Equivalent Expressions

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Equivalent Expressions · Grade 6 · Worksheet 2

1. (-12) × (-4) + 18 ÷ (-3) = ? Answer: 42 Solution: We have: (-12) × (-4) + 18 ÷ (-3) Handle the multiplication and division first (order of operations: multiplication and division before addition).
   Full step-by-step solution

   Let's solve step by step. We have: (-12) × (-4) + 18 ÷ (-3) Step 1: Handle the multiplication and division first (order of operations: multiplication and division before addition). First, (-12) × (-4) Multiplying two negative numbers gives a positive result: (-12) × (-4) = 48 Step 2: Now handle the division: 18 ÷ (-3) Dividing a positive by a negative gives a negative result: 18 ÷ (-3) = -6 Step 3: Now substitute back into the expression: 48 + (-6) Which is the same as: 48 - 6 = 42 Final answer: 42

2. Which expression is equivalent to 14(4x + 9)? Answer: 56x + 126 Solution: Step 1: Apply the distributive property: 14(4x + 9) = 14 × 4x + 14 × 9 Step 2: Multiply: 14 × 4x = 56x Step 3: Multiply: 14 × 9 = 126 Step 4: Combine: 56x + 126 The answer is 56x + 126.
   Full step-by-step solution

   Step 1: Apply the distributive property: 14(4x + 9) = 14 × 4x + 14 × 9 Step 2: Multiply: 14 × 4x = 56x Step 3: Multiply: 14 × 9 = 126 Step 4: Combine: 56x + 126 The answer is 56x + 126.

3. Harper is planting flowers. There are 5 rows, and each row has 5 flowers. Which expression shows the total number of flowers Harper planted? Answer: 25 Solution: The total number of flowers is the number of rows times the number of flowers per row: 5 * 5 = 25.
4. 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.

5. Emma is planning a school carnival and needs to calculate ticket sales. The carnival expects 450 attendees. If 2/5 of the attendees are students who pay $8 per ticket, and the remaining attendees are adults who pay $12 per ticket, how much total money will be collected from ticket sales? Answer: $4680 Solution: Find the number of student attendees: 2/5 of 450 = (2/5) × 450 = 900/5 = 180 students Find the number of adult attendees: Total attendees - student attendees = 450 - 180 = 270 adults Calculate money from students: 180 students × $8 per ticket = $1440 Calculate money from adults: 270 adults × $12…
   Full step-by-step solution

   Step 1: Find the number of student attendees: 2/5 of 450 = (2/5) × 450 = 900/5 = 180 students Step 2: Find the number of adult attendees: Total attendees - student attendees = 450 - 180 = 270 adults Step 3: Calculate money from students: 180 students × $8 per ticket = $1440 Step 4: Calculate money from adults: 270 adults × $12 per ticket = $3240 Step 5: Add both amounts: $1440 + $3240 = $4680 The total money collected from ticket sales is $4680.

6. Charlotte is collecting trading cards. Charlotte has 4 packs of cards, and each pack contains 8 cards. Which expression is equivalent to the total number of cards Charlotte has? Answer: 32 Solution: To find the total number of cards, multiply the number of packs by the number of cards per pack: 4 * 8 = 32.
7. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (8, 1), (8, 5), and (2, 5). If the gardener wants to plant flowers along the entire perimeter of the garden, how many units of fencing will she need to outline the garden? Answer: 20 Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) Points A and B have the same y-coordinate (y = 1), so AB is horizontal. Length = 8 - 2 = 6 units. Points B and C have the same x-coordinate (x = 8), so BC is vertical.
   Full step-by-step solution

   Let's solve this step by step. Step 1: Identify the coordinates of the rectangle The corners are: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) Step 2: Find the length of side AB Points A and B have the same y-coordinate (y = 1), so AB is horizontal. Length = 8 - 2 = 6 units. Step 3: Find the length of side BC Points B and C have the same x-coordinate (x = 8), so BC is vertical. Length = 5 - 1 = 4 units. Step 4: Find the length of side CD Points C and D have the same y-coordinate (y = 5), so CD is horizontal. Length = 8 - 2 = 6 units. Step 5: Find the length of side DA Points D and A have the same x-coordinate (x = 2), so DA is vertical. Length = 5 - 1 = 4 units. Step 6: Calculate the perimeter Perimeter = sum of all sides Perimeter = AB + BC + CD + DA Perimeter = 6 + 4 + 6 + 4 Perimeter = 20 units. Step 7: Interpret the result The gardener needs 20 units of fencing to outline the garden. Final answer: 20

8. (-3)² + 4 × (15 ÷ 3) = ? Answer: 29 Solution: Handle parentheses first. Inside parentheses: 15 ÷ 3 = 5 (-3)² + 4 × (5) Evaluate exponents. (-3)² means (-3) × (-3) = 9 9 + 4 × 5 Perform multiplication.
   Full step-by-step solution

   Let's solve step-by-step using the order of operations (PEMDAS/BODMAS). Step 1: Handle parentheses first. Inside parentheses: 15 ÷ 3 = 5 So the expression becomes: (-3)² + 4 × (5) Step 2: Evaluate exponents. (-3)² means (-3) × (-3) = 9 So the expression becomes: 9 + 4 × 5 Step 3: Perform multiplication. 4 × 5 = 20 So the expression becomes: 9 + 20 Step 4: Perform addition. 9 + 20 = 29 Final answer: 29

9. Liam is planning a school fundraiser and needs to buy supplies. He has a budget of $1500. He spends 35% of his budget on decorations, 0.4 of his budget on food, and the remaining money on prizes. How much money does Liam spend on prizes? Answer: 375 Solution: Liam’s total budget = $1500. - Decorations: 35% of budget - Food: 0.4 of budget (which is 40%) 35% of 1500 = 35/100 × 1500 = 0.35 × 1500 = 525 So decorations cost $525. 0.4 of 1500 = 0.4 × 1500 = 600 So food cost $600.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the budget and given percentages** Liam’s total budget = $1500. - Decorations: 35% of budget - Food: 0.4 of budget (which is 40%) - Remaining: prizes --- **Step 2: Calculate decorations cost** 35% of 1500 = 35/100 × 1500 = 0.35 × 1500 = 525 So decorations cost $525. --- **Step 3: Calculate food cost** 0.4 of 1500 = 0.4 × 1500 = 600 So food cost $600. --- **Step 4: Find total spent on decorations and food** Total spent on decorations + food = 525 + 600 = 1125 --- **Step 5: Find money left for prizes** Budget − spent = 1500 − 1125 = 375 --- **Step 6: Conclusion** Liam spends **$375** on prizes. --- **Final answer:** 375