Real-World Equations

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Answer Key & Explanations

Real-World Equations · Grade 6 · Worksheet 2

1. Aroha is saving money for a new bike. She already has $45 and plans to save $12 each week. Write an equation to represent the total amount of money (t) Aroha will have after w weeks. Answer: t = 45 + 12w Solution: Identify the starting amount: $45. Identify the weekly savings: $12 per week. Let w represent the number of weeks, so the total saved from weekly savings is 12w.
   Full step-by-step solution

   Step 1: Identify the starting amount: $45. Step 2: Identify the weekly savings: $12 per week. Step 3: Let w represent the number of weeks, so the total saved from weekly savings is 12w. Step 4: The total amount t is the starting amount plus the weekly savings: t = 45 + 12w. The answer is t = 45 + 12w.

2. Aroha has 2400 points in a video game. She earns 175 points for each level she completes. Write an equation for the total points (p) she will have after completing n levels. Answer: p = 2400 + 175n Solution: Identify the starting amount. Aroha begins with 2400 points, so this is a constant. Identify the rate of change.
   Full step-by-step solution

   Step 1: Identify the starting amount. Aroha begins with 2400 points, so this is a constant. Step 2: Identify the rate of change. She earns 175 points per level, so for n levels, she earns 175 × n points. Step 3: Write the equation. Total points (p) = starting points + points earned from levels. Step 4: p = 2400 + 175n The answer is p = 2400 + 175n.

3. Matiu earns $24 per hour. Write an equation for his total pay (p) after working h hours. Answer: p = 24h Solution: Identify the known information. Matiu earns $24 per hour. This is the constant rate.
   Full step-by-step solution

   Step 1: Identify the known information. Matiu earns $24 per hour. This is the constant rate. Step 2: Let h represent the number of hours worked. Step 3: Total pay is found by multiplying the hourly rate by the number of hours worked. Step 4: So the equation is p = 24 × h, which we write as p = 24h. The answer is p = 24h.

4. Mason is designing a rectangular patio on a coordinate grid. The corners of the patio are at (2, 7), (12, 7), (12, 17), and (2, 17). He wants to place a square garden bed in the center of the patio with sides of length 5 units, aligned with the grid. Write an equation for the area (A) of the remaining patio space after the garden bed is removed, using the dimensions of the patio and the garden bed. Answer: A = (10 x 10) - (5 x 5) or A = 100 - 25 Solution: Find the length of the patio. The x-coordinates are 2 and 12, so length = 12 - 2 = 10 units. Find the width of the patio.
   Full step-by-step solution

   Step 1: Find the length of the patio. The x-coordinates are 2 and 12, so length = 12 - 2 = 10 units. Step 2: Find the width of the patio. The y-coordinates are 7 and 17, so width = 17 - 7 = 10 units. Step 3: Calculate the area of the rectangular patio. Area = length x width = 10 x 10 = 100 square units. Step 4: Find the area of the square garden bed. Side length = 5 units, so area = 5 x 5 = 25 square units. Step 5: The remaining area is the patio area minus the garden bed area. So the equation is A = 100 - 25. The answer is A = 100 - 25.

5. Maya collects trading cards. Maya has 7 packs of cards. Each pack contains 26 cards. Write an equation to represent the total number of cards, t, that Maya has. Answer: 182 Solution: Maya has 7 packs. Each pack has 26 cards. Total cards = 7 × 26 = 182.
   Full step-by-step solution

   Step 1: Maya has 7 packs. Step 2: Each pack has 26 cards. Step 3: Total cards = 7 × 26 = 182. The equation is t = 7 × 26, so t = 182. The answer is 182 cards.

6. A recipe requires 3/4 cup of flour for 12 cookies. How many cups of flour are needed for 36 cookies? Answer: 2.25 Solution: Find how many times more cookies: 36 ÷ 12 = 3 times more cookies Multiply the flour amount by the same factor: 3/4 × 3 Calculate: 3/4 × 3 = 9/4 Convert to decimal: 9 ÷ 4 = 2.25 The answer is 2.25 cups of flour.
   Full step-by-step solution

   Step 1: Find how many times more cookies: 36 ÷ 12 = 3 times more cookies Step 2: Multiply the flour amount by the same factor: 3/4 × 3 Step 3: Calculate: 3/4 × 3 = 9/4 Step 4: Convert to decimal: 9 ÷ 4 = 2.25 Step 5: The answer is 2.25 cups of flour.

7. (-15) × 4 ÷ (-6) + 2³ = ? Answer: 18 Solution: Calculate the exponent first: 2³ = 2 × 2 × 2 = 8 Multiply: (-15) × 4 = -60 Divide: -60 ÷ (-6) = 10 (dividing two negatives gives a positive) Add: 10 + 8 = 18 The answer is 18.
   Full step-by-step solution

   Step 1: Calculate the exponent first: 2³ = 2 × 2 × 2 = 8 Step 2: Multiply: (-15) × 4 = -60 Step 3: Divide: -60 ÷ (-6) = 10 (dividing two negatives gives a positive) Step 4: Add: 10 + 8 = 18 The answer is 18.

8. Noah is planning a school fundraiser selling custom t-shirts. He orders 250 shirts for $8.75 each. The printing costs an additional $3.25 per shirt. If he sells all the shirts for $18.50 each, what will be his total profit from the fundraiser? Answer: 1625 Solution: Cost per shirt = purchase price + printing cost = $8.75 + $3.25 = $12.00 Total cost = 250 shirts × $12.00 per shirt = $3,000 Total revenue = 250 shirts × $18.50 per shirt = $4,625 Profit = total revenue - total cost = $4,625 - $3,000 = $1,625 The answer is 1625.
   Full step-by-step solution

   Step 1: Calculate total cost per shirt Cost per shirt = purchase price + printing cost = $8.75 + $3.25 = $12.00 Step 2: Calculate total cost for all shirts Total cost = 250 shirts × $12.00 per shirt = $3,000 Step 3: Calculate total revenue from selling all shirts Total revenue = 250 shirts × $18.50 per shirt = $4,625 Step 4: Calculate profit Profit = total revenue - total cost = $4,625 - $3,000 = $1,625 The answer is 1625.