Question

Lincoln went into a movie theater and bought 2 bags of popcorn and 4 candies, costing a total of $34. Zoey went into the same movie theater and bought 6 bags of popcorn and 5 candies costing a total of $74. Determine the price of each bag of popcorn and the price of each candy.

Answer 1

Let's denote the price of a bag of popcorn as \( p \) and the price of a candy as \( c \).

From the information given:

1. Lincoln's purchase gives us the equation:
\[
2p + 4c = 34
\]

2. Zoey's purchase gives us the equation:
\[
6p + 5c = 74
\]

Now we have a system of two equations:

\[
\begin{align*}
(1) & \quad 2p + 4c = 34 \\
(2) & \quad 6p + 5c = 74
\end{align*}
\]

We can simplify the first equation. Divide the entire equation by 2:

\[
p + 2c = 17 \quad \text{(3)}
\]

Now we will express \( p \) in terms of \( c \) from equation (3):

\[
p = 17 - 2c \quad \text{(4)}
\]

Next, we will substitute equation (4) into equation (2):

\[
6(17 - 2c) + 5c = 74
\]

Distributing the 6 gives:

\[
102 - 12c + 5c = 74
\]

Combining like terms results in:

\[
102 - 7c = 74
\]

Now, we will isolate \( c \):

\[
-7c = 74 - 102
\]
\[
-7c = -28
\]

Dividing both sides by -7:

\[
c = 4
\]

Now that we have the value of \( c \), we can substitute it back into equation (4) to find \( p \):

\[
p = 17 - 2(4)
\]
\[
p = 17 - 8
\]
\[
p = 9
\]

The solutions for the prices are:

\[
\text{Price of a bag of popcorn } (p) = 9
\]
\[
\text{Price of a candy } (c) = 4
\]

Finally, we conclude that the price of each bag of popcorn is $9 and the price of each candy is $4.