# 5.3 Multiplying and Dividing Fractions

## Unit Goals

• Students extend multiplication and division of whole numbers to multiply fractions by fractions and divide a whole number and a unit fraction.

### Section A Goals

• Recognize that $\frac{a}{b} \times \frac{c}{d}=\frac{a \ \times \ c}{b \ \times \ d}$ and use this generalization to multiply fractions numerically.
• Represent and describe multiplication of a fraction by a fraction using area concepts.

### Section B Goals

• Divide a unit fraction by a whole number using whole-number division concepts.
• Divide a whole number by a unit fraction using whole-number division concepts.

### Section C Goals

• Solve problems involving fraction multiplication and division.

### Problem 1

#### Pre-unit

Practicing Standards:  3.OA.A.2

There are 63 students in the cafeteria. There are 9 students at each table.

1. At how many tables are the students seated?

### Solution

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### Problem 2

#### Pre-unit

Practicing Standards:  3.MD.C.7

What is the area of this figure? Explain your reasoning.

### Solution

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### Problem 3

#### Pre-unit

Practicing Standards:  4.NF.B.4

Select all expressions that are equivalent to $$\frac{12}{5}$$.

A: $$6 \times \frac{2}{5}$$
B: $$5 \times \frac{1}{12}$$
C: $$12 \times \frac{1}{5}$$
D: $$8 \times \frac{4}{5}$$
E: $$4 \times \frac{3}{5}$$

### Solution

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### Problem 4

#### Pre-unit

Practicing Standards:  4.NF.B.4.c

Jada has 8 pennies. Each one weighs $$\frac{5}{2}$$ grams. How much do Jada’s pennies weigh altogether? Explain your reasoning.

### Solution

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### Problem 5

1. Shade $$\frac{1}{2}$$ of $$\frac{1}{5}$$ of the square.

2. Explain where you see $$\frac{1}{2}$$ of $$\frac{1}{5}$$ in your drawing.

### Solution

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### Problem 6

1. Write an expression for how much of the square is shaded.

2. Find the value of your expression.

### Solution

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### Problem 7

1. Write an equation representing the shaded part of the diagram.

2. Explain how the diagram shows each part of your equation.

### Solution

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### Problem 8

1. Write an expression for the shaded region of the square.

### Solution

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### Problem 9

1. Write an expression for the area of the shaded region.

2. Explain how the diagram shows your expression.

### Solution

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### Problem 10

1. Write a multiplication expression for the area of the shaded region. Explain your reasoning.

2. What is the area of the shaded region in square units?

### Solution

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### Problem 11

Find the value that makes each equation true.

1. $$\frac{7}{10} \times \frac{3}{5} = \underline{\hspace{0.7cm}}$$
2. $$\frac{2}{5} \times \underline{\hspace{0.7cm}} = \frac{8}{45}$$
3. $$\underline{\hspace{0.7cm}} \times \frac{4}{9} = \frac{28}{45}$$

### Solution

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### Problem 12

This flag of Sweden is $$3\frac{1}{5}$$ inches wide and 2 inches tall. The rectangle in the upper right is $$\frac{9}{5}$$ inches wide and $$\frac{4}{5}$$ inch tall.

1. What is the area of the whole flag?

2. What is the area of the rectangle in the upper right?

### Solution

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### Problem 13

#### Exploration

On this American flag the width of the blue rectangle is $$\frac{2}{5}$$ the width of the flag. What fraction of the area of the flag is the blue rectangle? Explain or show your reasoning.

### Solution

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### Problem 14

#### Exploration

Jada folded a square piece of paper in half many times, sometimes horizontally and sometimes vertically. She shaded the folded piece of paper and then unfolded it. Here is a picture.

What fraction of the paper did Jada shade? Explain how you know.

### Solution

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### Problem 1

For each pair of expressions, decide which is greater. Explain your choice without calculating the value of the expressions.

1. $$210 \div 3$$$$210 \div 5$$

2. $$210 \div 3$$$$75 \div 3$$

### Solution

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### Problem 2

A pan of macaroni and cheese is $$\frac{1}{3}$$ full. Four friends split the remaining macaroni and cheese equally.

