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Equivalent Expressions

Grade 7 · Algebra · Worksheet 3

  1. Emma is organizing a charity run and needs to calculate the total distance covered by all participants. There are 245 runners, and each runner completes 3.2 kilometers. If the charity donates $1.50 for every kilometer run, how much money will be raised in total? Answer: ______________
  2. (3/4 × 8/9) ÷ (2/3) = ? Answer: ______________
  3. Mere draws a rectangle on a coordinate plane with corners at (2, 2), (10, 2), (10, 6), and (2, 6). She then draws a second rectangle that has the same area but a different shape. The second rectangle has a length of 16 units. What is the width of the second rectangle? Show that the two rectangles are equivalent by writing an expression for the area of each rectangle in expanded and simplified form. Answer: ______________
  4. Rewrite 9(4x + 12) - 3(2x - 8) in simplest form. Answer: ______________
  5. Rewrite 5(2x + 10) - 25 in simplest form. Answer: ______________
  6. Rewrite 9(4x + 11) - 7x in simplest form. Answer: ______________
  7. A rectangular garden is drawn on a coordinate plane with vertices at A(-6, -1), B(6, -1), C(6, 11), and D(-6, 11). A square flower bed is placed inside the garden with its corners at (1, 4), (6, 4), (6, 9), and (1, 9). Write an expression in expanded form for the area of the garden that is not covered by the flower bed. Then, factor the expression completely and explain what each factor represents in terms of the garden's dimensions. Answer: ______________
  8. Rewrite 5(3x - 7) + 9x in simplest form. Answer: ______________
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Answer Key & Explanations

Equivalent Expressions · Grade 7 · Worksheet 3

  1. Emma is organizing a charity run and needs to calculate the total distance covered by all participants. There are 245 runners, and each runner completes 3.2 kilometers. If the charity donates $1.50 for every kilometer run, how much money will be raised in total? Answer: 1176 Solution: Calculate the total distance run by all participants. Number of runners = 245 Distance per runner = 3.2 km Total distance = 245 × 3.2 245 × 3 = 735 245 × 0.2 = 49 735 + 49 = 784 km Calculate the total donation amount.
    Full step-by-step solution

    Step 1: Calculate the total distance run by all participants. Number of runners = 245 Distance per runner = 3.2 km Total distance = 245 × 3.2 245 × 3 = 735 245 × 0.2 = 49 735 + 49 = 784 km Step 2: Calculate the total donation amount. Donation rate = $1.50 per km Total donation = 784 × 1.50 784 × 1 = 784 784 × 0.5 = 392 784 + 392 = 1176 The total amount raised is $1176.

  2. (3/4 × 8/9) ÷ (2/3) = ? Answer: 1 Solution: First, compute 3/4 × 8/9. Multiply the numerators: 3 × 8 = 24 Multiply the denominators: 4 × 9 = 36 So, 3/4 × 8/9 = 24/36 Simplify 24/36. The greatest common divisor of 24 and 36 is 12.
    Full step-by-step solution

    Let's solve step-by-step. Step 1: First, compute 3/4 × 8/9. Multiply the numerators: 3 × 8 = 24 Multiply the denominators: 4 × 9 = 36 So, 3/4 × 8/9 = 24/36 Step 2: Simplify 24/36. The greatest common divisor of 24 and 36 is 12. Divide numerator and denominator by 12: 24 ÷ 12 = 2 36 ÷ 12 = 3 So, 24/36 = 2/3 Step 3: Now the expression is (2/3) ÷ (2/3). Dividing by a fraction is the same as multiplying by its reciprocal. So, (2/3) ÷ (2/3) = (2/3) × (3/2) Step 4: Multiply numerators: 2 × 3 = 6 Multiply denominators: 3 × 2 = 6 So, 6/6 = 1 Final answer: 1

  3. Mere draws a rectangle on a coordinate plane with corners at (2, 2), (10, 2), (10, 6), and (2, 6). She then draws a second rectangle that has the same area but a different shape. The second rectangle has a length of 16 units. What is the width of the second rectangle? Show that the two rectangles are equivalent by writing an expression for the area of each rectangle in expanded and simplified form. Answer: The width of the second rectangle is 2 units. The area of the first rectangle is 8 * 4 = 32 square units, or (10-2) * (6-2) = 8 * 4 = 32. The area of the second rectangle is 16 * 2 = 32 square units. Solution: Find the dimensions of the first rectangle. The length is the difference in x-coordinates: 10 - 2 = 8 units. The width is the difference in y-coordinates: 6 - 2 = 4 units.
    Full step-by-step solution

    Step 1: Find the dimensions of the first rectangle. The length is the difference in x-coordinates: 10 - 2 = 8 units. The width is the difference in y-coordinates: 6 - 2 = 4 units. Step 2: Calculate the area of the first rectangle. Area = length * width = 8 * 4 = 32 square units. As an expression, this is (10 - 2) * (6 - 2) = 8 * 4 = 32. Step 3: The second rectangle has the same area (32 square units) and a length of 16 units. Step 4: Let w represent the width of the second rectangle. The area of the second rectangle is 16 * w. Step 5: Set the areas equal to each other: 16 * w = 32. Step 6: Solve for w: w = 32 / 16 = 2 units. Step 7: Write the area of the second rectangle as an expression: 16 * 2 = 32. Step 8: Show equivalence: Both rectangles have an area of 32 square units. The expression for the first rectangle (8 * 4) simplifies to 32, and the expression for the second rectangle (16 * 2) also simplifies to 32. Therefore, the two rectangles are equivalent because they have the same area. The answer is: The width of the second rectangle is 2 units.

