5th Grade · 5.NF.B.3

Fractions as division in 5th grade

Fractions as division is the idea that every fraction is also a division: $\frac{5}{8}$ means 5 divided by 8. In 5th grade, your child uses this idea to solve sharing problems where the answer is a fraction or a mixed number. When 5 pizzas are shared equally by 8 children, each child gets $\frac{5}{8}$ of a pizza.

What the standard says

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

5.NF.B.3 Interpret a fraction as division of the numerator by the denominator ( $\frac{a}{b}$ = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

What it means, in plain words

Until now, your child has read $\frac{5}{8}$ as 5 parts out of 8. In 5th grade, the same fraction gets a second meaning: 5 divided by 8. Reading a fraction as division works for every fraction: the top number divided by the bottom number. The fraction bar, the line between the two numbers, works like a division sign.

The two meanings fit together. Share 2 cakes equally among 3 people by cutting each cake into 3 equal pieces. That makes 6 pieces, so each person gets 2 pieces. Each piece is $\frac{1}{3}$ of a cake, so each person gets $\frac{2}{3}$ of a cake, and $2 \div 3 = \frac{2}{3}$ . Multiplying back checks it: $\frac{2}{3} \times 3 = 2$ .

Seeing fractions as division also changes what happens to a remainder. In a sharing story, a leftover that can be cut becomes a fraction instead of a remainder. Division word problems with fractions in 5th grade are mostly stories like this. Your child also says which two whole numbers the answer lies between, as a quick check on its size.

Your child should be able to:

Read a fraction as a division, such as $\frac{5}{8} = 5 \div 8$
Write a division as a fraction, such as $4 \div 9 = \frac{4}{9}$
Solve sharing problems where each share is a fraction or a mixed number
Say which two whole numbers an answer such as $\frac{20}{6}$ lies between

Example

5 pizzas are shared equally by 8 children. How much pizza does each child get?

The problem is $5 \div 8$ , which can be written as $\frac{5}{8}$ .
To see why, cut each pizza into 8 equal slices. That makes $5 \times 8 = 40$ slices.
Each child gets $40 \div 8 = 5$ slices, and each slice is $\frac{1}{8}$ of a pizza.
Check: $\frac{5}{8} \times 8 = 5$ , the number of pizzas.
So each child gets five-eighths of a pizza.

Example

A 20-foot board is cut into 6 pieces of equal length. How long is each piece, and between which two whole numbers does the length lie?

The problem is $20 \div 6 = \frac{20}{6}$ .
6 goes into 20 three times, since $6 \times 3 = 18$ , with 2 feet left over.
The 2 leftover feet are split into 6 equal parts, so each piece gets $\frac{2}{6} = \frac{1}{3}$ foot more.
Each piece is $3\frac{1}{3}$ feet, which lies between 3 and 4 feet.
So each piece is 3 and one-third feet long.

Where children usually get stuck

For 2 pizzas shared by 5 children, your child writes $\frac{5}{2}$ , or 2 and a half pizzas each.
The child puts the bigger number on top, out of a habit of dividing the bigger number by the smaller one. The amount being shared goes on top.
What helps. Ask whether each child gets more or less than a whole pizza. With fewer pizzas than children, each share must be less than 1, so the answer is $\frac{2}{5}$ .
Your child says 4 granola bars cannot be shared by 5 friends, or writes 0 R 4.
In earlier grades, division gave whole-number answers with remainders. Dividing a smaller number by a bigger one feels impossible.
What helps. Use 4 paper strips as the bars. Fold each into 5 equal parts and deal the parts out. Each friend gets 4 parts, or $\frac{4}{5}$ of a bar.
Asked how much water each of 5 jugs holds when 17 liters are shared equally, your child writes 3 R 2.
A remainder makes sense for things that cannot be cut, such as people or cars. Water can be split, so the 2 leftover liters are shared too.
What helps. Ask what happens to the 2 leftover liters. Shared among 5 jugs, each jug gets $\frac{2}{5}$ liter more, so each holds $3\frac{2}{5}$ liters.

