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The average (arithmetic mean) cost per book for the \(\mathrm{12}\) books on a certain table is \(\mathrm{k}\) dollars. If a book that costs \(\mathrm{18}\) dollars is removed from the table and replaced by a book that costs \(\mathrm{42}\) dollars, then in terms of \(\mathrm{k}\), what will be the average cost per book, in dollars, for the books in the table?
Let's start by understanding what we have and what we need to find.
We begin with 12 books on a table. The average cost of these books is k dollars per book. This means if we added up all the costs and divided by 12, we'd get k dollars.
Next, we're told that we remove one book that costs $18 and replace it with a book that costs $42. We still have 12 books total - we're just swapping one book for another.
Our goal is to find the new average cost per book, expressed in terms of k (the original average).
Process Skill: TRANSLATE - Converting the word problem into clear mathematical understanding of what we start with and what we need to find.
Now let's think about how the total cost changes when we swap the books.
Originally, we had 12 books with a total cost of 12k dollars (since average × number of books = total cost).
When we remove the $18 book and add the $42 book, we're essentially:
The net change in total cost = +$42 - $18 = +$24
So our new total cost = 12k + 24 dollars
Since we still have exactly 12 books, we can find the new average by dividing the new total cost by 12.
New average = (New total cost) ÷ 12
New average = \(\frac{12k + 24}{12}\)
New average = \(\frac{12k}{12} + \frac{24}{12}\)
New average = k + 2
This makes intuitive sense: we increased the total cost by $24, and when we spread that increase across 12 books, each book's average cost goes up by \($24 ÷ 12 = $2\).
We've already expressed our result in terms of k: the new average is k + 2 dollars.
Let's verify this makes sense: if the original average was k dollars per book, and we made a change that increases the total by $24 across 12 books, then each book's average cost should increase by $2, giving us k + 2.
The new average cost per book is k + 2 dollars.
Looking at our answer choices, this matches choice A) k + 2.
Answer: A
Students may think that removing one book and adding another changes the total number of books from 12 to something else (like 11 then 12, or just 11). This leads them to incorrectly calculate the new average using the wrong denominator. The key insight is that we're doing a replacement - the total count stays at 12 throughout.
2. Confusion about what changes and what stays constantStudents might try to recalculate everything from scratch instead of recognizing that only the total cost changes while the number of books remains the same. This leads to unnecessary complexity and potential errors. The strategic approach is to focus on the net change in total cost ($24 increase) rather than recalculating the entire scenario.
Students may calculate the change as $18 + $42 = $60 instead of $42 - $18 = $24. They might add both values thinking both contribute positively to the total, forgetting that removing the $18 book means subtracting $18 from the total cost.
2. Arithmetic error when distributing the changeEven if students correctly find that the total cost increases by $24, they may make errors when dividing by 12 to find the per-book increase. For example, calculating $24 ÷ 12 = $3 instead of $2, which would lead them to an answer of k + 3.
Students who calculated correctly but got confused about the direction of change might select 'k - 2' instead of 'k + 2'. They know the change is 2 but pick the wrong sign, thinking that since we removed something, the average should decrease, not recognizing that we replaced it with a more expensive book.
Step 1: Choose a convenient value for k
Let's set k = 10 dollars (choosing a round number makes calculations easier)
Step 2: Calculate the original total cost
With 12 books averaging $10 each:
Original total cost = \(12 \times 10 = \$120\)
Step 3: Calculate the new total cost after the swap
Remove the $18 book: $120 - $18 = $102
Add the $42 book: $102 + $42 = $144
New total cost = $144
Step 4: Calculate the new average
New average = \(\$144 ÷ 12 = \$12\)
Step 5: Express the result in terms of k
We started with k = 10 and got a new average of 12
The new average = k + 2 = 10 + 2 = 12 ✓
Step 6: Verify with the answer choices
Checking each choice with k = 10:
Only choice A matches our calculated result of $12.