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Equivalent Expressions

Grade 7 · Algebra · Worksheet 1

  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. Isabella draws a rectangular prism on a coordinate grid with vertices at (0,0,0), (9,0,0), (9,12,0), (0,12,0), (0,0,15), (9,0,15), (9,12,15), and (0,12,15). She then creates a new expression for the total surface area of the prism by expanding 2(9*12 + 9*15 + 12*15). Which of the following expressions is equivalent to the total surface area? A) 2*9*12 + 2*9*15 + 2*12*15 B) 2*9 + 2*12 + 2*15 C) 9*12 + 9*15 + 12*15 D) 2(9*12) + 2(9*15) + 2(12*15) Answer: ______________
  3. Rewrite 9(2x + 7) - 5(3x - 4) in simplest form. Answer: ______________
  4. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a $75 setup fee plus $12 per shirt. If Emma's budget for t-shirts is $1,500, what is the maximum number of shirts she can order without exceeding her budget? Answer: ______________
  5. Rewrite 5(4x + 10) - 25x in simplest form. Answer: ______________
  6. Rewrite 8(3x + 7) + 5(2x - 9) in simplest form. Answer: ______________
  7. Olivia is designing a rectangular community garden. The length of the garden is 7 meters more than three times its width. She writes two expressions to represent the area of the garden: one in expanded form and one in factored form. If the width of the garden is w meters, write two equivalent expressions for the area of the garden. Then, determine the area of the garden when the width is 9 meters using both expressions to verify they are equivalent. Answer: ______________
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Answer Key & Explanations

Equivalent Expressions · Grade 7 · Worksheet 1

  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. Isabella draws a rectangular prism on a coordinate grid with vertices at (0,0,0), (9,0,0), (9,12,0), (0,12,0), (0,0,15), (9,0,15), (9,12,15), and (0,12,15). She then creates a new expression for the total surface area of the prism by expanding 2(9*12 + 9*15 + 12*15). Which of the following expressions is equivalent to the total surface area? A) 2*9*12 + 2*9*15 + 2*12*15 B) 2*9 + 2*12 + 2*15 C) 9*12 + 9*15 + 12*15 D) 2(9*12) + 2(9*15) + 2(12*15) Answer: A) 2*9*12 + 2*9*15 + 2*12*15 Solution: The original expression is 2(9*12 + 9*15 + 12*15). This is a factored form of the surface area formula. Here, a = 2, b = 9*12, c = 9*15, and d = 12*15.
    Full step-by-step solution

    Step 1: The original expression is 2(9*12 + 9*15 + 12*15). This is a factored form of the surface area formula. Step 2: Apply the distributive property: a(b + c + d) = ab + ac + ad. Here, a = 2, b = 9*12, c = 9*15, and d = 12*15. Step 3: Distribute the 2 to each term inside the parentheses: 2 * (9*12) = 2*9*12 2 * (9*15) = 2*9*15 2 * (12*15) = 2*12*15 Step 4: The expanded expression is 2*9*12 + 2*9*15 + 2*12*15. Step 5: Compare with the options: Option A: 2*9*12 + 2*9*15 + 2*12*15 ✓ (matches exactly) Option B: 2*9 + 2*12 + 2*15 ✗ (incorrect - missing multiplication between dimensions) Option C: 9*12 + 9*15 + 12*15 ✗ (missing the factor of 2) Option D: 2(9*12) + 2(9*15) + 2(12*15) ✗ (while mathematically equivalent to A, it is not fully simplified as the parentheses are unnecessary; the problem asks for the simplest expanded form) Step 6: The simplest expanded form without unnecessary parentheses is option A. The answer is A) 2*9*12 + 2*9*15 + 2*12*15.

  3. Rewrite 9(2x + 7) - 5(3x - 4) in simplest form. Answer: 3x + 83 Solution: Expand 9(2x + 7) using the distributive property: 9 × 2x = 18x and 9 × 7 = 63, so 9(2x + 7) = 18x + 63. Expand 5(3x - 4) using the distributive property: 5 × 3x = 15x and 5 × (-4) = -20, so 5(3x - 4) = 15x - 20.
    Full step-by-step solution

    Step 1: Expand 9(2x + 7) using the distributive property: 9 × 2x = 18x and 9 × 7 = 63, so 9(2x + 7) = 18x + 63. Step 2: Expand 5(3x - 4) using the distributive property: 5 × 3x = 15x and 5 × (-4) = -20, so 5(3x - 4) = 15x - 20. Step 3: Now the expression is (18x + 63) - (15x - 20). Subtracting the second group means subtracting each term: 18x + 63 - 15x + 20. Step 4: Combine like terms: 18x - 15x = 3x, and 63 + 20 = 83. Step 5: The simplified expression is 3x + 83. The answer is 3x + 83.

