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the kesling middle school pta is planning a carnival to raise money for the schools 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 adission. 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 find out how many adults and children need to attend the carnival for the PTA to reach their goal of $3,500, we can set up a system of equations.

Let:

  • \( x \) = number of adults
  • \( y \) = number of children

We know two things:

  1. The total number of attendees is 500, so: \[ x + y = 500 \]

  2. The total admission fees need to be $3,500, with adults paying $10 and children paying $5: \[ 10x + 5y = 3500 \]

Now we have a system of equations:

  1. \( x + y = 500 \)
  2. \( 10x + 5y = 3500 \)

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

Now our system is:

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

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

Now, we can substitute \( x \) back into the first equation to find \( y \): \[ 200 + y = 500 \] \[ y = 500 - 200 = 300 \]

Thus, the solution is:

  • 200 adults
  • 300 children

To check:

  • Total number of attendees: \( 200 + 300 = 500 \) (correct)
  • Admission fees: \( 10(200) + 5(300) = 2000 + 1500 = 3500 \) (correct)

Therefore, the PTA needs 200 adults and 300 children to achieve their goal of $3,500.