Section C Practice Problems

Problem 1

Circle the greater fraction in each pair. Explain or show your reasoning.

$\frac25$ or $\frac26$
$\frac58$ or $\frac78$
$\frac{9}{10}$ or $\frac{103}{100}$

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Solution

$\frac25$ is greater because there are 2 parts in each fraction, but fifths are larger than sixths.
$\frac78$ is greater because the parts are the same size, but 7 is greater than 5.
$\frac{103}{100}$ is greater because it is greater than 1, and $\frac{9}{10}$ is less than 1.

Problem 2

Use $>$ , $<$ , or $=$ to make each statement true. Explain or show your reasoning.

$\frac{2}{3} \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{10}{15}$
$\frac15 \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{22}{100}$
$\frac{10}{4} \> \underline{\phantom{ \hspace{0.7cm} }} \> \frac{45}{20}$

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Solution

$\frac23 = \frac{10}{15}$ , because if we partition each third into 5 parts, there are 15 fifteenths in 1 whole, and 10 fifteenths in 2 thirds.
$\frac15 < \frac{22}{100}$ , because $\frac15 = \frac{1 \ \times \ 20}{5 \ \times \ 20}= \frac{20}{100}$ , and 22 hundredths is greater than 20 hundredths.
$\frac{10}{4} > \frac{45}{20}$ , because $\frac{10}{4} = \frac{5 \ \times \ 10}{5 \ \times \ 4}=\frac{50}{20}$ , and 50 twentieths is greater than 45 twentieths.

Problem 3

A water fountain is $\frac{7}{10}$ mile from the start of a hiking trail. A pond is $\frac{3}{5}$ mile from the start of the trail. A hiker begins walking at the start of the trail. Which will the hiker pass first, the water fountain or the pond? Explain your reasoning.

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Solution

The pond, because

\frac35 < \frac{7}{10}

. I know that

\frac35

is less than

\frac{7}{10}

because each

\frac15

is equivalent to

\frac{2}{10}

, so

\frac35

is equivalent to

\frac{6}{10}

, which is

\frac{1}{10}

less than

\frac{7}{10}

. The pond is

\frac{1}{10}

mile before the water fountain.

Problem 4

Tyler says he grew $\frac32$ centimeters since his height was measured 6 months ago.

Diego says, “Oh, you grew more than I did! My height went up by only $\frac78$ inch in the past 6 months.”

Explain why Tyler did not grow more than Diego did, even though $\frac32$ is greater than $\frac78$ .

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Solution

Sample response: The measurements were done using different units. One centimeter is not the same length as one inch. The ruler shows that centimeters are smaller than inches, and

{7 \over 8}

inch is larger than

{3 \over 2}

centimeters. from Unit 2, Lesson 14

Problem 5

List these fractions from least to greatest. Explain or show your reasoning.

$\frac13$
$\frac{5}{12}$
$\frac{2}{10}$

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Solution

\frac{2}{10}

\frac13

\frac{5}{12}

. Sample response: I know that

\frac{2}{10}

is equivalent to

\frac15

and

\frac13

is greater than

\frac15

. I know that

\frac13

is equivalent to

\frac{4}{12}

, so it is less than

\frac{5}{12}

Problem 6

List these fractions from least to greatest. Explain or show your reasoning.

$\frac{15}{8}$
$\frac{215}{100}$
$\frac{7}{4}$
$\frac{21}{10}$

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Solution

\frac{7}{4}

\frac{15}{8}

\frac{21}{10}

\frac{215}{100}

. Sample response:

\frac{7}{4}

and

\frac{15}{8}

are both less than 2, so they are less than

\frac{215}{100}

and

\frac{21}{10}

, which are both greater than 2. The fraction

\frac{7}{4}

\frac{1}{4}

less than 2 and

\frac{15}{8}

\frac{1}{8}

less than 2, so

\frac{7}{4}

is smaller.

\frac{21}{10}

is equivalent to

\frac{210}{100}

, which is less than

\frac{215}{100}

Problem 7

Jada lists these fractions that are all equivalent to $\frac12$ : $\quad \frac24, \frac{3}{6}, \frac{4}{8}, \frac{5}{10}$

She notices that each time the numerator increases by 1, the denominator increases by 2. Will this pattern continue? Explain your reasoning.

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Solution

Yes, it will continue because if a fraction is equivalent to

\frac12

, then the denominator, the number of parts in a whole, is twice the numerator, the number of parts in the fraction. If the numerator increases by 1 (one more part in a whole), then the denominator increases by 2 (two more parts in the whole).

Problem 8

Find a fraction that is between $\frac25$ and $\frac38$ . Explain or show your reasoning.

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Solution

Sample response:

\frac{31}{80}

. I know that

\frac25

is equivalent to

\frac{16}{40}

and

\frac{3}{8}

is equivalent to

\frac{15}{40}

. There is no whole number between 15 and 16, so I need to make the denominator bigger. Partitioning each

\frac{1}{40}

into two equal parts gives

\frac{2}{80}

. So,

\frac25

is equivalent to

\frac{32}{80}

and

\frac{3}{8}

is equivalent to

\frac{30}{80}

. That means

\frac{31}{80}

is a fraction between

\frac25

and

\frac38