Section A Practice Problems

Problem 1

Find the number that makes each equation true. Explain or show your reasoning.

  1. 000012=23\dfrac{\boxed{\phantom{\frac{0}{000}}}}{12} = \dfrac{2}{3}
  2. 56=000012\dfrac{5}{6} = \dfrac{\boxed{\phantom{\frac{0}{000}}}}{12}
Show Solution
Solution
  1. 812\frac{8}{12}. Sample response: There are 4 twelfths in each third.
  2. 1012\frac{10}{12}. Sample response: There are 2 twelfths in each sixth.

Problem 2

  1. The road around a lake is 15 kilometers long. What is the distance in meters?
  2. The length of an alligator is 4 meters. What is the length in centimeters?
Show Solution
Solution
  1. 15,000 m
  2. 400 cm

Problem 3

The value of the 6 in 618,204 is how many times the value of the 6 in 563? Explain or show your reasoning.

Show Solution
Solution

10,000. Sample response: The 6 in 563 has the value 60 while the 6 in 618,204 has the value 600,000. 600,000=10,000×60600,000 = 10,000 \times 60.

Problem 4

Find the value of each sum.

  1. 38+98\frac{3}{8} + \frac{9}{8}
  2. 315+353\frac{1}{5} + \frac{3}{5}
  3. 2410+17102\frac{4}{10} + 1\frac{7}{10}
Show Solution
Solution
  1. 128\frac{12}{8} (or equivalent)
  2. 3453\frac{4}{5} (or equivalent)
  3. 41104\frac{1}{10} (or equivalent)

Problem 5

Lin spends 5 minutes reading a story. Noah spends 3 times as long as Lin. How long does Noah spend reading the story? Explain or show your reasoning.

Show Solution
Solution

15. Sample response: 3×5=153 \times 5 = 15

Problem 6

  1. Write a multiplication expression for the shaded area, and find the value of the expression. Explain or show your reasoning.

    Diagram, square. Length and width, 1. Partitioned into 5 rows of 6 of the same size squares. In the top left corner, 3 rows of 5 squares shaded. 

  2. Find the value of 57×103\frac{5}{7} \times \frac{10}{3}.
Show Solution
Solution
  1. 35×56\frac{3}{5} \times \frac{5}{6} or 1530\frac{15}{30}. Sample response: 35\frac{3}{5} of 56\frac{5}{6} of the big square is shaded, or 15 of the 30 little squares are shaded.
  2. 5021\frac{50}{21}(or equivalent)

Problem 7

The line plot shows the lengths of some toothpicks, in inches.

Dot plot titled Toothpick Measurements from 0 to 3 by 1’s.
Dot plot titled Toothpick Measurements from 0 to 3 by 1’s. Hash marks by eighths. Horizontal axis, toothpick lengths, in inches. Beginning at 1 and 6 eighths, the number of X’s above each eighth increment is 5, 3, 4, 2, 0, 1.

  1. How many measurements are there?
  2. What is the difference in length between the longest toothpick and the shortest toothpick?
Show Solution
Solution
  1. 15
  2. 58\frac{5}{8} inch

Problem 8

  1. Write a multiplication equation relating the values of 0.5 and 0.05.
  2. Write a division equation relating the values of 0.5 and 0.05.
Show Solution
Solution
  1. 0.5=10×0.050.5 = 10 \times 0.05 or 0.05=0.1×0.50.05 = 0.1 \times 0.5
  2. 0.05=0.5÷100.05 = 0.5 \div 10

Problem 9

Write each number, using exponential form.

  1. 10×10×1010 \times 10 \times 10
  2. 100×100100 \times 100
  3. 100,000
  4. 1,000,000,000
Show Solution
Solution
  1. 10310^3
  2. 10410^4
  3. 10510^5
  4. 10910^9

Problem 10

  1. How many centimeters are in each measurement?

    0.12 m

    3.5 m

    19 m

  2. How many millimeters are in each measurement?

    3 m

    37 m

    1,915 m

  3. How does a whole number of meters change when it is represented in millimeters? Explain or show your reasoning.
Show Solution
Solution

Sample responses:

  1. 12 cm, 350 cm, 1,900 cm
  2. 3,000 mm, 37,000 mm, 1,915,000 mm
  3. Sample response: It gets much bigger. The number of meters is multiplied by 1,000 to get the measurement in millimeters, and this shifts each digit 3 places to the left and puts 3 zeros at the end of the number.

