Section B Practice Problems

Problem 1

Name 3 fractions that are equivalent to $\frac25$ . Explain or show your reasoning.

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Solution

Sample response:

\frac{4}{10}

\frac{8}{20}

\frac{16}{40}

. They are equivalent because each time the number of parts in the whole doubles, the number of parts also doubles.

Problem 2

Which of these fractions can be represented by the point on the number line? Explain your reasoning.

$\frac{86}{100}$

$\frac{90}{100}$

$\frac{94}{100}$

$\frac{101}{100}$

Number line. Scale 0 to 1. Evenly spaced by tenths. Unlabeled point between 9 tenth and 1, closer to 9 tenths.

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Solution

\frac{94}{100}

. Sample response: The fraction

\frac{9}{10}

is equivalent to

\frac{90}{100}

. The point is to the right of

\frac{90}{100}

but to the left of 1, so it is greater than

\frac{90}{100}

but less than

\frac{100}{100}

Problem 3

Explain why the fractions $\frac{10}{3}$ and $\frac{40}{12}$ are equivalent.

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Solution

For $\frac{10}{3}$ , there are 3 thirds in a whole. If I break each third into 4 equal parts, then there are 12 parts in the whole, or 12 twelfths. That means 10 thirds is the same size $10 \times 4$ , or 40 twelfths, so that’s $\frac{40}{12}$ .

Problem 4

Find 2 fractions equivalent to $\frac{10}{6}$ . Explain or show why they are equivalent to $\frac{10}{6}$ . Use the number line if you think it is helpful.

Number line. Scale 0 to 2. Evenly spaced by sixths.

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Solution

Sample response:

\frac53

and

\frac{20}{12}

. Two sixths make 1 third, so 10 sixths make 5 thirds. Two twelfths make 1 sixth, so 20 twelfths make 10 sixths.

Problem 5

Jada says that $\frac75$ is equivalent to $\frac{14}{10}$ because the numerator and denominator of $\frac{14}{10}$ are each 2 times the numerator and denominator of $\frac75$ .

Explain why Jada’s reasoning is correct.
Use Jada’s method to find another fraction equivalent to $\frac75$ .

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Solution

Sample response: There are two $\frac{1}{10}$ s in each $\frac15$ , so there are fourteen $\frac{1}{10}$ s in $\frac75$ .
Sample response: Multiplying the numerator and denominator of $\frac75$ by 10 gives the equivalent fraction $\frac{70}{50}$ .

Problem 6

Mai is thinking of a fraction. She gives several clues to help you guess her fraction. Try to guess Mai’s fraction after each clue.

My fraction is equivalent to $\frac23$ .
The numerator of my fraction is greater than 10.
A factor of my numerator is 8.
The numbers 8 and 5 are a factor pair of my numerator.

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Solution

Sample response: $\frac46$
Sample response: $\frac{20}{30}$
Sample response: $\frac{16}{24}$
Sample response: $\frac{40}{60}$

Problem 7

Think of a fraction: $\underline{\hspace{1.5cm}}$

Write several clues so a friend or family member can guess your fraction. Then present the clues one at a time and ask them to make a guess after each one.

My fraction is equivalent to $\underline{\hspace{1.5cm}}$ .
The numerator of my fraction is less than $\underline{\hspace{1.5cm}}$ .
One multiple of my numerator is $\underline{\hspace{1.5cm}}$ .
A factor pair of my denominator is $\underline{\hspace{1.5cm}}$ and $\underline{\hspace{1.5cm}}$ .

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Solution

Sample response:

Fraction: $\frac{16}{12}$

My fraction is equivalent to $\frac{4}{3}$
The numerator of my fraction is less than 20.
One multiple of my numerator is 80.
A factor pair of my denominator is 4 and 3.

Problem 8

Diego says he shaded $\frac{10}{20}$ of the diagram. Do you agree with Diego? Explain your reasoning.
Shade $\frac{18}{24}$ of the diagram. Explain how you know $\frac{18}{24}$ is shaded.

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Solution

Sample responses:

Diego shaded $\frac{1}{2}$ of the diagram. That is the same as $\frac{10}{20}$ because I can divide each half into ten equal pieces and then it’s $\frac{10}{20}$ .
- I did not want to divide the diagram into 24 equal pieces because they would be small and it would take a long time. But I know $\frac{18}{24} = \frac{6 \times 3}{6 \times 4}$ so I can shade $\frac{3}{4}$ .
- I thought about dividing the strip into 24 equal parts and so first I divided it into 4 equal parts. I was going to divide those parts into 6 equal parts (and then shade 18 of those) but I saw that 3 of the fourths is the same as $\frac{18}{24}$ so I just shaded 3 of the 4 equal parts.