Question

The $75 cost for a party was to be shared equally by all of those attending. Since five more people attended than was expected, the price per person dropped by 50 cents. How many people attended the party?

Answer 1

original cost per person = 75/x
new cost per person = 75/(x+5)

75/x - 75(x+5) = 1/2
multiply each term by x(x+5)
75(x+5) - 75x = (1/2)x(x+5)
times 2
150x + 750 - 150x = x^2 + 5x
x^2 + 5x - 750 = 0
(x-25)(x+30) = 0
x = 25 or x = -30 , but clearly x > 0

x=25

originally 25 had planned to attend, but 30 actually did.

check:
cost with 25 = 75/25 = $3.00
cost with 30 = 75/30 = $2.50
it cost 50 cents less.
Answer is correct.