5th Grade · 5.NF.B.7

Dividing whole numbers by fractions in 5th grade

Dividing whole numbers by fractions is new in 5th grade. It asks how many pieces of one size fit into an amount, such as how many half-cup scoops are in 6 cups. Your child also learns the reverse kind of problem, sharing a small fraction equally, and both kinds stay simple this year.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
5.NF.B.7.a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
5.NF.B.7.b Interpret division of a whole number by a unit fraction, and compute such quotients.
5.NF.B.7.c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

What it means, in plain words

In 5th grade, all fraction division uses unit fractions. A unit fraction is a fraction with 1 on top, such as $\frac{1}{3}$ or $\frac{1}{5}$ . Children divide a whole number by a unit fraction, and divide a unit fraction by a whole number. Dividing one fraction by another, such as $\frac{2}{3} \div \frac{3}{4}$ , comes in 6th grade.

Dividing whole numbers by fractions asks how many pieces fit. $3 \div \frac{1}{4}$ means "how many quarters fit into 3?" Each whole holds 4 quarters, so 3 wholes hold 12. The answer is bigger than the starting number, which surprises many children. Multiplication checks it: $12 \times \frac{1}{4} = 3$ .

Dividing unit fractions by whole numbers asks how big each share is. $\frac{1}{2} \div 3$ means "split one half into 3 equal shares". Each share is $\frac{1}{6}$ of the whole, and $3 \times \frac{1}{6} = \frac{1}{2}$ checks it. In a word problem, the first question is which kind it is: counting pieces, or finding the size of one share.

Your child should be able to:

Divide a whole number by a unit fraction, such as $2 \div \frac{1}{6}$
Divide a unit fraction by a whole number, such as $\frac{1}{5} \div 3$
Draw a bar or a circle that shows what a division means
Check a division answer with multiplication

Example

A rope 4 yards long is cut into pieces that are each $\frac{1}{5}$ yard long. How many pieces are there?

The question asks how many $\frac{1}{5}$ -yard pieces fit into 4 yards: $4 \div \frac{1}{5}$ .
Each yard holds 5 fifths.
4 yards hold $4 \times 5 = 20$ fifths.
Check: $20 \times \frac{1}{5} = 4$ .
The rope makes 20 pieces.

Example

Four friends share $\frac{1}{3}$ of a pan of brownies equally. What fraction of the whole pan does each friend get?

The question asks for the size of one share: $\frac{1}{3} \div 4$ .
Cut each third of the pan into 4 equal pieces. The whole pan now has $3 \times 4 = 12$ pieces.
One share is 1 of those pieces, $\frac{1}{12}$ of the pan.
Check: $4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$ .
Each friend gets one twelfth of the pan.

Where children usually get stuck

For $4 \div \frac{1}{2}$ , your child draws 4 circles, cuts the group in half and writes 2.
Dividing by $\frac{1}{2}$ got mixed up with dividing by 2. Taking half of 4 and counting the halves in 4 are different questions.
What helps. Read it aloud as "how many halves are in 4?" Draw 4 circles, cut each one in half and count the halves together. There are 8.
Your child works out $\frac{1}{2} \div 4$ as 8.
The child swapped the order and found $4 \div \frac{1}{2}$ instead. With whole numbers, children are used to dividing the bigger number by the smaller one.
What helps. Fold a paper strip in half, then fold one half into 4 equal parts and unfold it. Each small piece is $\frac{1}{8}$ of the strip. A share of half a strip cannot be 8 strips.
A 2-mile path is split into $\frac{1}{3}$ -mile sections. Your child writes $2 \div \frac{1}{3} = 6$ and answers "6 miles".
The arithmetic is right, but the child lost track of what the number counts. The 6 counts sections that are each one third of a mile long, not miles.
What helps. After every answer, ask "6 what?" and have your child finish the sentence with the unit from the problem.

