There were 155 people at a basketball game. Tickets for the game are [tex]$2.50$[/tex] for students and [tex]$4.00$[/tex] for adults. If the total money received for admission to the game was [tex]$492.50$[/tex], how much?
Answer (1)
Define variables: Let s be the number of students and a be the number of adults.
Set up equations: s + a = 155 and 2.50 s + 4.00 a = 492.50 .
Solve for a : Using elimination, find a = 70 .
Solve for s : Substitute a back into the first equation to find s = 85 .
The number of students is 85 and the number of adults is 70 .
Explanation
Understanding the Problem We are given the following information:
Total number of people at the basketball game: 155
Ticket price for students: $2.50
Ticket price for adults: $4.00
Total money received for admission: $492.50
We want to determine the number of students and adults at the basketball game.
Setting up the Equations Let's use variables to represent the unknowns:
Let s be the number of students.
Let a be the number of adults.
We can set up a system of two equations based on the given information:
The total number of people is 155: s + a = 155
The total money received is $492.50: $2.50s + 4.00a = 492.50
Solving for the Number of Adults Now we need to solve this system of equations. We can use the substitution or elimination method. Let's use the elimination method. Multiply the first equation by -2.50 to eliminate s :
− 2.50 ( s + a ) = − 2.50 ( 155 ) − 2.50 s − 2.50 a = − 387.50
Now add this modified equation to the second equation:
( − 2.50 s − 2.50 a ) + ( 2.50 s + 4.00 a ) = − 387.50 + 492.50 1.50 a = 105
Now, solve for a :
a = 1.50 105 = 70
Solving for the Number of Students Now that we know the number of adults, we can substitute it back into the first equation to solve for the number of students:
s + a = 155 s + 70 = 155 s = 155 − 70 = 85
Final Answer So, there were 85 students and 70 adults at the basketball game.
Examples
Imagine you're organizing a school play and need to figure out how many student tickets and adult tickets were sold. You know the total number of attendees and the total revenue from ticket sales. By setting up a system of equations similar to the basketball game problem, you can determine the exact number of student and adult tickets sold, helping you manage finances and plan future events more effectively. This method is useful in any scenario where you have two unknowns and two pieces of information relating them, such as calculating ingredient quantities in a recipe or determining investment allocations.
