15. A total of 2000 people attended a benefit concert was held to raise money for a children foundation. Student ticket cost $2 and an adult ticket cost $3. If the organizer raises a total of $5050, how many students attended the concert?

Question

Let : x = number of students
2000 - x = number of adults

Answer 1

Let's denote the number of student tickets as $x$ and the number of adult tickets as $2000-x$.

From the problem, we know:

1. The total number of attendees is 2000: $$x+(2000-x)=2000$$ This confirms our setup.

2. The total amount raised is $5050, which can be expressed with the ticket prices:

   • Student ticket price = $2
   • Adult ticket price = $3

   Therefore, the equation for total revenue is: $$2x+3(2000-x)=5050$$

Now, let's simplify this equation:

$$2x+6000-3x=5050$$

Combine like terms:

$$-1x+6000=5050$$

Subtract 6000 from both sides:

$$-x=5050-6000$$ $$-x=-950$$

Multiply both sides by -1:

$$x=950$$

So, the number of students who attended the concert is $\boxed{{\displaystyle 950}}$.

To verify, let's find the number of adults:

$$2000-950=1050$$

Now, we check the total revenue:

$$2(950)+3(1050)=1900+3150=5050$$

This confirms that our solution is correct, and 950 students attended the concert.