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A certain store purchased grills for \(\$\mathrm{50}\) each and lawn chairs for \(\$\mathrm{5}\) each and then sold each grill and each chair. The store's gross profit on each grill was \(\mathrm{30}\%\) of its purchase price, and the store's gross profit on each chair was \(\mathrm{50}\%\) of its purchase price. If the store sold 5 times as many chairs as grills and if the store's total gross profit on the grills and chairs was \(\$\mathrm{550}\), what was the store's total revenue from the sale of the grills and chairs?
Let's break down what we know and what we need to find in everyday terms:
What we know:
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
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.
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:
Since we know total profit was \(\$550\):
\(27.5\mathrm{g} = 550\)
Process Skill: SIMPLIFY - Breaking down the profit calculation into manageable pieces
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:
Total revenue:
\(\$1,500\) (purchase cost) + \(\$550\) (profit) = \(\$2,050\)
The store's total revenue from the sale of grills and chairs was \(\$2,050\).
This matches answer choice C. \(\$2,050\).
Verification:
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.
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 = \(\$1,500 + \$550 = \$2,050\).
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 \(\$2,050\).