At a used book sale, Sarah bought x paperback books priced at $0.25 each and y hardcover books priced at $0.45 each. She bought a total of 8 books and paid $2.60 in all. Solving which of the following equations would give the correct values of x and y?
x + y = 8 and 0.25x + 0.45y = 2.6
The correct equations represent the total number of books and the total cost of the books Sarah purchased. The first equation captures the total quantity of books, while the second equation accurately reflects the total amount spent based on the prices of the paperback and hardcover books.
This set of equations correctly models the problem. The equation \(x + y = 8\) accounts for the total number of books, and \(0.25x + 0.45y = 2.6\) represents the total amount spent, ensuring accurate calculations for both the number of books and the total cost.
While this option maintains the correct total number of books, it inaccurately represents the cost, swapping the prices of paperback and hardcover books. This error would lead to incorrect calculations for the total expenditure, making it invalid for solving the problem.
This choice incorrectly states the total number of books as 2.6, which is not possible since the number of books must be a whole number. Thus, this option cannot be used to determine the correct values of x and y.
This option sets up a completely different mathematical relationship that does not pertain to the problem at hand. The equations do not represent the total number of books or the total cost, making it irrelevant and incorrect for solving the problem.
The equations \(x + y = 8\) and \(0.25x + 0.45y = 2.6\) are essential for solving for the number of paperback and hardcover books Sarah purchased. The first equation ensures the total quantity is correct, while the second equation accurately captures the total expenditure based on the respective prices. The other choices either misrepresent the relationships or present erroneous equations, confirming that option A is the only suitable choice.
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