Section A Practice Problems

Problem 1

Find the value of each expression.

  1. 13×110\frac{1}{3} \times \frac{1}{10}
  2. 110×110\frac{1}{10} \times \frac{1}{10}
  3. 110×1100\frac{1}{10} \times \frac{1}{100}
Show Solution
Solution
  1. 130\frac{1}{30}
  2. 1100\frac{1}{100}
  3. 11,000\frac{1}{1,000}

Problem 2

  1. Write a multiplication equation to represent the shaded region of the diagram.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. 36 squares shaded. 

  2. What is the value of 710×510\frac{7}{10} \times \frac{5}{10}? Use the diagram if it is helpful.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. No squares shaded. 

Show Solution
Solution
  1. 910×410=36100\frac{9}{10} \times \frac{4}{10} = \frac{36}{100} or 410×910=36100\frac{4}{10} \times \frac{9}{10} = \frac{36}{100}
  2. 35100\frac{35}{100}. Sample response: I found the product of the numerators and the product of the denominators.

Problem 3

Find the value of 73×2873 \times 28. Use the diagram if it is helpful.

Diagram, rectangle partitioned vertically and horizontally into 4 rectangles.
Diagram, rectangle partitioned vertically and horizontally into 4 rectangles. Top left rectangle, vertical side, 20, horizontal side, 70. Top right rectangle, horizontal side, 3. Bottom 2 rectangles, vertical side, 8.

Show Solution
Solution

2,044. Sample response: I added up the partial products using the diagram.

area diagram

Problem 4

  1. What is the value of the digit 6 in 618,923?
  2. How many times greater is the value of the digit 6 in 618,923 than the value of the digit 6 in 27,652?
Show Solution
Solution
  1. 600,000
  2. 1,000

Problem 5

Find the value of 3,724÷73,724 \div 7. Explain or show your reasoning.

Show Solution
Solution

532. Sample response:

500×7=3,500500 \times 7 = 3,500
30×7=21030 \times 7 = 210
2×7=142 \times 7 = 14
532×7=3,724532 \times 7 = 3,724

Problem 6

Find the value of each sum or difference.

a.
Add. 13 thousand, 8 hundred, 17, minus, 6 thousand, 5 hundred, forty 4, equals.

b.
Subtract. eight thousand, seven hundred, ninety three, minus, four thousand six hundred, seventy five, equals.

Show Solution
Solution

  1. 20,361. Sample response:

    Addition algorithm.

  2. 4,118. Sample response:

    Subtraction algorithm

Problem 7

  1. What fraction of the whole square is shaded? Explain or show your reasoning.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. 1 square shaded.

  2. What fraction of the whole square is shaded? Explain or show your reasoning.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. Top left square partitioned into 10 rows. 1 row shaded.

Show Solution
Solution
  1. 1100\frac{1}{100} of the square is shaded. Sample response: There are 10 rows and 10 columns. So, there are 100 squares in the whole and 1 is shaded.
  2. 11,000\frac{1}{1,000} of the whole square is shaded. Sample response: There are 10 small rectangles in one square and 100 squares in the whole. So, that's 10×10010 \times 100 rectangles in the whole.

Problem 8

  1. Write a decimal number to represent how much of the square is shaded.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. 66 squares shaded. 

  2. Shade the square to represent the decimal one hundred fifteen thousandths.

    Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. No squares shaded. 

Show Solution
Solution
  1. 0.66

  2. hundredths grid

Problem 9

Write the decimal 0.418 as a fraction, in words, and in expanded form.

Show Solution
Solution

4181,000\frac{418}{1,000}, four hundred eighteen thousandths, (4×110)+(1×1100)+(8×11,000)\left(4 \times \frac{1}{10}\right) + \left(1 \times \frac{1}{100}\right) + \left(8 \times \frac{1}{1,000}\right)

Problem 10

  1. The gold nugget on the scale weighs 0.265 ounce. Name 2 different sets of 0.1-ounce, 0.01-ounce, and 0.001-ounce weights you can use to balance the nugget.

    Scale, tilted left-side down, right-side up. Left side, gold nuggets. Right side, nothing.

