Section C Practice Problems

Problem 1

  1. Andre ran 45\frac{4}{5} of a 7-mile trail. Did Andre run a distance greater than or less than 7 miles? Explain or show your reasoning.
  2. Clare ran 000010\frac{\boxed{\phantom{\frac{0}{000}}}}{\Large{10}} of a 7-mile trail. She ran a distance greater than 7 miles. Find a number that makes this statement true. Explain or show your reasoning.
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Solution
  1. Less. Sample response: 45\frac{4}{5} is less than 1 so that’s less than the full trail.
  2. Sample response: 11, because 1110\frac{11}{10} is more than 1, so Clare ran more than the full trail.

Problem 2

Point J on the number line shows how many miles Jada ran. Label the points on the number line to show each runner’s distance, in miles.

  1. Clare ran 85\frac{8}{5} as far as Jada.
  2. Tyler ran 43\frac{4}{3} as far as Jada.
  3. Lin ran 12\frac{1}{2} as far as Jada.
Number line. Tick mark, labeled 0. 4 dots plotted. From left to right, blank line, J, blank line, blank line. 
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Solution
Number line

Problem 3

Point A is labeled on the number line.

Number line. Tick mark, labeled 0. Point, labeled A to the right. 

Label each of these points on the number line.

  • 25×A\frac{2}{5} \times A
  • 1310×A\frac{13}{10} \times A
  • 74×A\frac{7}{4} \times A
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Solution
Number line

Problem 4

Use the equation 57=(1−27)\frac{5}{7} = \left(1 - \frac{2}{7}\right) to explain why 57×113<113\frac{5}{7} \times \frac{11}{3} < \frac{11}{3}.

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Solution

Sample response: (1−27)×113=(1×113)−(27×113)\left(1 - \frac{2}{7}\right) \times \frac{11}{3} = \left(1 \times \frac{11}{3}\right) - \left(\frac{2}{7} \times \frac{11}{3}\right)

Since 1×113=1131 \times \frac{11}{3} = \frac{11}{3} this means 57×113\frac{5}{7} \times \frac{11}{3} is less than 113\frac{11}{3} because it is 113\frac{11}{3} minus a number.

Problem 5

Explain why multiplying a fraction by a number less than 1 makes the fraction smaller.

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Solution

Sample response: Multiplying by 1 gives the same fraction so multiplying by less than 1 gives a smaller number.

Problem 6

Point P is labeled on the number line.

Number line. Tick mark, labeled 0. Point to the right of 0. Labeled P.

  1. Point P is 34\frac{3}{4} of a number A. Plot A on the number line. Explain or show your reasoning.
  2. Point P is 59\frac{5}{9} of a number B. Plot B on the number line. Explain or show your reasoning.
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Solution
Number line
  1. I know that 34\frac{3}{4} is 3 out of 4 equal parts so I divided the distance from 0 to P into 3 equal parts and then added 1 more of those parts to get A.
  2. I know that 95\frac{9}{5} is 9 equal parts, with 5 parts in a whole, so I divided the distance 0 to P into 5 equal parts and then took 9 of those parts to get B.

Problem 7

  1. About 10610^6 people live in Michigan. About 10410^4 of the people in Michigan live in Flint.

    1. How many times as many people live in Michigan as in Flint?

    2. How many times as many people live in Flint as in Michigan?

  2. There are about 1011 stars in the Milky Way. There are about 1024 stars in the universe.

    1. How many times as many stars are there in the universe than in the Milky Way?

    2. How many times as many stars are there in the Milky Way than in the universe?
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Solution
    1. 10210^{2} or 100. Sample response: 10610^{6} is 1,000,000 and it has 2 more zeros than 10410^{4} or 10,000.
    2. 1100\frac{1}{100}. Sample response: Dividing 10410^{4} by 10610^{6} will move the decimal point two places to the left, taking away 2 zeros.
    1. 10,000,000,000,000. Sample response: I need 13 factors of 10 because there are 13 more zeros in 102410^{24} compared to 101110^{11}. Thirteen factors of 10 means 13 zeros at the end of the number.
    2. 110,000,000,000,000\frac{1}{10,000,000,000,000}. Sample response: Dividing 101110^{11} by 102410^{24} will move the decimal point 13 places to the left and remove 13 of the 24 zeros.