1. Make a drawing that represents the situation.
2. Write a division expression representing how much of a pan each friend gets.
3. Explain how the drawing shows the division expression.

### Solution

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### Problem 3

1. Use the diagram to represent the expression $$\frac{1}{5} \div 2$$ .
2. Explain how the diagram shows $$\frac{1}{5} \div 2$$ .
3. What is the value of $$\frac{1}{5} \div2$$?

### Solution

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### Problem 4

Mai has a strip of paper that is 3 feet long. She cuts it into $$\frac{1}{4}$$ foot strips.

1. How many $$\frac{1}{4}$$ foot strips does Mai make? Explain or show your reasoning.

### Solution

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### Problem 5

Find the value of each expression.

1. $$5 \div \frac{1}{4}$$
2. $$6 \div \frac{1}{4}$$
3. $$3 \div \frac{1}{6}$$
4. $$3 \div \frac{1}{7}$$

### Solution

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### Problem 6

1. There are 4 liters of water. How many $$\frac{1}{2}$$ liter bottles of water is that?
2. 4 friends split $$\frac{1}{2}$$ pound of dried fruit equally. How many pounds of fruit does each friend get?

### Solution

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### Problem 7

Find the value of each expression. Explain or show your reasoning.

1. $$3 \div \frac{1}{4}$$
2. $$\frac{1}{5} \div 8$$

### Solution

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### Problem 8

#### Exploration

For each diagram, write a story problem that the tape diagram represents. Then use the diagram to solve the problem.

### Solution

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### Problem 9

#### Exploration

It takes Earth 1 year to go around the Sun.

1. During the time it takes Earth to go around around the Sun, Mercury goes around the Sun about 4 times. How many years does it take Mercury to make 1 full orbit of the Sun? Write an equation showing your answer.
2. During the time it takes Earth to go around the Sun, Saturn goes $$\frac{1}{29}$$ of the way around the Sun. How many years does it take Saturn to go around the Sun? Write an equation showing your answer.

### Solution

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### Problem 1

1. The container holds $$\frac{1}{2}$$ gallon of water. It is $$\frac{3}{4}$$ full. How many gallons of water are in the container?
2. The container has $$\frac{1}{2}$$ gallon of water. 6 friends split the water equally. How many gallons of water does each friend get?
3. The container has 1 gallon of water. Each bottle holds $$\frac{1}{8}$$ of a gallon. How many bottles of water does the container hold?

### Solution

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### Problem 2

Clare has 5 yards of ribbon. It takes $$\frac{1}{2}$$ yard to make a bow. How many bows can Clare make with the ribbon? Write a multiplication and a division equation showing the solution.

### Solution

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### Problem 3

Using the numbers 4, 5, 6, 7, 8, or 9, what is the largest product you can make?
$$\displaystyle \frac{\boxed{\phantom{\frac{000}{0}}}}{\boxed{\phantom{\frac{000}{0}}}} \times \frac{\boxed{\phantom{\frac{000}{0}}}}{\boxed{\phantom{\frac{000}{0}}}}$$
You can use each number at most once. Explain or show your reasoning.

### Solution

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### Problem 4

3 ounces is $$\frac{1}{4}$$ of the package of sunflower seeds. How many ounces of sunflower seeds are in the whole package? Explain or show your reasoning.

### Solution

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### Problem 5

A person drove 5 miles. That is $$\frac{1}{3}$$ of the distance from their home to work. How far is it from the person's home to work? Explain or show your reasoning.

### Solution

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### Problem 6

#### Exploration

1. Each millimeter is $$\frac{1}{1,000}$$ of a meter. There are 1,000 micrometers in a millimeter. How many meters is a micrometer? Explain or show your reasoning.
2. There are 1,000 nanometers in a micrometer. How many meters is a nanometer? (A single human hair can be about 50 micrometers thick. Nanometers can be used to describe the size of atoms.)

### Solution

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### Problem 7

#### Exploration

Jada wants to make a playpen for her dog with at least 70 square feet of space. She has 35 feet of fencing for the frame. Can Jada make a big enough playpen? Explain your reasoning.

### Solution

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