Try four practice questions

Real 5th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Lena has $\frac{1}{2}$ yard of fabric. She cuts it into $2$ equal pieces.
Use the dropdowns to complete the sentences.
The quotient is
. It describes
.
Show the answer and how to solve it
The correct completions are 'an amount' and 'one share' because dividing $\frac{1}{2}$ yard into $2$ equal pieces tells the size of one piece.
1. 1The story starts with $\frac{1}{2}$ yard of fabric being cut into $2$ equal pieces.
2. 2When a unit fraction is split equally, the quotient tells how much is in one piece — it is an amount.
3. 3So the quotient is an amount, and it describes one share.
4. 4Choose 'an amount' for the first dropdown and 'one share' for the second dropdown.
Common mistake. A common mistake is choosing 'a count' because the story mentions $2$ pieces. But $2$ is already given; the quotient tells how much fabric is in one piece.
The idea underneath. Division stories can ask for either the size of one equal part or the number of equal-size parts, and the action in the story tells which meaning the quotient has.
2On level
A class has 8 cups of trail mix. They pack it into small bags that each hold $\frac{1}{4}$ cup.
Marcus's claim: "The quotient tells how many $\frac{1}{4}$ -cup bags of trail mix can be filled." Chloe's claim: "The quotient tells the total amount of trail mix, which is 8 cups."
Select the statement that best evaluates the students' claims.
A
Only Marcus is correct
B
Only Chloe is correct
C
Both students are correct
D
Neither student is correct
Show the answer and how to solve it
Only Marcus is correct because the quotient tells how many $\frac{1}{4}$ -cup bags can be filled from $8$ cups of trail mix.
1. 1The story gives $8$ cups of trail mix and says each bag holds $\frac{1}{4}$ cup. The quotient answers how many bags fit into the total.
2. 2One whole cup fills $4$ quarter-cup bags, since $4 \times \frac{1}{4} = 1$ .
3. 3So $8$ cups fill $8 \times 4 = 32$ quarter-cup bags. The quotient is a count of bags, not the starting amount of trail mix.
4. 4Marcus correctly describes that count. Chloe restates the $8$ cups already given in the story — that is the starting amount, not what the division produces.
Common mistake. A common mistake is thinking the quotient restates the starting amount because the story names $8$ cups. But $8$ cups is already given; the quotient tells how many bags of that size can be filled.
The idea underneath. When a whole amount is divided into unit-fraction pieces, the quotient tells how many pieces of that size fit in the whole.
3On level
A cook cuts 1 tray of cornbread into pieces that each hold $\frac{1}{4}$ of a tray. The tape diagram below shows the pieces.
Select what the quotient tells in this situation.
A
The quotient tells how much is in each share.
B
The quotient tells how many shares there are.
Show the answer and how to solve it
The correct choice is "The quotient tells how many shares there are." because each piece is already $\frac{1}{4}$ of a tray, and the division tells how many pieces fit in 1 tray.
1. 1The brace over the tape diagram shows a total of 1 tray of cornbread.
2. 2Each small segment in the diagram is $\frac{1}{4}$ of a tray, so the size of one piece is already given.
3. 3That means the quotient tells how many $\frac{1}{4}$ -tray pieces can be cut from 1 tray — it counts the pieces.
Common mistake. A common mistake is choosing "how much is in each share" just because $\frac{1}{4}$ is named in the story. Here $\frac{1}{4}$ is already the size of each piece, so the quotient must tell how many pieces there are.
The idea underneath. When the story already gives the size of each share, the quotient tells how many shares fit; when it already gives the number of shares, the quotient tells how much is in each share.
4Stretch
Lena has $\frac{1}{4}$ yard of fabric. She cuts it into $6$ equal strips.
Use the dropdowns to complete the sentences.
The quotient is
. It describes
.
Show the answer and how to solve it
The correct completions are 'an amount' and 'one share' because dividing $\frac{1}{4}$ yard equally into $6$ strips tells the length of one strip.
1. 1The story starts with $\frac{1}{4}$ yard of fabric, and that fabric is being cut into $6$ equal strips.
2. 2When a unit fraction amount is split equally, the quotient tells how much is in one piece — that is an amount, not a count.
3. 3So the quotient is an amount, and it describes one share — the length of one strip.
4. 4Choose 'an amount' for the first dropdown and 'one share' for the second dropdown.
Common mistake. A common mistake is choosing 'a count' because the story mentions $6$ strips. But $6$ strips is already given; the quotient tells how much fabric is in one strip.
The idea underneath. Division stories can ask for either the size of one equal part or the number of equal-size parts, and the action in the story tells which meaning the quotient has.