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.
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Section A: Fraction Multiplication

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?
  2. Write a division equation to represent your answer.

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.

6-sided shape.

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.

    Diagram, square. Length and width, 1. 
  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.

    Diagram, square. Length and width, 1. Partitioned into 5 rows of 4 of the same size rectangles. 1 rectangle 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.

    Diagram, square. Length and width, 1. Partitioned into 6 rows of 3 of the same size rectangles. 1 rectangle shaded.
  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.

    Diagram, square. Length and width, 1. Partitioned into 3 rows of 4 of the same size rectangles. 3 rectangles shaded.
  2. Explain how your expression matches the shaded region.

Solution

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Problem 9

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

    Diagram, square. Length and width, 1. Partitioned into 5 rows of 4 of the same size rectangles. 4 rectangles shaded.
  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?

    Blue rectangle. Partitioned into two rows and two columns by yellow stripes.
  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.

American flag. 13 stripes, 7 red, 6 white. Blue rectangle, top left corner. 50 white stars in blue rectangle. 

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.

Diagram, square. Length and width, 1. Rectangular portion shaded with length, about 1 eighth, width, about 1 sixteenth. 

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

Solution

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Section B: Fraction Division

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 \(\frac{1}{5} \div 2\) .
    Diagram. 1 part. Total length, 1. 
  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.
  2. Write a division equation to represent your answer.

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

Solve each problem. Write an equation showing your answer.

  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.

ADiagram. 4 equal parts, each labeled question mark. Total length, 1.

BDiagram. 3 equal parts. 1 part divided into 4 equal parts, 1 shaded. Total length, 1.

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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Section C: Problem Solving with Fractions

Problem 1

Solve each problem. Write an equation showing your answer.

  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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