Question 12·Hard·Linear Equations in One Variable
In a certain month, a streaming service offered two subscription tiers: a basic tier costing $15 per month and a premium tier costing $24 per month. During that month, the number of premium subscriptions was 60 fewer than twice the number of basic subscriptions. If the service collected $16,200 in subscription revenue that month, how many premium subscriptions were sold?
(Express the answer as an integer)
For word problems that describe prices and quantities, first define clear variables for the unknown counts, then carefully translate key phrases like "60 fewer than twice" as an equation (e.g., ). Next, write a revenue equation by multiplying each price by its quantity and adding them to match the total given. Use substitution to turn the situation into a single linear equation in one variable, solve it step by step, and finally plug back in to find the other quantity—always double-check that your final number answers exactly what the question is asking (basic vs. premium, items vs. total, etc.).
Hints
Set up your variables
Let one variable represent the number of basic subscriptions and another represent the number of premium subscriptions. What should each variable stand for?
Translate the relationship between basic and premium
Focus on the phrase "60 fewer than twice the number of basic subscriptions." How do you write "twice the number of basic subscriptions" with your variable, and then how do you adjust it to show "60 fewer"?
Use the revenue information
Write an equation for the total revenue using the cost of each subscription type and the number of each type. How can you express the total revenue in terms of your two variables?
Reduce to one variable
Use the relationship between the two subscription types to substitute into the revenue equation so that you have only one variable. Solve that equation, then use your relationship again to find the other quantity and answer the question being asked.
Desmos Guide
Enter the system of equations
In Desmos, enter the two equations using for basic subscriptions and for premium subscriptions:
15x + 24y = 16200y = 2x - 60
Make sure both are turned on so both lines are visible.
Find the intersection point
Look for the point where the two lines intersect. Tap or click that intersection point; Desmos will display its coordinates . The -coordinate of this point is the number of premium subscriptions, and the -coordinate is the number of basic subscriptions.
Step-by-step Explanation
Define variables and translate the relationship
Let be the number of basic subscriptions and be the number of premium subscriptions.
The phrase "the number of premium subscriptions was 60 fewer than twice the number of basic subscriptions" means:
This is because you first take twice the number of basic subscriptions () and then subtract 60 to make it "60 fewer."
Write the revenue equation
Each basic subscription brings in $15 and each premium subscription brings in $24.
So the total revenue from all subscriptions is:
- Revenue from basics:
- Revenue from premiums:
We are told the total revenue is $16,200, so:
Substitute to get an equation in one variable
Use the relationship in the revenue equation so everything is in terms of .
Start with:
Substitute :
Distribute the 24:
Combine like terms:
Solve for the number of basic subscriptions
Solve the linear equation for .
Add 1440 to both sides:
Divide both sides by 63:
So there were 280 basic subscriptions.
Find the number of premium subscriptions and check
Use the relationship with :
Check with the revenue:
- Basic revenue:
- Premium revenue:
- Total: , which matches the problem.
So the number of premium subscriptions sold was 500.