Question 68·Medium·Linear Inequalities in One or Two Variables
A theater sells adult tickets for $15 each and student tickets for $10 each. For an evening show, the theater must sell at least 50 tickets, and the total revenue from ticket sales can be no more than $700.
What is the maximum number of adult tickets the theater can sell for that show?
For word problems with inequalities and a maximum or minimum, always start by defining variables and translating each sentence into an inequality ("at least" → , "no more than" → ). Identify what you are trying to maximize or minimize, then use the other condition(s) to rewrite everything in terms of that one variable (usually by solving one equation or inequality for the other variable and substituting). Simplify to a single inequality, solve it carefully, and remember to choose the largest or smallest integer that satisfies the inequality, since ticket counts and similar quantities must be whole numbers.
Hints
Set up variables and inequalities
Try defining variables for the number of adult and student tickets. What inequality represents "at least 50 tickets"? What inequality represents "no more than $700" in revenue?
Focus on what you are maximizing
You are asked for the maximum number of adult tickets. How could you use the ticket-count condition to write the number of student tickets in terms of the number of adult tickets?
Use substitution to get one inequality in one variable
Once you have written in terms of , substitute that expression into the revenue inequality. Then you will have just one inequality involving that you can solve.
Think about the inequality solution
After solving the inequality for , remember that must be a whole number (you cannot sell a fraction of a ticket). Which value should you choose to make as large as possible while still satisfying the inequality?
Desmos Guide
Model the situation with one expression
Because we want to maximize adult tickets and the theater must sell at least 50 tickets, assume the total is exactly 50. In Desmos, define an expression for revenue in terms of adult tickets :
- Type:
R(a) = 15a + 10(50 - a)
Here, a is the number of adult tickets and 50 - a is the number of student tickets.
Use a slider to explore values of a
Desmos will offer to create a slider for a. Turn it on, and set the slider so that a takes integer values in a reasonable range (for example, from 0 to 50).
Find the largest a that keeps revenue within the limit
Look at the value of R(a) as you increase a. Find the largest integer value of a for which R(a) <= 700 is still true (the revenue is at most $700). That value of a is the maximum number of adult tickets that meets all the conditions.
Step-by-step Explanation
Translate the words into inequalities
Let be the number of adult tickets and be the number of student tickets.
- "At least 50 tickets" means the total number of tickets is at least 50:
- "No more than $700" in revenue means the total ticket income is at most $700:
Now you have two inequalities that must both be true.
Use the ticket-count condition to reduce to one variable
We want to maximize (adult tickets). To allow as many adult tickets as possible while staying under the money limit, we should not sell extra unnecessary tickets.
That means we should assume the total number of tickets is exactly 50 (the minimum allowed), so
From this equation, solve for in terms of :
Substitute into the revenue inequality and simplify
Substitute into the revenue inequality :
Simplify step by step:
- Distribute the 10:
- Combine like terms ():
- Subtract 500 from both sides:
This inequality shows the restriction on that comes from the money limit.
Solve for the maximum number of adult tickets
Solve by dividing both sides by 5:
Since is the number of tickets, it must be a whole number. The largest whole number that is less than or equal to 40 is , so the maximum number of adult tickets the theater can sell is 40.