Equivalent Expressions

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Answer Key & Explanations

Equivalent Expressions · Grade 6 · Worksheet 2

1. (-3)² + 4 × (24 ÷ 6) = ? Answer: 25 Solution: Calculate the exponent first: (-3)² = 9 Calculate the division inside parentheses: 24 ÷ 6 = 4 Perform the multiplication: 4 × 4 = 16 Add the results: 9 + 16 = 25 The answer is 25.
   Full step-by-step solution

   Step 1: Calculate the exponent first: (-3)² = 9 Step 2: Calculate the division inside parentheses: 24 ÷ 6 = 4 Step 3: Perform the multiplication: 4 × 4 = 16 Step 4: Add the results: 9 + 16 = 25 The answer is 25.

2. Noah is planning a school field trip to a science museum. The museum charges $15 per student and $25 per adult chaperone. There are 36 students going on the trip, and the ratio of students to chaperones is 9:2. If the school also needs to pay a $150 bus fee, what is the total cost of the field trip? Answer: $1040 Solution: Find the number of chaperones using the ratio. The ratio of students to chaperones is 9:2, and there are 36 students. Since 36 ÷ 9 = 4, multiply both parts of the ratio by 4: 9 × 4 = 36 students and 2 × 4 = 8 chaperones.
   Full step-by-step solution

   Step 1: Find the number of chaperones using the ratio. The ratio of students to chaperones is 9:2, and there are 36 students. Since 36 ÷ 9 = 4, multiply both parts of the ratio by 4: 9 × 4 = 36 students and 2 × 4 = 8 chaperones. Step 2: Calculate the student ticket cost: 36 students × $15 per student = $540. Step 3: Calculate the chaperone ticket cost: 8 chaperones × $25 per chaperone = $200. Step 4: Add the bus fee: $150. Step 5: Calculate the total cost: $540 (students) + $200 (chaperones) + $150 (bus) = $890 + $150 = $1040. The total cost of the field trip is $1040.

3. (-12) + 8 - (-5) = ? Answer: 1 Solution: (-12) + 8 - (-5) Subtracting a negative number is the same as adding its positive: - (-5) = + 5 (-12) + 8 + 5 First: (-12) + 8 Think of this as 8 - 12 = -4 So now we have: -4 + 5 Add -4 + 5 This is the same as 5 - 4 = 1 The result is 1.
   Full step-by-step solution

   Let's solve step-by-step: Step 1: Write the problem clearly (-12) + 8 - (-5) Step 2: Interpret the subtraction of a negative number Subtracting a negative number is the same as adding its positive: - (-5) = + 5 So the expression becomes: (-12) + 8 + 5 Step 3: Add the numbers from left to right First: (-12) + 8 Think of this as 8 - 12 = -4 So now we have: -4 + 5 Step 4: Add -4 + 5 This is the same as 5 - 4 = 1 Step 5: Final answer The result is 1.

4. A rectangular garden has a length of 12.5 meters and a width of 8.4 meters. The gardener wants to put a fence around the entire garden. Which expression correctly calculates the perimeter of the garden? Answer: B. 2 × (12.5 + 8.4) Solution: 1. The perimeter is the total distance around the garden. 2.
   Full step-by-step solution

   Step-by-step solution: 1. Understand the problem: We have a rectangular garden with length = 12.5 meters and width = 8.4 meters. The perimeter is the total distance around the garden. 2. Recall the formula for the perimeter of a rectangle: Perimeter = 2 × (length + width) This is because a rectangle has two equal lengths and two equal widths. 3. Apply the formula to our numbers: Perimeter = 2 × (12.5 + 8.4) 4. Check each option: - Option A: 2 × 12.5 + 8.4 This equals 25 + 8.4 = 33.4, but the correct perimeter is 2 × (12.5 + 8.4) = 2 × 20.9 = 41.8, so this is wrong. - Option B: 12.5 × 8.4 This is the area, not the perimeter. - Option C: 2 × (12.5 + 8.4) This matches the correct formula. - Option D: 12.5 + 8.4 + 12.5 + 8.4 This is also correct, but it is not the simplified expression given in the problem. The question asks for the expression that correctly calculates the perimeter, and both C and D are mathematically correct, but C is the standard factored form. 5. Conclusion: The expression 2 × (12.5 + 8.4) is the standard and most efficient way to represent the perimeter of a rectangle. Final answer: 2 × (12.5 + 8.4)

5. (-3)² + 4 × (18 ÷ 3) = ? Answer: 33 Solution: Calculate the exponent first: (-3)² = 9 Calculate the division inside the parentheses: 18 ÷ 3 = 6 Perform the multiplication: 4 × 6 = 24 Add the results: 9 + 24 = 33 The answer is 33.
   Full step-by-step solution

   Step 1: Calculate the exponent first: (-3)² = 9 Step 2: Calculate the division inside the parentheses: 18 ÷ 3 = 6 Step 3: Perform the multiplication: 4 × 6 = 24 Step 4: Add the results: 9 + 24 = 33 The answer is 33.

6. Which expression is equivalent to 6(2x + 11) - 2(3x - 1)? Answer: 6x + 68 Solution: Combine the results: (12x + 66) + (-6x + 2). Combine like terms: 12x - 6x = 6x, and 66 + 2 = 68. The equivalent expression is 6x + 68.
   Full step-by-step solution

   Step 1: Apply the distributive property to the first term: 6(2x + 11) = 12x + 66. Step 2: Apply the distributive property to the second term: -2(3x - 1) = -6x + 2. Step 3: Combine the results: (12x + 66) + (-6x + 2). Step 4: Combine like terms: 12x - 6x = 6x, and 66 + 2 = 68. Step 5: The equivalent expression is 6x + 68. The answer is 6x + 68.

7. A rectangular prism is described with dimensions: length = 12 cm, width = 8 cm, and height = 5 cm. What is the total surface area of this prism in square centimeters? Answer: 392 Solution: A rectangular prism has 6 faces: 2 length×width faces, 2 length×height faces, and 2 width×height faces.
   Full step-by-step solution

   Step 1: A rectangular prism has 6 faces: 2 length×width faces, 2 length×height faces, and 2 width×height faces. Step 2: Calculate area of one length×width face: 12 × 8 = 96 cm² There are 2 of these: 2 × 96 = 192 cm² Step 3: Calculate area of one length×height face: 12 × 5 = 60 cm² There are 2 of these: 2 × 60 = 120 cm² Step 4: Calculate area of one width×height face: 8 × 5 = 40 cm² There are 2 of these: 2 × 40 = 80 cm² Step 5: Add all the areas: 192 + 120 + 80 = 392 cm² The total surface area is 392 square centimeters.