Question

The rental of a movie theatre for Jason’s birthday was to be shared cost amongst all his friends ( excluding the birthday boy). The total rental cost was $600 for three-hour period. Sixteen more of his friends than anticipated showed up that day to rent the theatre, reducing the cost per person by $6.25. How many friends showed up to rent the theatre?

Answer 1

Let's say the number of friends who were anticipated to show up is x.
If all the expected friends showed up, each would pay $600/x for the rental.
16 more friends showed up, bringing the total number of friends to x+16.
Since the cost per person reduced $6.25, the new cost per person is $600/(x+16).
Since the cost per person reduced by $6.25, we can set up an equation: $600/x - $600/(x+16) = $6.25
Let's multiply through by x(x+16) to eliminate the denominators: 600(x+16)-600(x) = 6.25x(x+16)
Expanding, we get: 9600 = 6.25x^2 + 100x
Rearranging the equation and simplifying, we get: 6.25x^2 + 100x - 9600 = 0
Dividing through by 6.25, we get: x^2 + 16x - 1536 = 0
Factoring the quadratic, we get: (x + 48)(x - 32) = 0
So, x = 32 or x = -48
Since it doesn't make sense to have a negative number of anticipated friends, we discard x = -48.
Therefore, the number of friends who were anticipated to show up is x = 32.
The total number of friends who showed up is x + 16 = 32 + 16 = <<32+16=48>>48. Answer: \boxed{48}.