  4. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a $75 setup fee plus $12 per shirt. If Emma's budget for t-shirts is $1,500, what is the maximum number of shirts she can order without exceeding her budget? Answer: 118 Solution: Let x represent the number of shirts Emma can order. The total cost is the setup fee plus the cost per shirt: 75 + 12x Set up the inequality: 75 + 12x ≤ 1500 Subtract 75 from both sides: 12x ≤ 1425 Divide both sides by 12: x ≤ 1425 ÷ 12 Calculate: 1425 ÷ 12 = 118.75 Since Emma cannot order a…
    Full step-by-step solution

    Step 1: Let x represent the number of shirts Emma can order. Step 2: The total cost is the setup fee plus the cost per shirt: 75 + 12x Step 3: Set up the inequality: 75 + 12x ≤ 1500 Step 4: Subtract 75 from both sides: 12x ≤ 1425 Step 5: Divide both sides by 12: x ≤ 1425 ÷ 12 Step 6: Calculate: 1425 ÷ 12 = 118.75 Step 7: Since Emma cannot order a fraction of a shirt, the maximum number is 118. The answer is 118 shirts.

  5. Rewrite 5(4x + 10) - 25x in simplest form. Answer: -5x + 50 Solution: Distribute the 5: 5(4x + 10) = 5 × 4x + 5 × 10 = 20x + 50. Subtract 25x: 20x + 50 - 25x. Combine like terms: 20x - 25x = -5x, so the expression becomes -5x + 50.
    Full step-by-step solution

    Step 1: Distribute the 5: 5(4x + 10) = 5 × 4x + 5 × 10 = 20x + 50. Step 2: Subtract 25x: 20x + 50 - 25x. Step 3: Combine like terms: 20x - 25x = -5x, so the expression becomes -5x + 50. The final answer is -5x + 50.

  6. Rewrite 8(3x + 7) + 5(2x - 9) in simplest form. Answer: 34x + 11 Solution: Distribute 8 into (3x + 7): 8 × 3x = 24x, 8 × 7 = 56, so 8(3x + 7) = 24x + 56. Distribute 5 into (2x - 9): 5 × 2x = 10x, 5 × (-9) = -45, so 5(2x - 9) = 10x - 45.
    Full step-by-step solution

    Step 1: Distribute 8 into (3x + 7): 8 × 3x = 24x, 8 × 7 = 56, so 8(3x + 7) = 24x + 56. Step 2: Distribute 5 into (2x - 9): 5 × 2x = 10x, 5 × (-9) = -45, so 5(2x - 9) = 10x - 45. Step 3: Add the two results: (24x + 56) + (10x - 45) = 24x + 10x + 56 - 45. Step 4: Combine like terms: 24x + 10x = 34x, and 56 - 45 = 11. Step 5: The simplified expression is 34x + 11. The answer is 34x + 11.

  7. Olivia is designing a rectangular community garden. The length of the garden is 7 meters more than three times its width. She writes two expressions to represent the area of the garden: one in expanded form and one in factored form. If the width of the garden is w meters, write two equivalent expressions for the area of the garden. Then, determine the area of the garden when the width is 9 meters using both expressions to verify they are equivalent. Answer: Area = 306 square meters Solution: Express the length. The length is 7 meters more than three times the width: L = 3w + 7. Write the area in expanded form.
    Full step-by-step solution

    Step 1: Express the length. The length is 7 meters more than three times the width: L = 3w + 7. Step 2: Write the area in expanded form. Area = length * width = (3w + 7) * w = 3w^2 + 7w. Step 3: Write the area in factored form. The common factor in 3w^2 + 7w is w: Area = w(3w + 7). Step 4: Substitute w = 9 into the expanded form: 3(9)^2 + 7(9) = 3(81) + 63 = 243 + 63 = 306. Step 5: Substitute w = 9 into the factored form: 9(3*9 + 7) = 9(27 + 7) = 9 * 34 = 306. Step 6: Both forms give 306, so they are equivalent. The area of the garden is 306 square meters. The answer is 306 square meters.