Section C Practice Problems

Problem 1

Andre is building a tower out of different foam blocks. These blocks come in three different thicknesses: $\frac{1}{2}$ foot, $\frac{1}{4}$ foot, and $\frac{1}{8}$ foot.

Andre stacks two $\frac{1}{2}$ -foot blocks, two $\frac{1}{4}$ -foot blocks, and two $\frac{1}{8}$ -foot blocks to create a tower. What is the height of the tower in feet? Explain or show how you know.

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Solution

1 \frac{3}{4}

feet tall. Sample response:

\frac{1}{2} + \frac{1}{2} = 1

and

\frac{1}{8} + \frac{1}{8} = \frac{2}{8}

, and I know that

\frac{2}{8}

is the same as

\frac{1}{4}

. Now I have

\frac{1}{4} + \frac{1}{4} + \frac{1}{4}

, which is

\frac{3}{4}

. So the tower is

1 + \frac{3}{4}

1\frac{3}{4}

feet tall.

Problem 2

Find the value of each of the following sums. Show your reasoning. Use number lines if you find them helpful.

$\frac{1}{10} + \frac{3}{100}$
$\frac{24}{100} + \frac{4}{10}$
$\frac{7}{10} + \frac{13}{100}$

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Solution

$\frac{13}{100}$ , because $\frac{1}{10} = \frac{10}{100}$ , so $\frac{10}{100} + \frac{3}{100} = \frac{13}{100}$ .
$\frac{64}{100}$ , because $\frac{4}{10} = \frac{40}{100}$ , so $\frac{24}{100} + \frac{40}{100} = \frac{64}{100}$ .
$\frac{83}{100}$ , because $\frac{7}{10} = \frac{70}{100}$ , so $\frac{70}{100} + \frac{13}{100} = \frac{83}{100}$ .

Problem 3

Is the value of each expression greater than, less than, or equal to 1? Explain how you know.

$\frac{3}{10} + \frac{7}{100}$
$\frac{13}{10} + \frac{7}{100}$
$\frac{30}{100} + \frac{7}{10}$

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Solution

Less than 1. I know that $\frac{3}{10}$ needs $\frac{7}{10}$ to make 1, but I only have $\frac{7}{100}$ , which is less than $\frac{7}{10}$ .
Greater than 1. I know that $\frac{13}{10}$ is already greater than 1. Adding something else to it makes it even greater.
Equal to 1. I know that $\frac{7}{10}$ needs $\frac{3}{10}$ to make 1, and $\frac{30}{100}$ is the same as $\frac{3}{10}$ .

Problem 4

Diego and Lin continue to play with their coins.

Diego says he has exactly 3 coins whose thickness adds up to $\frac{50}{100}$ centimeters (cm). What coins does Diego have? Explain or show your reasoning.

coin	thickness (cm)
1 centavo	$\frac{12}{100}$
10 centavos	$\frac{22}{100}$
1 peso	$\frac{16}{100}$
2 pesos	$\frac{14}{100}$
5 pesos	$\frac{2}{10}$
20 pesos	$\frac{25}{100}$

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Solution

Sample responses:

Diego has a 1-peso coin, 2-peso coin, and a 5-peso coin. The combined thickness of the 1-peso and 2-peso coins is $\frac{30}{100}$ cm, because $\frac{16}{100} + \frac{14}{100} = \frac{30}{100}$ . Combining that with the 5-peso coin gives $\frac{30}{100} + \frac{2}{10}$ or $\frac{30}{100} + \frac{20}{100}$ , which is $\frac{50}{100}$ cm.
Diego has two 2-peso coins and one 10-centavo coins, because $\frac{14}{100} + \frac{14}{100} + \frac{22}{100} = \frac{50}{100}$ cm.
Diego has a 1-centavo coin, a 10-centavo coin, and a 1-peso coin, because $\frac{12}{100} + \frac{22}{100} + \frac{16}{100} = \frac{50}{100}$ cm.

