \times \frac{1}{3} = 2$ pieces. It fails because $6 \times \frac{1}{3}$ finds one third of $6$ feet. That is $2$ feet, a length and not a count of pieces. And $2$ pieces would use only $\frac{2}{3}$ foot of string.

The story asks how many $\frac{1}{3}$ -foot pieces fit in $6$ feet, so divide. Each foot holds $3$ pieces, so $6 \div \frac{1}{3} = 6 \times 3 = 18$ pieces. The test agrees. A whole number greater than 0 is divided by a unit fraction less than 1, so the quotient must be greater than $6$ . The answer $18$ fits and $2$ does not.

Worked example

Find and fix the mistake

A baker has $6$ cups of sugar. She uses $2$ cups for frosting. Each batch of cookies needs $\frac{1}{3}$ cup of sugar. How many batches can she make with the sugar that is left? Sam writes $6 - 2 = 4$ and then $\frac{1}{3} \div 4 = \frac{1}{12}$ . He says she can make $\frac{1}{12}$ of a batch. Find and fix Sam's mistake.

Check Sam's first step. The cookies use only the sugar that is left. The baker had 6 cups and used 2 cups, so 4 cups are left. Sam's subtraction is correct.
$6 - 2 = 4$
Test Sam's quotient against the story. The 4 cups get split into batches of one third of a cup. That is a whole number greater than 0 divided by a unit fraction less than 1, so the quotient should be greater than 4. One twelfth is less than 4, so Sam's answer is not reasonable.
$\frac{1}{12} < 4$
Find the mistake. Sam divided in the wrong order. He split one third of a cup into 4 shares. The story asks how many one-third-cup batches fit in 4 cups. So 4 is the dividend and one third is the divisor.
$4 \div \frac{1}{3}$
Fix it. One cup holds 3 thirds, so 4 cups hold 4 groups of 3 thirds. That is 12 batches. The quotient 12 is greater than 4, so it passes the test.
$4 \div \frac{1}{3} = 4 \times 3 = 12$
Check with multiplication. The 12 batches each use one third of a cup, so together they should use all 4 cups. They do, because 12 thirds make 4 wholes.
$12 \times \frac{1}{3} = \frac{12}{3} = 4$

Answer

Sam divided in the wrong order. The correct division is $4 \div \frac{1}{3} = 12$ , so the baker can make $12$ batches of cookies.

Here is the idea to keep. In a multi-step problem, plan before you calculate. Find the amount that gets cut up or shared, find what you divide by, and do any other step the story needs. Then test your quotient against the dividend:

A whole number greater than 0 divided by a unit fraction less than 1 gives a quotient greater than the dividend.
A unit fraction divided by a whole number greater than 1 gives a quotient less than the dividend.

If an answer does not fit, check the order of the division, and check whether the story really divides. For both kinds of division in this lesson, confirm your answer with multiplication: the quotient times the divisor equals the dividend.

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