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
2 sandwiches are shared equally among 4 friends. Mia chose the tape diagram below to model the story. Each row stands for 1 sandwich.
Select the statement that best describes Mia's model.
A
The model cuts each whole into the wrong number of equal parts.
B
The model shows too few wholes, so it does not represent all the items being shared.
C
The model matches the story.
D
The model uses the number of people as the number of wholes.
Show the answer and how to solve it
The correct statement is that the model uses the number of people as the number of wholes, because 2 sandwiches shared by 4 friends should show 2 rows, not 4.
1. 1The story says 2 sandwiches are being shared. Since each row stands for 1 sandwich, the diagram should show 2 rows.
2. 2The story also says 4 friends are sharing. That means each sandwich should be cut into 4 equal parts.
3. 3Mia's diagram shows 4 rows, not 2. Each row is split into 4 equal parts, which does match the 4 friends. The error is in the number of whole sandwiches shown.
4. 4The diagram used 4 (the number of friends) as the number of whole sandwiches instead of 2. The model uses the number of people as the number of wholes.
Common mistake. A common mistake is checking only the 4 parts per row and not the number of rows. A correct equal-sharing model must match both the items being shared (2 sandwiches) and the number of people sharing (4 friends).
The idea underneath. In an equal-sharing model, the number of whole items comes from what is being shared, and the number of equal parts in each whole comes from how many people share it.
2On level
A fraction also means division. Multiplying a fraction by its denominator gives back the numerator.
$\displaystyle \frac{6}{7} \times 7 =$
Type the missing number that completes the equation.
Show the answer and how to solve it
The answer is $6$ because $6 \div 7$ , then multiplied by $7$ , returns $6$ .
1. 1Start with the equation. The box on the right is the missing number.
2. 2A fraction can also be read as division. Here, $\frac{6}{7}$ means $6 \div 7$ .
3. 3Multiplying by $7$ undoes dividing by $7$ , so the result is the numerator $6$ .
4. 4So the missing number is $6$ .
Common mistake. A common mistake is typing $7$ because it is the denominator. In this equation, multiplying the fraction by $7$ gives back the numerator $6$ , not the denominator.
The idea underneath. A fraction keeps track of a division problem, so multiplying by the denominator gives back the numerator.
3On level
Liam is rewriting the fraction below as a division expression. His work is shown.
Fraction:
$\displaystyle \frac{3}{8}$
Liam wrote:
$\displaystyle 8 \div 3$
Select the statement that best describes what Liam did.
A
Liam switched the numerator and denominator.
B
Liam rewrote the fraction correctly.
C
Liam used multiplication instead of division.
D
Liam changed one of the numbers from the fraction.
Show the answer and how to solve it
The correct statement is that Liam switched the numerator and denominator. The fraction $\frac{3}{8}$ should be written as $3 \div 8$ , not $8 \div 3$ .
1. 1Look at the original fraction and Liam's division expression.
2. 2A fraction can be written as division by using the top number first and the bottom number second.
3. 3Liam wrote $8 \div 3$ . He used the same two numbers, but he put them in the opposite order.
4. 4So the best description is that Liam switched the numerator and denominator.
Common mistake. A common mistake is putting the bottom number first because it is the divisor. When a fraction is written as division, the top number 3 is divided by the bottom number 8.
The idea underneath. The fraction bar means division, so a/b and a ÷ b use the same two numbers in the same order.

4Stretch

Luis is finding how much each person gets when 3 brownies are shared equally by 7 classmates. Luis's goal: "Find how much of a brownie each classmate gets." His work is shown below.

\displaystyle 3 \text{ brownies} \div 7 \text{ classmates}

\displaystyle = \frac{7}{3} \text{ brownie for each classmate}

Select the statement that best describes what Luis did.

The student's work is correct.

The student switched the number of items and the number of people in the fraction.

The student found the share for 1 item,

\frac{1}{7}

, instead of all the items.

The student gave the total number of items instead of the amount each person gets.

Show the answer and how to solve it

The correct statement is: The student switched the number of items and the number of people in the fraction.

1First, compare Luis's work to the goal in the story.
2For equal sharing, divide the number of items by the number of people.
3Luis wrote $\frac{7}{3}$ . That puts $7$ classmates in the numerator and $3$ brownies in the denominator.
4So Luis did not find the correct share; he switched the item count and the person count in the fraction.

Common mistake. A common mistake is placing the person count in the numerator. In this story, $3$ brownies shared by $7$ classmates means each classmate gets $\frac{3}{7}$ brownie, not $\frac{7}{3}$ .

The idea underneath. In an equal-sharing fraction, the numerator counts the items being shared and the denominator counts the people or groups sharing them.