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Question

Difficulty: MediumLinear Equations in One Variable

An online streaming service offers the two monthly subscription plans described in the table below:

PlanMonthly feeCost per premium movie rental
Plan A$12$1.50
Plan B$27 (includes first 4 rentals)$0.75 (for each rental after the first 4)

If a user rented mm premium movies in a month, where m>4m > 4, and the total cost for both plans would be the same, what is the value of mm?

Answer: 16

Answer

The value of mm that results in the same total monthly cost for both plans is 16.
The correct answer of 16 represents the exact number of premium movie rentals where the total monthly cost for both plans is equal to $36. Any other number of movie rentals will result in different costs for the two plans.

Step-by-Step Solution

1
Define the cost equation for Plan A
CostA=12+1.50m\text{Cost}_A = 12 + 1.50m
Plan A charges a flat 12monthlyfeeplus12 monthly fee plus 1.50 for each of the mm movies rented.
2
Define the cost equation for Plan B
CostB=27+0.75(m−4)\text{Cost}_B = 27 + 0.75(m - 4)
Plan B charges a 27monthlyfeethatcoversthefirst4movies,and27 monthly fee that covers the first 4 movies, and 0.75 for each of the m−4m - 4 additional movies rented because m>4m > 4.
3
Equate the two cost expressions and solve for mm
12+1.50m=27+0.75(m−4)  ⟹  12+1.50m=24+0.75m  ⟹  0.75m=12  ⟹  m=1612 + 1.50m = 27 + 0.75(m - 4) \implies 12 + 1.50m = 24 + 0.75m \implies 0.75m = 12 \implies m = 16
To find when the costs are identical, set the two algebraic expressions equal to each other and isolate the variable mm.

Key Concept

Setting up and solving linear equations in one variable from context

Alternative Method

Instead of setting up full equations, we can look at the cost difference at m=4m = 4. At 44 movies, Plan A costs 12+1.50(4)=1812 + 1.50(4) = 18 dollars, and Plan B costs 2727 dollars (since 44 movies are included). The price difference is 27−18=927 - 18 = 9 dollars. For each movie rented beyond 44, the cost of Plan A increases by 1.501.50 dollars while Plan B only increases by 0.750.75 dollars. The rate of change difference is 1.50−0.75=0.751.50 - 0.75 = 0.75 dollars per movie. To bridge the initial 99 dollar difference, the user needs to rent 90.75=12\frac{9}{0.75} = 12 more movies. Thus, the total number of movies is 4+12=164 + 12 = 16.
Estimated Time:1m 30s
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Topics

Linear Equations in One Variable

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