A certain store purchased grills for $50 each and lawn chairs for $5 each and then sold each grill and...

1. Translate the problem requirements: We need to find total revenue. We know purchase prices (\(\$50\) grills, \(\$5\) chairs), profit percentages (\(30\%\) on grills, \(50\%\) on chairs), quantity relationship (5 times as many chairs as grills), and total profit (\(\$550\)). Revenue = Purchase price + Profit for all items combined.
2. Set up variables for quantities: Let the number of grills = g, then chairs = 5g. This keeps our algebra simple and avoids fractions.
3. Calculate individual profits and create total profit equation: Profit per grill = \(\$15\), profit per chair = \(\$2.50\). Total profit = \(15\mathrm{g} + 2.50(5\mathrm{g}) = \$550\).
4. Solve for quantities and calculate total revenue: Find g, then calculate total purchase cost and add total profit to get revenue.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know and what we need to find in everyday terms:

What we know:

• Grills cost the store \(\$50\) each to buy
• Chairs cost the store \(\$5\) each to buy
• The store makes \(30\%\) profit on each grill (\(30\%\) of \(\$50\))
• The store makes \(50\%\) profit on each chair (\(50\%\) of \(\$5\))
• The store sold 5 times as many chairs as grills
• Total profit from both grills and chairs was \(\$550\)

What we need to find: Total revenue from selling grills and chairs

Key insight: Revenue = What the store paid to buy everything + Total profit they made

Process Skill: TRANSLATE - Converting the business scenario into clear mathematical relationships

2. Set up variables for quantities

To keep our math simple, let's use a variable for the smaller quantity:

Let \(\mathrm{g}\) = number of grills sold
Then \(5\mathrm{g}\) = number of chairs sold (since they sold 5 times as many chairs)

This approach avoids fractions and keeps our calculations clean. For example, if they sold 10 grills, they sold \(5 \times 10 = 50\) chairs.

3. Calculate individual profits and create total profit equation

Now let's figure out how much profit the store made on each item:

Profit per grill:
\(30\%\) of \(\$50 = 0.30 \times \$50 = \$15\) per grill

Profit per chair:
\(50\%\) of \(\$5 = 0.50 \times \$5 = \$2.50\) per chair

Total profit equation:

• Profit from all grills: \(\mathrm{g}\) grills × \(\$15\) per grill = \(15\mathrm{g}\)
• Profit from all chairs: \(5\mathrm{g}\) chairs × \(\$2.50\) per chair = \(12.5\mathrm{g}\)
• Total profit: \(15\mathrm{g} + 12.5\mathrm{g} = 27.5\mathrm{g}\)

Since we know total profit was \(\$550\):
\(27.5\mathrm{g} = 550\)

Process Skill: SIMPLIFY - Breaking down the profit calculation into manageable pieces

4. Solve for quantities and calculate total revenue

Finding the number of grills:
\(27.5\mathrm{g} = 550\)
\(\mathrm{g} = 550 \div 27.5 = 20\)

So the store sold 20 grills and \(5 \times 20 = 100\) chairs.

Calculating total revenue:
Revenue = Purchase cost + Profit

Total purchase cost:

• Cost of grills: \(20 \times \$50 = \$1,000\)
• Cost of chairs: \(100 \times \$5 = \$500\)
• Total purchase cost: \(\$1,000 + \$500 = \$1,500\)

Total revenue:
\(\$1,500\) (purchase cost) + \(\$550\) (profit) = \(\$2,050\)

5. Final Answer

The store's total revenue from the sale of grills and chairs was \(\$2,050\).

This matches answer choice C. \(\$2,050\).

Verification:

• 20 grills sold at \(\$65\) each (cost \(\$50\) + profit \(\$15\)) = \(\$1,300\)
• 100 chairs sold at \(\$7.50\) each (cost \(\$5\) + profit \(\$2.50\)) = \(\$750\)
• Total revenue: \(\$1,300 + \$750 = \$2,050\) ✓

Common Faltering Points

Errors while devising the approach

1. Confusing profit percentage with selling price percentage
Students often misinterpret "\(30\%\) profit on purchase price" as meaning the selling price is \(30\%\) of the purchase price, rather than understanding that profit is \(30\%\) of purchase price. This leads them to think grills sell for \(\$15\) instead of \(\$65\) (\(\$50 + \$15\) profit).

2. Setting up the relationship between chairs and grills incorrectly
When the problem states "sold 5 times as many chairs as grills," students may set up the variable relationship backwards (\(5\mathrm{g}\) grills and \(\mathrm{g}\) chairs) or use two separate variables without establishing the 5:1 relationship, making the problem unnecessarily complex.

3. Confusing revenue with profit in the problem setup
Students may not clearly distinguish between what the question is asking for (total revenue) versus what information is given (total profit = \(\$550\)). This confusion can lead to treating profit and revenue as the same thing throughout their solution approach.

Errors while executing the approach

1. Arithmetic errors in percentage calculations
Students frequently make calculation mistakes when computing \(30\%\) of \(\$50\) or \(50\%\) of \(\$5\), potentially getting \(\$1.50\) instead of \(\$15\) for grill profit, or \(\$0.25\) instead of \(\$2.50\) for chair profit.

2. Errors in division when solving for the number of grills
When solving \(27.5\mathrm{g} = 550\), students may struggle with dividing by the decimal 27.5, potentially getting \(\mathrm{g} = 2\) instead of \(\mathrm{g} = 20\), or making other computational errors that throw off all subsequent calculations.

3. Miscalculating total costs or forgetting to include both purchase cost and profit
Students may calculate only the profit (\(\$550\)) as their final answer, or only the purchase costs (\(\$1,500\)), forgetting that revenue = purchase cost + profit = \(\$1,500 + \$550 = \$2,050\).

Errors while selecting the answer

1. Selecting the profit amount instead of revenue
After calculating correctly that total profit is \(\$550\), students may mistakenly select this as their final answer if there's a choice close to this value, forgetting that the question asks for total revenue, not total profit.

2. Selecting the total purchase cost instead of revenue
Students who correctly calculate the total purchase cost as \(\$1,500\) may select this amount if it appears in the answer choices, without adding the profit to get the total revenue of \(\$2,050\).