The total cost for a bucket of popcorn and 4 movie tickets is $56. The total cost for the same size bucket of popcorn and 6 movie tickets is $80. The cost of a bucket of popcorn is $8. Which equation represents the relationship between $y$, the total cost of the popcorn and movie tickets, and $x$, the number of movie tickets that are purchased?
A. $y=12x+8$
B. $y=12x-8$
C. $y=14x+8$
D. $y=14x-8
Answer (2)
Define variables: Let p be the cost of popcorn and t be the cost of a ticket.
Set up equations: p + 4 t = 56 and p + 6 t = 80 , with p = 8 .
Solve for t : Substitute p = 8 into the equations to find t = 12 .
Determine the equation: y = 12 x + 8 , where y is the total cost and x is the number of tickets. The final answer is y = 12 x + 8 .
Explanation
Problem Analysis Let's analyze the given information to determine the relationship between the total cost, the number of movie tickets, and the cost of popcorn.
Given Information We are given that the total cost for a bucket of popcorn and 4 movie tickets is $56 , and the total cost for the same bucket of popcorn and 6 movie tickets is $80 . We also know that the cost of a bucket of popcorn is $8 . We want to find an equation that represents the total cost y in terms of the number of movie tickets x .
Setting up Equations Let p be the cost of a bucket of popcorn and t be the cost of a movie ticket. We can write two equations based on the given information:
p + 4 t = 56
p + 6 t = 80
We are also given that p = 8 .
Substituting the Value of Popcorn Cost Substitute p = 8 into the two equations:
8 + 4 t = 56
8 + 6 t = 80
Solving for the Ticket Cost Solve for t in each equation:
4 t = 56 − 8
4 t = 48
t = 4 48
t = 12
And:
6 t = 80 − 8
6 t = 72
t = 6 72
t = 12
Both equations give us t = 12 , so the cost of a movie ticket is $12 .
Expressing Total Cost The total cost y is the cost of the popcorn plus the cost of the movie tickets, which can be represented as:
y = p + x t
where x is the number of movie tickets.
Finding the Equation Substitute p = 8 and t = 12 into the equation:
y = 8 + 12 x
So, the equation representing the relationship between the total cost y and the number of movie tickets x is y = 12 x + 8 .
Final Answer Therefore, the equation that represents the relationship between y , the total cost of the popcorn and movie tickets, and x , the number of movie tickets that are purchased is y = 12 x + 8 .
Examples
Imagine you're planning a movie night with friends. Knowing the cost of a bucket of popcorn and each movie ticket, you can use the equation y = 12 x + 8 to quickly calculate the total cost based on how many friends are joining. For example, if 5 friends are coming, you can calculate the total cost as y = 12 × 5 + 8 = 68 , meaning the total cost for popcorn and tickets would be $68 . This helps in budgeting and splitting costs fairly among your friends, ensuring everyone enjoys the movie night without any financial surprises. This equation is a practical way to manage expenses for entertainment activities.
The equation that relates the total cost of popcorn and movie tickets is y = 12 x + 8 . This equation shows that for each movie ticket purchased, the cost increases by $12, in addition to the $8 for the popcorn. The correct answer is option A: y = 12 x + 8 .
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