Question 130·Easy·Linear Inequalities in One or Two Variables
A community center is selling advance tickets for a concert. Adult tickets cost $15 each, and student tickets cost $10 each. The center needs to collect at least $1,300 from ticket sales. If 60 student tickets are sold, what is the minimum number of adult tickets that must be sold to meet the center's goal?
For linear inequality word problems like this, first define a clear variable (here, the number of adult tickets). Translate the wording into an inequality: fixed or known quantities get substituted right away (such as the 60 student tickets), and phrases like “at least” become . Solve the resulting linear inequality as you would a linear equation, then check whether your answer must be a whole number. If you get a decimal but need a minimum count (people, tickets, items), always round up and quickly verify by plugging back into the original money expression.
Hints
Set up the revenue expression
Let represent the number of adult tickets. How much money comes from adult tickets, and how much comes from the 60 student tickets? Add these together for the total revenue.
Use the phrase “at least” correctly
Translate “at least $1300” into an inequality symbol. Should the total money be , , , or $1300?
Solve and handle the whole-number requirement
After you solve the inequality for , think about whether can be a decimal and what to do if your solution is not a whole number but represents a minimum amount.
Desmos Guide
Compute the adult ticket requirement with an expression
In Desmos, type the expression (1300 - 10*60)/15 to represent solving for (after subtracting the student-ticket income).
Interpret the decimal output
Look at the value Desmos gives for this expression. Since it represents the minimum number of adult tickets needed and the number of tickets must be a whole number, think about whether you should round this value up or down to still reach at least $1300.
Step-by-step Explanation
Define the variable and translate the situation
Let be the number of adult tickets sold.
Each adult ticket brings in $15, and each student ticket brings in $10. The community center wants to collect at least $1300, so the total money from adult and student tickets combined must be greater than or equal to $1300.
Write an inequality for the total revenue
Money from adult tickets: .
Money from student tickets: $10 times the number of student tickets.
The problem says 60 student tickets are sold, so the total income is
This must be at least $1300, so we write the inequality
Substitute and simplify the inequality
First compute the student-ticket income:
So the inequality becomes
Subtract 600 from both sides:
Solve for the number of adult tickets and interpret the result
Now divide both sides by 15 to solve for :
Compute the fraction:
This means the number of adult tickets must be at least about 46.7. Since you cannot sell a fraction of a ticket and the total must still reach at least $1300, you must round up to the next whole number.
So the minimum number of adult tickets that must be sold is 47.