Section A Practice Problems

Problem 1

There are 63 students in the cafeteria. There are 9 students at each table.

  1. At how many tables are the students seated?
  2. Write a division equation to represent your answer.
Show Solution
Solution
  1. 7
  2. 63÷9=763 \div 9 = 7 or 63÷7=963 \div 7 = 9. 

Problem 2

What is the area of this figure? Explain your reasoning.

6-sided shape.
6-sided shape. Straight sides. All side lengths meet at right angles. Side lengths. Bottom, 10 centimeters. Right side rises question mark, then goes left 4 centimeters, and goes up 6 centimeters. Top side length, question mark. Left side length, 10 centimeters.

Show Solution
Solution
76 square centimeters. Sample Response: I can draw a 10 cm by 10 cm square around the figure which has area 100 square centimeters. Then I subtract the 6 cm by 4 cm rectangle in the top right which has area 24 square centimeters. 100−24=76100 - 24 = 76.

Problem 3

Select all expressions that are equivalent to 125\frac{12}{5}.

Show Solution
Solution
A,C,E

Problem 4

Jada has 8 pennies. Each one weighs 52\frac{5}{2} grams. How much do her pennies weigh altogether? Explain your reasoning.

Show Solution
Solution
20 grams. Sample response: Since 8×52=4028 \times \frac{5}{2} = \frac{40}{2}, the pennies weigh 402\frac{40}{2} grams which is 20 grams.
 

Problem 5

  1. Shade 12\frac{1}{2} of 15\frac{1}{5} of the square.

    Diagram, square. Length and width, 1. 

  2. Explain where you see 12\frac{1}{2} of 15\frac{1}{5} in your drawing.
Show Solution
Solution

Sample response:

  1.  
    Diagram

  2. First, I divided the squares into fifths vertically and then I shaded 12\frac{1}{2} of one of those pieces.

Problem 6

  1. Write an expression for how much of the square is shaded.

    Diagram, square. Length and width, 1. Partitioned into 5 rows of 4 of the same size rectangles. 1 rectangle shaded.

  2. Find the value of your expression.
Show Solution
Solution
  1. 14×15\frac{1}{4} \times \frac{1}{5} or 15×14\frac{1}{5} \times \frac{1}{4}
  2. 120\frac{1}{20}

Problem 7

  1. Write an equation representing the shaded part of the diagram.

    Diagram, square. Length and width, 1. Partitioned into 6 rows of 3 of the same size rectangles. 1 rectangle shaded.

  2. Explain how the diagram shows each part of your equation.
Show Solution
Solution

Sample responses:

  1. 13×16=118\frac{1}{3} \times \frac{1}{6} = \frac{1}{18} or 16×13=118\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}
  2. The square is divided in sixths horizontally so that's 16\frac{1}{6}. Then 13\frac{1}{3} of one of those sixths is shaded. The value is 118\frac{1}{18} because 1 of 18 rectangles is shaded. 

Problem 8

  1. Write an expression for the shaded region of the square.

    Diagram, square. Length and width, 1. Partitioned into 3 rows of 4 of the same size rectangles. 3 rectangles shaded.

  2. Explain how your expression matches the shaded region.
Show Solution
Solution

Sample response:

  1. 34×13\frac{3}{4} \times \frac{1}{3}
  2. The top row is 13\frac{1}{3} of the square and 34\frac{3}{4} of that 13\frac{1}{3} is shaded.

Problem 9

  1. Write an expression for the area of the shaded region.

    Diagram, square. Length and width, 1. Partitioned into 5 rows of 4 of the same size rectangles. 4 rectangles shaded.

  2. Explain how the diagram shows your expression.
Show Solution
Solution

Sample response:

  1. 45×14\frac{4}{5} \times \frac{1}{4} or 14×45\frac{1}{4} \times\frac{4}{5}
  2. The first column is 14\frac{1}{4} of the square and 45\frac{4}{5} of that is shaded.

Problem 10

  1. Write a multiplication expression for the area of the shaded region. Explain your reasoning.

  2. What is the area of the shaded region in square units?
Show Solution
Solution
  1. 56×34\frac{5}{6} \times \frac{3}{4} or 34×56\frac{3}{4} \times \frac{5}{6}. Sample response: The shaded region is 56\frac{5}{6} of 34\frac{3}{4} of the whole square.
  2. 1524\frac{15}{24} (or equivalent)

Problem 11

Find the value that makes each equation true.

  1. 710×35=‾\frac{7}{10} \times \frac{3}{5} = \underline{\hspace{0.7cm}}
  2. 25×‾=845\frac{2}{5} \times \underline{\hspace{0.7cm}} = \frac{8}{45}
  3. ‾×49=2845\underline{\hspace{0.7cm}} \times \frac{4}{9} = \frac{28}{45}
Show Solution
Solution
  1. 2150\frac{21}{50}
  2. 49\frac{4}{9}
  3. 75\frac{7}{5}

Problem 12

Lin says that 456×6124\frac{5}{6}\times6\frac{1}{2} is greater than 24.

  1. Do you agree with Lin? Explain or show your reasoning.
  2. What is the value of 456×6124\frac{5}{6}\times6\frac{1}{2}?
Show Solution
Solution
  1. Sample response: Yes, 4×64\times 6 is 24. Since 4564\frac{5}{6} is close to 5 and 6126\frac{1}{2} is more than 6, the product of 456×6124\frac{5}{6}\times6\frac{1}{2} will be more than 30.
  2. 3151231 \frac{5}{12} (or equivalent expression or value). Sample response: I multiplied 4×64 \times 6, 4×124 \times\frac{1}{2}, 56×6\frac{5}{6}\times6, and 56×12\frac{5}{6}\times\frac{1}{2} and added their products together.​​​

Problem 13

This flag of Sweden is 3153\frac{1}{5} inches wide and 2 inches tall. The rectangle in the top right is 95\frac{9}{5} inches wide and 45\frac{4}{5} inch tall.

  1. What is the area of the whole flag?

    Blue rectangle. Partitioned into two rows and two columns by yellow stripes.

  2. What is the area of the rectangle in the top right?
Show Solution
Solution
  1. 6256\frac{2}{5} square inches
  2. 3625\frac{36}{25} square inches

Problem 14

On this American flag the width of the blue rectangle is 25\frac{2}{5} the width of the flag. What fraction of the area of the flag is the blue rectangle? Explain or show your reasoning.

American flag. 13 stripes, 7 red, 6 white. Blue rectangle, top left corner. 50 white stars in blue rectangle. 

Show Solution
Solution
Sample response: 1465\frac{14}{65}. The height of the blue rectangle is 713\frac{7}{13} the height of the flag because there are 13 stripes and it’s 7 of them. So, its area is 25×713\frac{2}{5} \times \frac{7}{13} of the area of the entire flag.

Problem 15

Jada folded a square piece of paper in half many times, sometimes horizontally and sometimes vertically. She shaded the folded piece of paper and then unfolded it. Here is a picture.

Diagram, square. Length and width, 1. Rectangular portion shaded with length, about 1 eighth, width, about 1 sixteenth. 

What fraction of the paper did Jada shade? Explain how you know.

Show Solution
Solution
1128\frac{1}{128}. Sample response: By making tick marks I checked that the height of the square is 18\frac{1}{8} the height of the square. She folded 3 times in that direction. The width is half the height so 116\frac{1}{16} the width of the square. She folded 4 times in this direction. The fraction of the shaded rectangle is 18×116\frac{1}{8} \times \frac{1}{16} or 1128\frac{1}{128} of the whole square.