Section A Practice Problems

Problem 1

Locate and label $\frac{6}{4}$ on the number line.
Explain or show why your point represents $\frac{6}{4}$ .

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Solution

Sample response: The tick marks are equally spaced and there are 4 of them in 1. So, each tick mark is $\frac{1}{4}$ and $\frac{6}{4}$ is at the sixth tick mark.

Problem 2

Shade $\frac{3}{4}$ of the rectangle. Explain or show your reasoning.

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Solution

Sample response:

There are 4 equal parts and 3 of them are shaded. So, that is

\frac{3}{4}

of the rectangle.

Problem 3

Explain or show why $\frac{4}{3} = 4 \times \frac{1}{3}$ .

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Solution

Sample response:

The drawing shows that there are 4 equal groups of $\frac{1}{3}$ in $\frac{4}{3}$ so $\frac{4}{3} = 4\times\frac{1}{3}$ .

Problem 4

Each book is $\frac{3}{8}$ inch thick. How many inches thick is a stack of 5 books? Explain or show your reasoning.

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Solution

\frac{15}{8}

inches. Sample response:

5 \times \frac{3}{8}

\frac{5 \times 3}{8}

Problem 5

There are 36 fish in 4 aquariums. There are the same number of fish in each aquarium. How many fish are in each aquarium? Explain or show your reasoning.
There are 24 dogs at a shelter. There are 4 times as many dogs as cats at the shelter. How many cats are at the shelter? Explain or show your reasoning.

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Solution

9 fish. Sample response: $4 \times 9 = 36$
6 cats. Sample response: $24 \div 4 = 6$

Problem 6

A bottle holds $\frac{7}{10}$ liter of water. How much water do 6 bottles hold? Explain or show your reasoning.

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Solution

$\frac{42}{10}$ or $4\frac{2}{10}$ liters (or equivalent). Sample responses:

$6 \times 7 = 42$ , so there are $\frac{42}{10}$ liters of water in the 6 bottles.

Problem 7

What is the area of the rectangle? Explain or show your reasoning.

Rectangle. Horizontal side: 12 centimeters. Vertical side: 8 centimeters.

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Solution

96 square centimeters. Sample response:

12 \times 8 = 96

Problem 8

There are 3 students. They equally share 18 sheets of construction paper for an art project. How much paper does each student get? Explain or show your reasoning.
There are 3 students. They equally share 1 tube of clay for an art project. How much clay does each student get? Explain or show your reasoning.

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Solution

6 sheets of paper. Sample response: $18 \div 3 = 6$ .
$\frac{1}{3}$ of a tube of glue. Sample response:

Problem 9

There are 4 hikers. They equally share 3 liters of water. How much water does each hiker get? Explain or show your reasoning.
There are 4 hikers. They equally share 5 liters of water. How much water does each hiker get? Explain or show your reasoning.

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Solution

$\frac{3}{4}$ liter (or equivalent). Sample response: Each hiker gets $\frac{1}{4}$ of each liter of water.
$\frac{5}{4}$ liter (or equivalent). Sample response: Each hiker gets $\frac{1}{4}$ of each liter of water.

Problem 10

Jada cuts an 11-inch strip of paper into 5 equal parts. How many inches long is each part?
Jada cuts a strip of paper into 5 equal parts. Each part is $\frac{7}{5}$ inches long. How long was the strip of paper?

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Solution

$\frac{11}{5}$ inches (or equivalent).
$\frac{35}{5}$ inches or 7 inches (or equivalent).

Problem 11

<p>3 diagrams of equal lengths. 5 equal parts. 1 part shaded. Total length, 1</p>

Describe a division situation that the diagram could represent.
Write an equation that represents the diagram and the situation.

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Solution

Sample responses:

There are 3 liters of juice that are split evenly between 5 people. How many liters of juice does each person get?
$3 \div 5 = \frac{3}{5}$ .

Problem 12

Decide whether each equation is true or false. Explain or show your reasoning.

$3 \div 7 = \frac{3}{7}$
$18 \div 5 = \frac{5}{18}$
$15 \div 6 = 2 \frac{1}{2}$

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Solution

True. Sample response: I can divide the 3 wholes into 7 equal parts by taking $\frac{1}{7}$ of each whole and putting those parts together. That's the same as $\frac{3}{7}$ of one whole.
False. Sample response: This is not true since $\frac{5}{18}$ is less than 1 and $18 \div 5$ is greater than 1.
True. Sample response: I can first take 12 and make 6 equal portions of 2 and then there are 3 left which makes 6 equal portions of $\frac{1}{2}$ .

Problem 13

Describe a situation in the classroom or at home where you share something equally that results in fractional size parts.
Draw a picture or diagram to represent the situation.
Write a division equation to represent the situation.

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Solution

Sample responses:

My mom used 1 pound of meat to make burgers. She made 5 equal burgers. How much meat is in each burger?
The whole tape represents 1 pound.
$1 \div 5 = \frac{1}{5}$ .

Problem 14

Elena travels to visit her grandparents. Their house is 125 miles from Elena’s.

Elena stops for lunch $\frac{2}{3}$ of the way to her grandparents’ house. How far has Elena traveled? Explain or show your reasoning.
Elena reaches the city where her grandparents live after 110 miles. Is she more or less than $\frac{9}{10}$ of the way to their house? Explain or show your reasoning.

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Solution

$83 \frac{1}{3}$ miles. Sample response: $\frac{1}{3}$ of 120 is 40, so $\frac{2}{3}$ of 120 is 80. Then $\frac{1}{3}$ of 5 is $\frac{5}{3}$ , so $\frac{2}{3}$ of 5 is $\frac{10}{3}$ which is the same as $3 \frac{1}{3}$ .
Less. Sample response: $\frac{1}{10}$ of 125 is $\frac{1}{10}$ of 120, 12, and $\frac{1}{10}$ of 5, $\frac{1}{2}$ . So, when Elena is $\frac{9}{10}$ of the way to her grandparents there will be $12 \frac{1}{2}$ miles left. After 110 miles, there are 15 miles left so she is not yet $\frac{9}{10}$ of the way there.

Problem 15

Describe a situation that represents the equation $4 \div 6 = \frac{4}{6}$ .
Draw a diagram to represent the situation.

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Solution

Sample responses:

There are 4 quarts of apple cider that 6 friends split equally. How many quarts of apple cider does each friend get?