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The Kesling Middle School PTA is planning a carnival to raise money for the school's art department They estimate that the event will be very popular and that they will have 500 people attend. They plan to charge adults $10 and children $5 for admission. The PTA wants to earn $3,500 from admission charges. How many adults and how many children need to attend for the PTA to reach their goal of $3,500?

1 answer

To solve this problem, we can use a system of equations.

Let's define:

  • \( x \) as the number of adults attending.
  • \( y \) as the number of children attending.

We have the following two equations based on the problem:

  1. The total number of attendees (adults and children) is 500: \[ x + y = 500 \]

  2. The total revenue from admissions should be $3,500. Adults pay $10 and children pay $5: \[ 10x + 5y = 3500 \]

Now we can solve this system of equations.

First, we can simplify the second equation by dividing everything by 5: \[ 2x + y = 700 \]

Now we have a simpler system:

  1. \( x + y = 500 \) (Equation 1)
  2. \( 2x + y = 700 \) (Equation 2)

Next, we can subtract Equation 1 from Equation 2: \[ (2x + y) - (x + y) = 700 - 500 \] This simplifies to: \[ x = 200 \]

Now that we have \( x \), we can substitute it back into Equation 1 to find \( y \): \[ 200 + y = 500 \] \[ y = 500 - 200 = 300 \]

Therefore, the PTA needs to have 200 adults and 300 children attend to reach their goal of $3,500 in admissions.

To summarize:

  • Number of adults: \( \boxed{200} \)
  • Number of children: \( \boxed{300} \)