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A manufacturer makes and sells 2 products, P and Q. The revenue from the sale of each unit of P is \(\$20.00\) and the revenue from the sale of each unit of Q is \(\$17.00\). Last year the manufacturer sold twice as many units of Q as P. What was the manufacturer's average (\(\mathrm{arithmetic\ mean}\)) revenue per unit sold of these 2 products last year?
Let's break down what we know in everyday terms:
Think of this like finding the average price of items in a shopping cart where you bought different quantities of each item. The average isn't just the simple average of the two prices \((\$20 + \$17) \div 2 = \$18.50\), because we sold different amounts of each product.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical relationships
Since we sold twice as many Q units as P units, let's pick simple numbers that make our calculations easy.
Let's say we sold:
Why choose these numbers? Because they're the simplest whole numbers that give us the 2:1 ratio we need. We could use 10 and 20, or 100 and 200, but 1 and 2 make the arithmetic much easier.
This concrete approach helps us see exactly what's happening: for every 1 unit of P sold, we sold 2 units of Q.
Now let's calculate the money earned from each product:
Revenue from P:
\(1 \text{ unit} \times \$20 \text{ per unit} = \$20\)
Revenue from Q:
\(2 \text{ units} \times \$17 \text{ per unit} = \$34\)
Total revenue:
\(\$20 + \$34 = \$54\)
Total units sold:
\(1 \text{ unit} + 2 \text{ units} = 3 \text{ units}\)
Notice how the Q units contribute more to the total revenue ($34) than the P units ($20), even though Q has a lower price per unit. This happens because we sold more Q units.
The average revenue per unit is simply:
Average = Total Revenue ÷ Total Units
\(\text{Average} = \$54 \div 3 \text{ units} = \$18 \text{ per unit}\)
This makes intuitive sense: our average ($18) is closer to the price of Q ($17) than to the price of P ($20) because we sold twice as many Q units. The larger quantity of Q units 'pulls' the average toward Q's price.
The manufacturer's average revenue per unit sold was $18.00
Looking at our answer choices, this matches option E. $18.00
Verification: Our answer ($18) falls between the two individual prices ($17 and $20) and is closer to $17, which makes sense since we sold more of the lower-priced product Q.
Students often think they need to find the simple average of the two prices: \((\$20 + \$17) \div 2 = \$18.50\). This is incorrect because it ignores the different quantities sold. The correct approach requires finding the weighted average where the weights are the number of units sold of each product.
The problem states "sold twice as many units of Q as P." Some students might incorrectly interpret this as "sold twice as many units of P as Q" or get confused about which product has the higher sales volume. This reversal would lead to incorrect calculations throughout the problem.
Students might set up the problem as "let P units = x, then Q units = 2x" and work with algebraic expressions. While this approach can work, it's more error-prone and time-consuming than simply choosing concrete numbers like 1 and 2 that satisfy the ratio requirement.
When calculating total revenue, students might make arithmetic mistakes such as: \(2 \times \$17 = \$35\) (instead of $34), or adding incorrectly: \(\$20 + \$34 = \$55\) (instead of $54). These small errors cascade into wrong final answers.
Students might accidentally calculate Total Units ÷ Total Revenue instead of Total Revenue ÷ Total Units, which would give them \(3 \div 54 = 0.056\) instead of \(54 \div 3 = 18\). This type of error often happens when students rush through the final calculation.
After doing all the work correctly and getting $18.00, students might second-guess themselves and choose option D ($18.50) because it matches the simple average of $20 and $17. This happens when students don't trust their weighted average calculation and revert to the incorrect simple average approach.
Step 1: Choose convenient concrete values
Since we know that twice as many units of Q were sold compared to P, let's choose simple numbers that reflect this 2:1 ratio:
These numbers are chosen because they directly satisfy the given relationship and make calculations straightforward.
Step 2: Calculate total revenue
Step 3: Calculate total units sold
Total units = 1 + 2 = 3 units
Step 4: Find average revenue per unit
Average revenue per unit = Total revenue ÷ Total units = \(\$54.00 \div 3 = \$18.00\)
Answer: E
Note: This smart numbers approach works because the ratio relationship (2:1) allows us to choose specific values that maintain the mathematical relationship while making calculations concrete and manageable.