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On a certain day, a bakery produced a batch of rolls at a total production cost of \(\$300\). On that day, \(\frac{4}{5}\) of the rolls in the batch were sold, each at a price that was \(50\%\) greater than the average (arithmetic mean) production cost per roll. The remaining rolls in the batch were sold the next day, each at a price that was \(20\%\) less than the price of the day before. What was the bakery's profit on this batch of rolls?
Let's break down what we're looking for step by step. We need to find the bakery's profit on this batch of rolls.
Profit is simply: Money earned from selling - Money spent to make the product
We know the bakery spent \(\$300\) to produce the entire batch. Now we need to figure out how much money they earned from selling all the rolls.
The rolls were sold in two groups:
Process Skill: TRANSLATE - Converting the problem's business language into clear mathematical relationships
To understand the selling prices, we first need to find the average production cost per roll.
Let's say there were N rolls total in the batch.
Average production cost per roll = \(\$300 \div N\)
Now, on the first day, each roll was sold at a price that was 50% greater than this average production cost.
This means: First day selling price = Average production cost + 50% of average production cost
First day selling price = Average production cost \(\times 1.5\)
First day selling price = \((\$300 \div N) \times 1.5 = \$450 \div N\)
On the second day, each roll was sold at 20% less than the first day's price.
Second day selling price = First day price - 20% of first day price
Second day selling price = First day price \(\times 0.8\)
Second day selling price = \((\$450 \div N) \times 0.8 = \$360 \div N\)
Now let's calculate how much money the bakery earned from each group of sales.
Revenue from first day (\(\frac{4}{5}\) of the rolls):
Number of rolls sold = \(\frac{4}{5} \times N\)
Price per roll = \(\$450 \div N\)
Revenue = \((\frac{4}{5} \times N) \times (\$450 \div N) = \frac{4}{5} \times \$450 = \$360\)
Revenue from second day (\(\frac{1}{5}\) of the rolls):
Number of rolls sold = \(\frac{1}{5} \times N\)
Price per roll = \(\$360 \div N\)
Revenue = \((\frac{1}{5} \times N) \times (\$360 \div N) = \frac{1}{5} \times \$360 = \$72\)
Notice how the N cancels out in both calculations - this means our answer doesn't depend on the actual number of rolls, which makes sense!
Total Revenue = \(\$360 + \$72 = \$432\)
Now we can calculate the bakery's profit:
Profit = Total Revenue - Total Production Cost
Profit = \(\$432 - \$300 = \$132\)
This matches answer choice C.
The bakery's profit on this batch of rolls was \(\$132\), which corresponds to answer choice C.
1. Misinterpreting the profit calculation requirement: Students may confuse profit with revenue and attempt to calculate only the total selling price without subtracting the production cost. The question asks for "profit," but students might focus solely on the sales calculations and forget that profit = revenue - cost.
2. Confusion about the pricing relationships: Students may misunderstand that the second day's price is 20% less than the first day's selling price (not 20% less than the production cost). The phrase "20 percent less than the price of the day before" requires careful attention to what baseline is being referenced.
3. Overlooking the fraction split: Students might miss that exactly \(\frac{4}{5}\) of rolls were sold on day one and \(\frac{1}{5}\) on day two, potentially assuming equal splits or other proportions. This fraction split is crucial for calculating the weighted revenue correctly.
1. Arithmetic errors in percentage calculations: When calculating 50% greater than production cost (\(\times 1.5\)) and 20% less than first day price (\(\times 0.8\)), students commonly make multiplication errors or incorrectly apply the percentage changes (e.g., using \(\times 0.2\) instead of \(\times 0.8\) for the 20% reduction).
2. Incorrect fraction arithmetic: Students may struggle with multiplying fractions like \(\frac{4}{5} \times \$450\) or \(\frac{1}{5} \times \$360\), particularly when converting between mixed operations involving fractions and dollar amounts.
3. Variable cancellation confusion: When working with the unknown number of rolls N, students might panic when they see N in both numerator and denominator, not recognizing that N should cancel out, leading them to assume they need to find the actual number of rolls.
No likely faltering points - once students correctly calculate the total revenue as \(\$432\) and subtract the production cost of \(\$300\), the final answer of \(\$132\) directly corresponds to answer choice C without additional interpretation required.
Step 1: Choose a convenient number of rolls
Since we need to work with \(\frac{4}{5}\) of the rolls being sold on day 1, let's choose 100 rolls total. This makes the fractions easy to work with:
Step 2: Calculate production cost per roll
Total production cost = \(\$300\)
Production cost per roll = \(\$300 \div 100 = \$3\) per roll
Step 3: Determine selling prices
Day 1 selling price = \(\$3 + 50\%\) of \(\$3 = \$3 + \$1.50 = \$4.50\) per roll
Day 2 selling price = \(\$4.50 - 20\%\) of \(\$4.50 = \$4.50 - \$0.90 = \$3.60\) per roll
Step 4: Calculate total revenue
Revenue from day 1 = \(80 \text{ rolls} \times \$4.50 = \$360\)
Revenue from day 2 = \(20 \text{ rolls} \times \$3.60 = \$72\)
Total revenue = \(\$360 + \$72 = \$432\)
Step 5: Calculate profit
Profit = Total revenue - Total production cost
Profit = \(\$432 - \$300 = \$132\)
The smart numbers approach works perfectly here because we can choose any convenient number of rolls (the profit will be the same regardless), and 100 makes the fraction calculations straightforward.