Problem 11

  1. How many meters are in each measurement?

    16 millimeters

    1,375 millimeters

    57 millimeters

  2. How does a whole number of millimeters change when you represent the measurement in meters? Explain or show your reasoning.
Show Solution
Solution
  1. 0.016 m, 1.375 m, 0.057 m
  2. Sample response: It gets much smaller. The number of millimeters is divided by 1,000 to get the measurement in meters, and that means each digit moves 3 places to the right so ones move to thousandths, tens move to hundredths, and so on.

Problem 12

A track is 366 meters around. An athlete runs 15 laps. What is the distance in kilometers? Explain or show your reasoning.

Show Solution
Solution

5.490 km (or equivalent). Sample response: I found 366×15=5,490366 \times 15 = 5,490. Then I divide by 1,000 to get the number of kilometers, so that's 5.490.

Problem 13

Clare drinks 8 glasses of water each day. There are 235 milliliters in each glass. How many liters of water does Clare drink each day? Explain or show your reasoning.

Show Solution
Solution

1.880 L (or equivalent). Sample response: 8×235=1,8808 \times 235 = 1,880. There are 1,000 milliliters in each liter, so I need to divide by 1,000 to get the number of liters. 1,880÷1,000=1.8801,880 \div 1,000 = 1.880. Clare drinks 1.880 liters of water each day.

Problem 14

A track is 400 yards around. How many full laps does Tyler run if he runs at least 2 miles?

Show Solution
Solution

9 laps. Sample response: There are 5,280 feet in a mile so that’s 5,280÷35,280 \div 3 or 1,760 yards. So 2 miles is 2×1,7602 \times 1,760 or 3,520 yards. 9 laps is 3,600 yards so 9 laps is enough. 8 laps is only 3,200 yards, so Tyler needs to run 9 laps.

Problem 15

  1. Write each number, using exponential form. 

    1. 1,000,000,000 (the approximate combined population of North America and South America, in 2023)
    2. 100,000,000,000 (the estimated number of stars in the Milky Way)

    3. 100,000,000,000,000 (the denomination of a bill in Zimbabwean (zim-BAHB-way-yen) dollars, from 2008)

      Photograph of 1 hundred trillion dollars bill from Zimbabwe.

    4. 10,000,000,000,000,000,000 (the estimated number of grains of sand on Earth)
    5. 1,000,000,000,000,000,000,000,000 (the estimated number of stars in the universe)
    6. 10,000,000,000,000,000,000,000,000,000,000,000,000,000 (the estimated number of living cells that ever existed on Earth)
    7. 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
      000,000,000,000,000,000,000,000,000,000 (the estimated number of atoms in the universe!)
  2. How is exponential form helpful for writing these numbers?
Show Solution
Solution
    1. 10910^{9}
    2. 101110^{11}
    3. 101410^{14}
    4. 101910^{19}
    5. 102410^{24}
    6. 104010^{40}
    7. 108010^{80}
  2. Sample response: Exponential notation is helpful because I don't have to count all of the zeros. The numbers are still unimaginably large though. The notation does not help to imagine how big these numbers are.

Problem 16

You have a piggy bank with 1 kilogram of coins inside. Which coins do you want to have in the piggy bank? Explain or show your reasoning.

coin approximate weight (grams)
penny 2.5
nickel 5
dime 2.3
quarter 5.7

Show Solution
Solution

Sample response: A dime is worth more than a penny or a nickel and is lighter, so there would be more of them in the piggy bank. So dimes are definitely better than pennies or nickels. To check between dimes and quarters, a quarter is worth 2.5 times as much as a dime. It looks like the quarters are a little bit better because 2 quarters weigh 11.4 grams and 5 dimes weigh 11.5 grams while both are worth 50 cents. It is very close though, so my hope would be that the piggy bank has all dimes and quarters and no pennies and nickels.