Equivalent Expressions

lessonbunny.com

Answer Key & Explanations

Equivalent Expressions · Grade 6 · Worksheet 1

1. Nikau is collecting trading cards. Nikau has 6 packs of cards, and each pack contains 15 cards. Which expression is equivalent to the total number of cards Nikau has? Answer: 90 Solution: To find the total number of cards, multiply the number of packs by the number of cards per pack: 6 * 15 = 90.
2. Emma is planning a school bake sale and needs to calculate her profit. She buys ingredients for $48.75 and spends 4 hours baking. If she sells all her baked goods for $126.50 and values her time at $12 per hour, what is Emma's total profit from the bake sale? Answer: $28.75 Solution: - Ingredient cost: $48.75 - Time cost: 4 hours × $12/hour = $48 - Total costs: $48.75 + $48 = $96.75 - Revenue from sales: $126.50 - Profit = Revenue - Total costs - Profit = $126.50 - $96.75 - Profit = $28.75 Emma's total profit from the bake sale is $28.75.
   Full step-by-step solution

   Step 1: Calculate Emma's total costs - Ingredient cost: $48.75 - Time cost: 4 hours × $12/hour = $48 - Total costs: $48.75 + $48 = $96.75 Step 2: Calculate profit - Revenue from sales: $126.50 - Profit = Revenue - Total costs - Profit = $126.50 - $96.75 - Profit = $28.75 Emma's total profit from the bake sale is $28.75.

3. Liam is designing a rectangular garden for his school project. The length of the garden is 12.5 meters, and the width is 8.4 meters. He needs to buy fencing to go around the entire garden. How many meters of fencing does Liam need to purchase? Answer: 41.8 Solution: To find how much fencing Liam needs, we need to calculate the perimeter of the rectangular garden. Recall the formula for the perimeter of a rectangle. The perimeter P is given by: P = 2 * (length + width) Identify the given measurements.
   Full step-by-step solution

   To find how much fencing Liam needs, we need to calculate the perimeter of the rectangular garden. Step 1: Recall the formula for the perimeter of a rectangle. The perimeter P is given by: P = 2 * (length + width) Step 2: Identify the given measurements. Length = 12.5 meters Width = 8.4 meters Step 3: Substitute the values into the formula. P = 2 * (12.5 + 8.4) Step 4: Perform the addition inside the parentheses. 12.5 + 8.4 = 20.9 Step 5: Multiply the result by 2. P = 2 * 20.9 = 41.8 Step 6: State the final answer. Therefore, Liam needs to purchase 41.8 meters of fencing.

4. 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 rectangular prism? Answer: 392 Solution: Calculate the area of the front and back faces (length × height) Area = 12 cm × 5 cm = 60 cm² Since there are 2 such faces: 2 × 60 cm² = 120 cm² Calculate the area of the left and right faces (width × height) Area = 8 cm × 5 cm = 40 cm² Since there are 2 such faces: 2 × 40 cm² = 80 cm² Calculate…
   Full step-by-step solution

   Step 1: Calculate the area of the front and back faces (length × height) Area = 12 cm × 5 cm = 60 cm² Since there are 2 such faces: 2 × 60 cm² = 120 cm² Step 2: Calculate the area of the left and right faces (width × height) Area = 8 cm × 5 cm = 40 cm² Since there are 2 such faces: 2 × 40 cm² = 80 cm² Step 3: Calculate the area of the top and bottom faces (length × width) Area = 12 cm × 8 cm = 96 cm² Since there are 2 such faces: 2 × 96 cm² = 192 cm² Step 4: Add all the areas together 120 cm² + 80 cm² + 192 cm² = 392 cm² The total surface area is 392 cm².

5. (-3)² + 4 × (20 ÷ 5) = ? Answer: 25 Solution: Calculate the exponent first: (-3)² = 9 Calculate the division inside the parentheses: 20 ÷ 5 = 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 the parentheses: 20 ÷ 5 = 4 Step 3: Perform the multiplication: 4 × 4 = 16 Step 4: Add the results: 9 + 16 = 25 The answer is 25.

6. Liam is designing a rectangular garden with a length-to-width ratio of 5:3. If the total 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 \( 3x \), since 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 \( 3x \), since 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 \( x^2 \)** \[ x^2 = 240 / 15 \] \[ x^2 = 16 \] --- **Step 5: Solve for \( x \)** \[ x = \sqrt{16} = 4 \] (We take the positive root since dimensions are positive.) --- **Step 6: Find the width** Width = \( 3x = 3 \times 4 = 12 \) meters. --- **Final Answer:** 12

7. A rectangular prism is drawn with dimensions: length = 12 cm, width = 8 cm, and height = 5 cm. The prism is then cut into 1 cm cubes. How many of these small cubes can be made from the original prism? Answer: 480 Solution: The prism has dimensions: length = 12 cm, width = 8 cm, height = 5 cm Since we're cutting into 1 cm cubes, we need to find how many cubes fit along each dimension: - Along the length: 12 cubes - Along the width: 8 cubes - Along the height: 5 cubes To find the total number of cubes, multiply the…
   Full step-by-step solution

   Step 1: The prism has dimensions: length = 12 cm, width = 8 cm, height = 5 cm Step 2: Since we're cutting into 1 cm cubes, we need to find how many cubes fit along each dimension: - Along the length: 12 cubes - Along the width: 8 cubes - Along the height: 5 cubes Step 3: To find the total number of cubes, multiply the number of cubes along each dimension: 12 × 8 × 5 Step 4: First multiply 12 × 8 = 96 Step 5: Then multiply 96 × 5 = 480 Step 6: Therefore, 480 cubes can be made from the original prism.

8. Which expression is equivalent to 9(2x + 7)? Answer: 18x + 63 Solution: Step 1: Apply the distributive property: 9(2x + 7) = 9 × 2x + 9 × 7 Step 2: Multiply: 9 × 2x = 18x Step 3: Multiply: 9 × 7 = 63 Step 4: Combine: 18x + 63 The answer is 18x + 63.
   Full step-by-step solution

   Step 1: Apply the distributive property: 9(2x + 7) = 9 × 2x + 9 × 7 Step 2: Multiply: 9 × 2x = 18x Step 3: Multiply: 9 × 7 = 63 Step 4: Combine: 18x + 63 The answer is 18x + 63.

9. Liam is planning a school fundraiser and needs to buy supplies. He has a budget of $1200. He spends 35% of his budget on decorations and 0.3 of his budget on food. What fraction of his total budget does Liam have left after these purchases? Answer: 7/20 Solution: Liam's total budget = $1200. - Decorations: 35% of the budget Amount spent on decorations = 35/100 × 1200 = 0.35 × 1200 = $420. - Food: 0.3 of the budget 0.3 means 30% (since 0.3 = 30/100).
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the budget and the amounts spent** Liam's total budget = $1200. - Decorations: 35% of the budget Amount spent on decorations = 35/100 × 1200 = 0.35 × 1200 = $420. - Food: 0.3 of the budget 0.3 means 30% (since 0.3 = 30/100). Amount spent on food = 0.3 × 1200 = $360. --- **Step 2: Find total spent and amount left** Total spent = 420 + 360 = $780. Amount left = 1200 − 780 = $420. --- **Step 3: Fraction of budget left** Fraction left = (Amount left) / (Total budget) = 420 / 1200. Simplify: Divide numerator and denominator by 60: 420 ÷ 60 = 7, 1200 ÷ 60 = 20. So, 420/1200 = 7/20. --- **Step 4: Conclusion** Liam has 7/20 of his budget left after buying decorations and food. --- **Final answer:** 7/20