  4. Rewrite 9(4x + 12) - 3(2x - 8) in simplest form. Answer: 30x + 132 Solution: 9 × 4x = 36x 9 × 12 = 108 So, 9(4x + 12) = 36x + 108 Apply the distributive property to expand -3(2x - 8). -3 × 2x = -6x -3 × (-8) = 24 So, -3(2x - 8) = -6x + 24 Combine the two expanded expressions.
    Full step-by-step solution

    Step 1: Apply the distributive property to expand 9(4x + 12). 9 × 4x = 36x 9 × 12 = 108 So, 9(4x + 12) = 36x + 108 Step 2: Apply the distributive property to expand -3(2x - 8). -3 × 2x = -6x -3 × (-8) = 24 So, -3(2x - 8) = -6x + 24 Step 3: Combine the two expanded expressions. (36x + 108) + (-6x + 24) = 36x + 108 - 6x + 24 Step 4: Combine like terms. 36x - 6x = 30x 108 + 24 = 132 Step 5: Write the simplified expression. 30x + 132 The answer is 30x + 132.

  5. Rewrite 5(2x + 10) - 25 in simplest form. Answer: 10x + 25 Solution: Distribute the 5 to each term inside the parentheses: 5(2x) = 10x and 5(10) = 50. So 5(2x + 10) - 25 becomes 10x + 50 - 25. Combine the constant terms: 50 - 25 = 25.
    Full step-by-step solution

    Step 1: Distribute the 5 to each term inside the parentheses: 5(2x) = 10x and 5(10) = 50. So 5(2x + 10) - 25 becomes 10x + 50 - 25. Step 2: Combine the constant terms: 50 - 25 = 25. Step 3: The simplified expression is 10x + 25. The answer is 10x + 25.

  6. Rewrite 9(4x + 11) - 7x in simplest form. Answer: 29x + 99 Solution: Distribute the 9 to each term inside the parentheses: 9(4x + 11) = 9 × 4x + 9 × 11 = 36x + 99. Now the expression is 36x + 99 - 7x. Combine like terms (the terms with x): 36x - 7x = 29x.
    Full step-by-step solution

    Step 1: Distribute the 9 to each term inside the parentheses: 9(4x + 11) = 9 × 4x + 9 × 11 = 36x + 99. Step 2: Now the expression is 36x + 99 - 7x. Step 3: Combine like terms (the terms with x): 36x - 7x = 29x. Step 4: The constant term 99 remains. Step 5: The simplified expression is 29x + 99. Final answer: 29x + 99.

  7. A rectangular garden is drawn on a coordinate plane with vertices at A(-6, -1), B(6, -1), C(6, 11), and D(-6, 11). A square flower bed is placed inside the garden with its corners at (1, 4), (6, 4), (6, 9), and (1, 9). Write an expression in expanded form for the area of the garden that is not covered by the flower bed. Then, factor the expression completely and explain what each factor represents in terms of the garden's dimensions. Answer: Expanded: 12(12) - 5(5) = 144 - 25 = 119 square units; Factored: (12 - 5)(12 + 5) = 7 * 17 = 119 square units; The factors 12 - 5 and 12 + 5 represent the difference and sum of the garden's side length and the flower bed's side length. Solution: Identify the garden dimensions. The vertices are A(-6, -1), B(6, -1), C(6, 11), D(-6, 11). The length along the x-axis from -6 to 6 is 12 units.
    Full step-by-step solution

    Step 1: Identify the garden dimensions. The vertices are A(-6, -1), B(6, -1), C(6, 11), D(-6, 11). The length along the x-axis from -6 to 6 is 12 units. The height along the y-axis from -1 to 11 is 12 units. So the garden is a 12 by 12 square. Area of garden = 12 * 12 = 144 square units. Step 2: Identify the flower bed dimensions. The square has corners at (1, 4), (6, 4), (6, 9), and (1, 9). The side length from x=1 to x=6 is 5 units. The side length from y=4 to y=9 is also 5 units. Area of flower bed = 5 * 5 = 25 square units. Step 3: The area not covered by the flower bed is the garden area minus the flower bed area. In expanded form: 12(12) - 5(5) = 144 - 25 = 119 square units. Step 4: Notice that both areas are perfect squares: 12^2 and 5^2. So the expression is 12^2 - 5^2, which is a difference of squares. Factor using a^2 - b^2 = (a - b)(a + b): 12^2 - 5^2 = (12 - 5)(12 + 5) = 7 * 17 = 119 square units. Step 5: Interpretation of factors: 12 - 5 = 7 represents the difference between the garden's side length and the flower bed's side length. 12 + 5 = 17 represents the sum of the garden's side length and the flower bed's side length. Their product gives the remaining area. The answer is 119 square units.

  8. Rewrite 5(3x - 7) + 9x in simplest form. Answer: 24x - 35 Solution: Multiply 5 by each term inside: 5 * 3x = 15x and 5 * (-7) = -35. So, 5(3x - 7) = 15x - 35. Now the expression is 15x - 35 + 9x.
    Full step-by-step solution

    Step 1: Apply the distributive property to 5(3x - 7). Multiply 5 by each term inside: 5 * 3x = 15x and 5 * (-7) = -35. So, 5(3x - 7) = 15x - 35. Step 2: Now the expression is 15x - 35 + 9x. Step 3: Combine like terms. The terms with x are 15x and 9x. Add them: 15x + 9x = 24x. Step 4: The constant term is -35. So the simplified expression is 24x - 35. The answer is 24x - 35.