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?
Let's assume the number of adults attending the carnival is A, and the number of children attending is C.
From the problem, we know that the total number of attendees is 500, so A + C = 500.
The adults are charged $10 each and children $5 each, so the total amount earned from adults is 10A and from children is 5C.
The PTA wants to earn $3,500 from admission charges, so 10A + 5C = $3,500.
From the first equation, we can rewrite it as A = 500 - C.
Substituting this into the second equation, we get 10(500 - C) + 5C = $3,500.
Expanding, we have 5000 - 10C + 5C = $3,500.
Combining like terms, we get -5C = $3,500 - $5000 = -$1,500.
Dividing both sides of the equation by -5, we get C = 300.
Substituting this back into the first equation, we have A + 300 = 500.
Subtracting 300 from both sides, we get A = 500 - 300 = 200.
Therefore, there need to be 200 adults and 300 children attending for the PTA to reach their goal of $3,500.