Question
Gillian purchased 25 books at the library book sale. Each hardcover book cost $1.50, and each paperback book cost $0.50. Gillian spent a total of $26.50. The book costs can be represented by the system of equations below.
h + p = 25
1.50h + 0.50p = 26.50
How many paperback books did Gillian buy?
11
12
13
14
h + p = 25
1.50h + 0.50p = 26.50
How many paperback books did Gillian buy?
11
12
13
14
Answers
GPT-4o mini
Let's denote the number of hardcover books as \( h \) and the number of paperback books as \( p \). We have the following system of equations:
1. \( h + p = 25 \) (Equation representing the total number of books)
2. \( 1.50h + 0.50p = 26.50 \) (Equation representing the total cost of the books)
We can solve this system of equations step by step.
From the first equation, we can express \( h \) in terms of \( p \):
\[
h = 25 - p
\]
Now, we can substitute this expression for \( h \) into the second equation:
\[
1.50(25 - p) + 0.50p = 26.50
\]
Expanding this, we have:
\[
37.50 - 1.50p + 0.50p = 26.50
\]
Combining like terms:
\[
37.50 - 1.00p = 26.50
\]
Now, we subtract 37.50 from both sides:
\[
-1.00p = 26.50 - 37.50
\]
\[
-1.00p = -11.00
\]
Dividing both sides by -1, we get:
\[
p = 11
\]
So, Gillian bought **11 paperback books**.
To verify, we can find the number of hardcover books using \( h = 25 - p \):
\[
h = 25 - 11 = 14
\]
Now we can check the total cost:
\[
1.50 \times 14 + 0.50 \times 11 = 21.00 + 5.50 = 26.50
\]
The solution satisfies both equations, confirming that Gillian bought **11 paperback books**.
1. \( h + p = 25 \) (Equation representing the total number of books)
2. \( 1.50h + 0.50p = 26.50 \) (Equation representing the total cost of the books)
We can solve this system of equations step by step.
From the first equation, we can express \( h \) in terms of \( p \):
\[
h = 25 - p
\]
Now, we can substitute this expression for \( h \) into the second equation:
\[
1.50(25 - p) + 0.50p = 26.50
\]
Expanding this, we have:
\[
37.50 - 1.50p + 0.50p = 26.50
\]
Combining like terms:
\[
37.50 - 1.00p = 26.50
\]
Now, we subtract 37.50 from both sides:
\[
-1.00p = 26.50 - 37.50
\]
\[
-1.00p = -11.00
\]
Dividing both sides by -1, we get:
\[
p = 11
\]
So, Gillian bought **11 paperback books**.
To verify, we can find the number of hardcover books using \( h = 25 - p \):
\[
h = 25 - 11 = 14
\]
Now we can check the total cost:
\[
1.50 \times 14 + 0.50 \times 11 = 21.00 + 5.50 = 26.50
\]
The solution satisfies both equations, confirming that Gillian bought **11 paperback books**.