Maria is renting identical folding chairs for an event. The rental company charges a flat handling fee of $60, plus $15 for each chair rented. Maria’s budget allows her to spend no more than $360 on the chairs.
What is the maximum number of chairs Maria can rent?
For budget word problems with a flat fee plus a per-item cost, first define a variable for the number of items, then write an inequality of the form (flat fee) + (per-item cost)·(number of items) ≤ (budget). Carefully translate phrases like "no more than" to the correct inequality symbol (≤), solve the inequality step by step (subtract the constant, then divide by the coefficient), and finally remember that counts of items must be whole numbers—choose the largest integer that still satisfies the inequality to answer "maximum" questions efficiently.
Hints
Identify the fixed and variable costs
Ask yourself: what part of the cost is a one-time fee, and what part depends on how many chairs Maria rents? Try writing an expression for the total cost in terms of the number of chairs.
Turn the budget phrase into an inequality
The phrase "no more than $360" describes a relationship between the total cost and 360. Should the total cost be greater than, less than, equal to, or less than or equal to 360? Which inequality symbol matches that idea?
Solve the inequality step by step
Once you have an inequality for the total cost, isolate the variable by first subtracting the constant term from both sides, then dividing both sides by the coefficient of the variable.
Think about whole numbers and "maximum"
Remember that the number of chairs must be a whole number. After solving the inequality, which whole number is the largest value that still satisfies the inequality?
Desmos Guide
Graph the total cost as a function of chairs
In Desmos, type y = 60 + 15x to graph the total cost (on the -axis) as a function of the number of chairs (on the -axis). This line shows how the cost increases as more chairs are rented.
Graph the budget limit
On a new line, type y = 360 to draw a horizontal line representing Maria’s maximum budget. This is the highest cost she can pay.
Find the largest allowed number of chairs from the graph
Click or tap the intersection point of the two lines. The -coordinate of this point tells you how many chairs make the total cost exactly equal to the budget. Any -value to the left (smaller) on the first line gives a cost that is below the budget. Use this to determine the largest whole-number value of that does not exceed the intersection’s -coordinate, and match that number to the correct answer choice.
Step-by-step Explanation
Define the variable and set up the inequality
Let be the number of chairs Maria rents.
The total cost is the flat handling fee plus the cost per chair:
- Handling fee: $60
- Chair cost: $15 per chair, so dollars for chairs
So the total cost is .
Maria can spend no more than $360, which means the total cost must be less than or equal to 360:
Isolate the term with the variable
Start solving the inequality by moving the constant term to the other side.
Subtract from both sides of :
So now you have . This relates the cost per chair to the budget after the handling fee is paid.
Solve for the number of chairs in terms of a division
To isolate , divide both sides of the inequality by .
Because is positive, the direction of the inequality stays the same:
This simplifies the left side to and leaves a division to compute on the right side.
Compute, interpret the inequality, and choose the maximum whole number
Now simplify the division on the right side:
So the inequality becomes
This means Maria can rent any number of chairs up to and including 20 without going over her $360 budget.
The maximum number of chairs she can rent is 20, which corresponds to answer choice C) 20.