Problem 5

A chocolate cake recipe calls for 2 cups of flour. You gather your measuring cups and notice you have the following sizes: $\frac{1}{2}$ cup, $\frac{1}{3}$ cup, $\frac{1}{4}$ cup, and $\frac{1}{6}$ cup.

What are the different ways you could use all 4 measuring cups to measure 2 cups of flour?
What are other ways you could use just some of the 4 measuring cups to measure exactly 2 cups of flour?

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Solution

Sample responses:

All measuring cups:
- Two $\frac{1}{2}$ -cup, two $\frac{1}{4}$ -cup, two $\frac{1}{3}$ -cup, and two $\frac{1}{6}$ -cup measures
- One $\frac{1}{2}$ -cup, two $\frac{1}{4}$ -cup, one $\frac{1}{3}$ -cup, and four $\frac{1}{6}$ -cup measures
- Two $\frac{1}{2}$ -cup, two $\frac{1}{4}$ -cup, one $\frac{1}{3}$ -cup, and two $\frac{1}{6}$ -cup measures
Some of the measuring cups:
- Four $\frac{1}{2}$ -cup measures
- One $\frac{1}{2}$ -cup, two $\frac{1}{4}$ -cup, and three $\frac{1}{3}$ -cup measures
- One $\frac{1}{3}$ -cup, and ten $\frac{1}{6}$ -cup measures
- Four $\frac{1}{4}$ -cup, one $\frac{1}{3}$ -cup, four $\frac{1}{6}$ -cup measures

Problem 6

A dime is worth $\frac{1}{10}$ of a dollar and a penny is worth $\frac{1}{100}$ of a dollar.

What are different combinations of dimes and pennies that represent $\frac{89}{100}$ of a dollar? Use equations to show your reasoning.
A nickel is worth $\frac{5}{100}$ of a dollar. What are some different combinations of dimes, nickels, and pennies that represent $\frac{89}{100}$ of a dollar? Use equations to show your reasoning.

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Solution

Sample responses:
- 89 pennies. $89 \times \frac{1}{100} = \frac{89}{100}$
- 79 pennies and 1 dime: $79 \times \frac{1}{100} + 1 \times \frac{10}{100} = \frac{89}{100}$
- 69 pennies and 2 dimes: $69 \times \frac{1}{100} + 2 \times \frac{10}{100} = \frac{89}{100}$
- 59 pennies and 3 dimes: $59 \times \frac{1}{100} + 3 \times \frac{10}{100} = \frac{89}{100}$
- 49 pennies and 4 dimes: $49 \times \frac{1}{100} + 4 \times \frac{10}{100} = \frac{89}{100}$
- 39 pennies and 5 dimes: $39 \times \frac{1}{100} + 5 \times \frac{10}{100} = \frac{89}{100}$
- 29 pennies and 6 dimes: $29 \times \frac{1}{100} + 6 \times \frac{10}{100} = \frac{89}{100}$
- 19 pennies and 7 dimes: $19 \times \frac{1}{100} + 7 \times \frac{10}{100} = \frac{89}{100}$
- 9 pennies and 8 dimes: $9 \times \frac{1}{100} + 8 \times \frac{10}{100} = \frac{89}{100}$
Sample responses:
- 4 pennies, 1 nickel, and 8 dimes: $4 \times \frac{1}{100} + 1 \times \frac{5}{100}+ 8 \times \frac{10}{100} = \frac{89}{100}$
- 9 pennies, 6 nickels, 5 dimes: $9 \times \frac{1}{100} + 6 \times \frac{5}{100} + 5\times \frac{10}{100} = \frac{89}{100}$
- 4 pennies, 9 nickels, and 4 dimes: $4 \times \frac{1}{100} + 9 \times \frac{5}{100}+ 4 \times \frac{10}{100} = \frac{89}{100}$