Question

Ten granola bars and twelve bottles of water cost $23. Five granola bars and four bottles of water cost $10. How much do one granola bar and one bottle of water cost?

Answer 1

Let the cost of one granola bar be \( x \) and the cost of one bottle of water be \( y \). We can set up the following system of equations based on the information given:

1. For the cost of ten granola bars and twelve bottles of water:
\[
10x + 12y = 23
\]
2. For the cost of five granola bars and four bottles of water:
\[
5x + 4y = 10
\]

We can simplify and solve these equations step by step.

First, we can simplify the first equation by dividing all terms by 2:
\[
5x + 6y = 11.5 \quad \text{(1)}
\]

The second equation remains the same:
\[
5x + 4y = 10 \quad \text{(2)}
\]

Next, we will eliminate \( 5x \) from equation (1) and equation (2). We can subtract equation (2) from equation (1):
\[
(5x + 6y) - (5x + 4y) = 11.5 - 10
\]
This simplifies to:
\[
2y = 1.5
\]

Now, we solve for \( y \):
\[
y = \frac{1.5}{2} = 0.75
\]

Now that we have the value of \( y \), we can substitute it back into one of the original equations to find \( x \). We will use equation (2):
\[
5x + 4(0.75) = 10
\]
This simplifies to:
\[
5x + 3 = 10
\]

Subtracting 3 from both sides gives:
\[
5x = 7
\]

Now, we solve for \( x \):
\[
x = \frac{7}{5} = 1.4
\]

We now have:
\[
x = 1.4 \quad \text{(cost of one granola bar)}
\]
\[
y = 0.75 \quad \text{(cost of one bottle of water)}
\]

The total cost of one granola bar and one bottle of water is:
\[
x + y = 1.4 + 0.75 = 2.15
\]

Thus, the cost of one granola bar and one bottle of water together is:
\[
\boxed{2.15}
\]