Expression Properties

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Answer Key & Explanations

Expression Properties · Grade 6 · Worksheet 2

1. Use the distributive property to rewrite: 9(8x + 11) Answer: 72x + 99 Solution: Here, a = 9, b = 8x, c = 11. Multiply 9 by 8x: 9 × 8x = 72x. Multiply 9 by 11: 9 × 11 = 99.
   Full step-by-step solution

   Step 1: Apply the distributive property: a(b + c) = ab + ac. Here, a = 9, b = 8x, c = 11. Step 2: Multiply 9 by 8x: 9 × 8x = 72x. Step 3: Multiply 9 by 11: 9 × 11 = 99. Step 4: Combine the terms: 72x + 99. The answer is 72x + 99.

2. Tane is helping his school's gardening club set up a rectangular herb garden. The length of the garden is 14 meters, and the width is 9 meters. They want to plant a border of flowers around the entire garden. The number of flower plants needed for the border is given by the expression 2(14 + 9). Use the distributive property to rewrite this expression in an equivalent form without parentheses, then calculate how many flower plants they need. Answer: 46 Solution: Start with the expression 2(14 + 9). Calculate each product: 2×14 = 28 and 2×9 = 18. Add the results: 28 + 18 = 46.
   Full step-by-step solution

   Step 1: Start with the expression 2(14 + 9). Step 2: Apply the distributive property: 2(14 + 9) = 2×14 + 2×9. Step 3: Calculate each product: 2×14 = 28 and 2×9 = 18. Step 4: Add the results: 28 + 18 = 46. The answer is 46 flower plants.

3. (-4)³ ÷ 8 + 2 × (-5) = ? Answer: -18 Solution: Calculate the exponent: (-4)³ = (-4) × (-4) × (-4) = 16 × (-4) = -64 Perform the division: -64 ÷ 8 = -8 Perform the multiplication: 2 × (-5) = -10 Add the results: -8 + (-10) = -18 The answer is -18.
   Full step-by-step solution

   Step 1: Calculate the exponent: (-4)³ = (-4) × (-4) × (-4) = 16 × (-4) = -64 Step 2: Perform the division: -64 ÷ 8 = -8 Step 3: Perform the multiplication: 2 × (-5) = -10 Step 4: Add the results: -8 + (-10) = -18 The answer is -18.

4. Noah is helping his school's art club design a display for their annual art show. They have a large rectangular board that is 6x + 11 inches wide and 4x + 6 inches tall. Noah wants to write an equivalent expression for the area of the board using the distributive property. If he rewrites the area as (6x + 11)(4x + 6), what is the equivalent expression in expanded form? Answer: 24x^2 + 80x + 66 Solution: Write the expression for area: (6x + 11)(4x + 6) Apply the distributive property (also known as FOIL): First: 6x * 4x = 24x^2 Outer: 6x * 6 = 36x Inner: 11 * 4x = 44x Last: 11 * 6 = 66 Combine like terms: 36x + 44x = 80x Write the equivalent expression: 24x^2 + 80x + 66 The answer is 24x^2 + 80x…
   Full step-by-step solution

   Step 1: Write the expression for area: (6x + 11)(4x + 6) Step 2: Apply the distributive property (also known as FOIL): First: 6x * 4x = 24x^2 Outer: 6x * 6 = 36x Inner: 11 * 4x = 44x Last: 11 * 6 = 66 Step 3: Combine like terms: 36x + 44x = 80x Step 4: Write the equivalent expression: 24x^2 + 80x + 66 The answer is 24x^2 + 80x + 66.

5. Liam is designing a rectangular garden with a length-to-width ratio of 5:3. If the area of the garden is 240 square meters, what is the width of the garden in meters? Answer: 12 Solution: Let the length be \( 5x \) and the width be \( 3x \), because the ratio of length to width is 5:3.
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Define variables for length and width** Let the length be \( 5x \) and the width be \( 3x \), because the ratio of length to width is 5:3. --- **Step 2: Write the area equation** Area of a rectangle = length × width So: \( (5x) \times (3x) = 240 \) --- **Step 3: Simplify the equation** \( 15x^2 = 240 \) --- **Step 4: Solve for \( x^2 \)** Divide both sides by 15: \( x^2 = 240 / 15 \) \( x^2 = 16 \) --- **Step 5: Solve for \( x \)** \( x = \sqrt{16} \) \( x = 4 \) (We take the positive root since length and width are positive.) --- **Step 6: Find the width** Width = \( 3x = 3 \times 4 = 12 \) meters. --- **Final Answer:** 12

6. Liam is planning a school fundraiser selling custom t-shirts. He found a supplier that charges $8.50 per shirt for orders up to 100 shirts, but offers a 15% discount on all shirts if he orders more than 100. If Liam wants to order 120 shirts, how much will he pay in total after the discount? Answer: 867 Solution: 120 shirts × $8.50 per shirt = $1,020 15% of $1,020 = 0.15 × 1,020 = $153 $1,020 - $153 = $867 The answer is $867.
   Full step-by-step solution

   Step 1: Calculate the original cost without discount 120 shirts × $8.50 per shirt = $1,020 Step 2: Calculate the discount amount 15% of $1,020 = 0.15 × 1,020 = $153 Step 3: Subtract the discount from the original cost $1,020 - $153 = $867 The answer is $867.

7. Use the distributive property to rewrite: 12(2x + 5) = ? Answer: 24x + 60 Solution: Here, a = 12, b = 2x, c = 5. Multiply 12 by 2x: 12 × 2x = 24x. Multiply 12 by 5: 12 × 5 = 60.
   Full step-by-step solution

   Step 1: Apply the distributive property: a(b + c) = ab + ac. Here, a = 12, b = 2x, c = 5. Step 2: Multiply 12 by 2x: 12 × 2x = 24x. Step 3: Multiply 12 by 5: 12 × 5 = 60. Step 4: Add the results: 24x + 60. The answer is 24x + 60.

8. Use the distributive property to rewrite: 7(3x + 11) = ? Answer: 21x + 77 Solution: Identify the expression: 7(3x + 11) Apply the distributive property: a(b + c) = ab + ac Multiply 7 by 3x: 7 × 3x = 21x Multiply 7 by 11: 7 × 11 = 77 Combine the results: 21x + 77 The answer is 21x + 77.
   Full step-by-step solution

   Step 1: Identify the expression: 7(3x + 11) Step 2: Apply the distributive property: a(b + c) = ab + ac Step 3: Multiply 7 by 3x: 7 × 3x = 21x Step 4: Multiply 7 by 11: 7 × 11 = 77 Step 5: Combine the results: 21x + 77 The answer is 21x + 77.