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?
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}.