  2. One gold nugget weighs 0.008 ounce. A second gold nugget weighs 0.8 ounce.

    • How many times as much as the first nugget does the second nugget weigh?
    • How many times as much as the second nugget does the first nugget weigh?
Show Solution
Solution
  1. Sample response: 265 of the 0.001 ounce weights or two 0.1 ounce, six 0.01 ounce, and five 0.001 ounce weights
    • 100
    • 1100\frac{1}{100}

Problem 11

Noah threw the ball 4.89 yards.

  1. Noah threw the ball farther than Lin. How far could Lin have thrown it?
  2. Andre threw the ball farther than Noah but less than 4.9 yards. How far could Andre have thrown it? Explain or show your reasoning.
Show Solution
Solution

Sample responses:

  1. 4 yards, because 4.89 has 89 hundredths more.
  2. 4.895 yards. I needed thousandths because there are no hundredths between 4.89 and 4.90.

Problem 12

  1. Label the tick marks. Use the number line to explain your reasoning.

    Number line. Eleven evenly spaced tick marks. First tick mark, 65 hundredths. Last tick mark, 66 hundredths.

  2. Which is greater, 0.654 or 0.658? Explain or show your reasoning.
Show Solution
Solution

  1. Number line
  2. 0.658. Sample response: It's further to the right on the number line.

Problem 13

A $5 gold coin weighs 8.359 grams.

  1. Locate 8.359 on the number line.

  2. A scale measures to the nearest 0.01 gram. What will the scale show for the weight of the coin? Explain or show your reasoning.
Show Solution
Solution
  1. point on a number line
  2. 8.36 grams. Sample response: It's between 8.35 and 8.36 but much closer to 8.36 than to 8.35.

Problem 14

  1. What is 0.374 rounded to the nearest hundredth? Explain or show your reasoning. Use the number line if it's helpful.

  2. What is 9.893 rounded to the nearest tenth? To the nearest hundredth? Explain or show your reasoning.
Show Solution
Solution
  1. 0.37. Sample response: It's closer to 0.37 than to 0.38. 
    point on a number line
  2. 9.9. Sample response: It's between 9.8 and 9.9 but closer to 9.9 and 9.89 to the nearest hundredth. It is between 9.89 and 9.90 but closer to 9.89

Problem 15

List the decimals from least to greatest: 6.95, 6.895, 6.598, 6.985, 5.986

Show Solution
Solution
5.986, 6.598, 6.895, 6.95, 6.985

Problem 16

To the nearest hundredth of a mile per hour, a luge rider's top speed was 81.73 mph. What are some possible speeds to the thousandth of a mile per hour? Use the number line if it is helpful.

Number line. Scale, 81 and 72 hundredths to 81 and 74 hundredths, by hundredths.

Show Solution
Solution
Sample response: 81.725, 81.726, 81.727, 81.728, 81.729, 81.730, 81.731, 81.732, 81.733, 81.734

Problem 17

  1. Jada has 3 doubloons. She knows that 2 have the same weight and 1 is heavier than the other 2. Jada also has a balance to compare the weights of coins. Explain or show how Jada can use the balance to figure out which doubloon is heavier and which 2 weigh the same.
  2. What if Jada has 5 doubloons and knows that 4 have the same weight and 1 is heavier?
Show Solution
Solution

Sample response:

  1. She can put one doubloon on one side of the balance and one doubloon on the other side of the balance. If the balance tilts, that tells Jada which doubloon is heavier.
  2. She can put 2 doubloons on one side of the balance and 2 doubloons on the other side. If it balances, then the doubloon not on the scale is the heavier one. If the scale tilts, she can weigh the two doubloons on the heavier side against each other and one of them will be the heavier one.

Problem 18

There are 2 packages of ground beef at the store. One package says it has 1 pound of beef. The second package says it has 0.97 pound of beef. Jada says that the 1-pound package has more beef. Do you agree? Explain or show your reasoning.

Show Solution
Solution

Sample responses:

  • Yes, 1>0.971 > 0.97. So, the 1 pound package has more beef than the 0.97 pound package.
  • It's possible that the 1 pound is rounded. If it is rounded to the nearest hundredth then it has more meat because it has between 0.995 and 1.004 pounds. If the package is rounded to the nearest tenth, then it has between 0.95 and 1.04 pounds. So, it could have less than 0.97 pounds.