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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

Let the number of adults be \( a \) and the number of children be \( c \).

We have two equations based on the problem:

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

  2. The total revenue from admissions should be $3,500. Adults are charged $10 and children $5: \[ 10a + 5c = 3500 \]

Now, we have a system of equations:

  1. \( a + c = 500 \) (1)
  2. \( 10a + 5c = 3500 \) (2)

We can simplify equation (2) by dividing all terms by 5: \[ 2a + c = 700 \]

Now we have the simplified system:

  1. \( a + c = 500 \) (1)
  2. \( 2a + c = 700 \) (3)

Next, we can eliminate \( c \) by subtracting equation (1) from equation (3): \[ (2a + c) - (a + c) = 700 - 500 \] \[ 2a + c - a - c = 200 \] \[ a = 200 \]

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

Thus, the number of adults and children that need to attend is:

  • Adults: \( 200 \)
  • Children: \( 300 \)

To verify, we can check the revenue: \[ 10a + 5c = 10(200) + 5(300) = 2000 + 1500 = 3500 \] And the total attendance: \[ a + c = 200 + 300 = 500 \]

Therefore, the solution is correct. The Kesling Middle School PTA needs 200 adults and 300 children to attend to reach their goal of $3,500.