15. 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?
Answer: A
x + y = 8 and 0.25x + 0.45y = 2.6
To determine the values of x and y, the equations must represent both the total number of books and the total cost of the books. The equation x + y = 8 accounts for the total number of books, while 0.25x + 0.45y = 2.6 accurately represents the total cost based on the prices of the paperback and hardcover books.
A) x + y = 8 and 0.25x + 0.45y = 2.6
This option is correct because it includes both necessary equations: the first equation signifies the total number of books purchased, while the second equation reflects the total expenditure based on the prices of the books. Thus, solving these two equations will yield the correct values for x and y.
B) x + y = 8 and 0.45x + 0.25y = 2.6
This option is incorrect as it misplaces the coefficients related to the prices of the books. The first equation remains valid for the total number of books, but the second equation erroneously assigns the price of the hardcover books to x and the paperback books to y, which does not match the given prices.
C) x + y = 2.6 and 0.25x + 0.45y = 8
This option is incorrect because the first equation incorrectly states the total number of books as 2.6, which is not possible since x and y must be whole numbers representing the count of books. The second equation also does not reflect the correct relationship between the total cost and the number of books.
D) xy = 8 and 2.6xy = 1
This option is incorrect as it presents an entirely different format that does not reflect the problem's conditions. The equations involving the product of x and y do not pertain to the total number of books or the total cost, thus failing to provide any valuable information regarding the values of x and y.
Conclusion
Option A is definitively correct as it accurately encompasses both the total number of books and the total cost equations, essential for solving for x and y. The other options fail to align with the requirements of the problem, either by misrepresenting the relationships or providing incorrect total values.