Expression Properties

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Answer Key & Explanations

Expression Properties · Grade 6 · Worksheet 3

1. Hana draws a rectangular prism with dimensions: length = 12 cm, width = 6 cm, and height = 4 cm. She then draws a net of the prism. Use the distributive property to write an equivalent expression for the total surface area of the prism, and then calculate the area. Answer: 288 Solution: Identify the three pairs of identical faces and their areas. - Pair 1 (length × height): 12 × 4 = 48 square cm each, two faces: 2 × 48 = 96 - Pair 2 (width × height): 6 × 4 = 24 square cm each, two faces: 2 × 24 = 48 - Pair 3 (length × width): 12 × 6 = 72 square cm each, two faces: 2 × 72 = 144…
   Full step-by-step solution

   Step 1: Identify the three pairs of identical faces and their areas. - Pair 1 (length × height): 12 × 4 = 48 square cm each, two faces: 2 × 48 = 96 - Pair 2 (width × height): 6 × 4 = 24 square cm each, two faces: 2 × 24 = 48 - Pair 3 (length × width): 12 × 6 = 72 square cm each, two faces: 2 × 72 = 144 Step 2: Write the sum of all face areas: 96 + 48 + 144. Step 3: Factor out the common factor of 2 using the distributive property: 96 + 48 + 144 = 2(48 + 24 + 72) Step 4: Calculate inside the parentheses: 48 + 24 + 72 = 144. Step 5: Multiply: 2 × 144 = 288. The total surface area is 288 square cm.

2. 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.

3. Olivia is organizing a school art fair and needs to create an equivalent expression for the total cost of supplies. She buys 7 boxes of paint brushes, each costing (3p + 5) dollars, where p represents the price of a single paintbrush in dollars. Olivia remembers that she can use the distributive property to rewrite the total cost expression in a different form. What is the equivalent expression for the total cost of the paint brushes using the distributive property? Answer: 21p + 35 Solution: The original expression for the total cost is 7(3p + 5). Multiply 7 by 3p: 7 × 3p = 21p. Multiply 7 by 5: 7 × 5 = 35.
   Full step-by-step solution

   Step 1: The original expression for the total cost is 7(3p + 5). Step 2: Apply the distributive property: a(b + c) = ab + ac, where a = 7, b = 3p, and c = 5. Step 3: Multiply 7 by 3p: 7 × 3p = 21p. Step 4: Multiply 7 by 5: 7 × 5 = 35. Step 5: Combine the terms: 21p + 35. The equivalent expression is 21p + 35.

4. Maria is making a fruit punch recipe that requires a 3:2:1 ratio of orange juice, pineapple juice, and cranberry juice. If she wants to make 12 liters of punch total, how many liters of cranberry juice does she need? Answer: 2 Solution: The ratio is 3:2:1 for orange:pineapple:cranberry juice. Add the ratio parts: 3 + 2 + 1 = 6 total parts. The total punch is 12 liters, so each ratio part equals 12 ÷ 6 = 2 liters.
   Full step-by-step solution

   Step 1: The ratio is 3:2:1 for orange:pineapple:cranberry juice. Step 2: Add the ratio parts: 3 + 2 + 1 = 6 total parts. Step 3: The total punch is 12 liters, so each ratio part equals 12 ÷ 6 = 2 liters. Step 4: Cranberry juice is 1 part, so 1 × 2 = 2 liters. Maria needs 2 liters of cranberry juice.

5. Emma drew a diagram showing a large rectangle divided into two smaller rectangles side by side. The large rectangle has a total width represented by the expression (7 + 3y) inches and a height of 5 inches. The left smaller rectangle has a width of 7 inches and the right smaller rectangle has a width of 3y inches. Using the visual model of the area of the large rectangle, write an equivalent expression for the total area by applying the distributive property. Answer: 5(7) + 5(3y) or 35 + 15y Solution: The area of a rectangle is found by multiplying its length (or width) by its height. Here, the large rectangle has a total width of (7 + 3y) inches and a height of 5 inches.
   Full step-by-step solution

   Step 1: The area of a rectangle is found by multiplying its length (or width) by its height. Here, the large rectangle has a total width of (7 + 3y) inches and a height of 5 inches. So, the total area is 5(7 + 3y). Step 2: To apply the distributive property, multiply the height, 5, by each term inside the parentheses: 5 * 7 = 35 and 5 * 3y = 15y. Step 3: The equivalent expression for the total area is 35 + 15y. This represents the sum of the areas of the left rectangle (5 * 7 = 35 square inches) and the right rectangle (5 * 3y = 15y square inches). Final Answer: 5(7) + 5(3y) or 35 + 15y

6. Sophia is organizing a school donation drive. She has collected 1200 notebooks and wants to pack them into boxes. Each box can hold either 15 notebooks or 20 notebooks. She has already filled 25 boxes with 15 notebooks each. Use the distributive property to write an equivalent expression for the total number of notebooks she has packed so far, and then find that total. Answer: 375 Solution: Identify the expression for the total notebooks packed. The total is 25 boxes × 15 notebooks per box = 25 × 15. Use the distributive property to rewrite 25 × 15.
   Full step-by-step solution

   Step 1: Identify the expression for the total notebooks packed. The total is 25 boxes × 15 notebooks per box = 25 × 15. Step 2: Use the distributive property to rewrite 25 × 15. 25 × 15 = (20 + 5) × 15 Using the distributive property: (20 × 15) + (5 × 15) Step 3: Calculate each part. 20 × 15 = 300 5 × 15 = 75 Step 4: Add the parts together. 300 + 75 = 375 The total number of notebooks packed is 375.

7. Mason draws a diagram of a large rectangle divided into two smaller rectangles side by side to model the expression 17(x + 22). The large rectangle has a length of 17 units and a width of (x + 22) units. Using the distributive property, write an equivalent expression for the area of the large rectangle. Answer: 17x + 374 Solution: The area of the large rectangle is given by the expression 17(x + 22). Step 2: Apply the distributive property: multiply 17 by each term inside the parentheses. 17 * x = 17x.
   Full step-by-step solution

   Step 1: The area of the large rectangle is given by the expression 17(x + 22). Step 2: Apply the distributive property: multiply 17 by each term inside the parentheses. 17 * x = 17x. 17 * 22 = 374. Step 3: Write the sum of these products: 17x + 374. The equivalent expression is 17x + 374.