Question 213451
x = number of people who were going to attend.
x-8 = number of people who eventually attended.
-----
y = original cost per person = 42240/x.
multiply both sides of this by x to get:
x*y = 42240.
-----
y+22 = increased cost per person = 42240/(x-8).
multiply both sides of this by (x-8) to get:
(x-8)*y = 42240.
-----
since they both equal 42240, we have:
x*y = (x-8)*(y+22)
-----
we remove parentheses on the right side of this equation to get:
xy = xy + 22x -8y - 176
-----
we subtract xy from both sides of this equation to get:
0 = 22x - 8y - 176
-----
we add 8y to both sides of this equation to get:
8y = 22x - 176
-----
we divide both sides of this equation to get:
y = (22x - 176)/8
-----
we substitute in the equation xy = 42240 to get:
x * (22x-176)/8 = 42240
-----
we multiply both sides of this equation by 8 to get:
x * (22x - 176) = 42240 * 8
-----
we remove parentheses to get:
22x^2 - 176x = 42240 * 8
-----
we divide both sides of this equation by 22 to get:
x^2 - 8x = 15360
-----
we place this equation into standard quadratic equation form of ax^2 + bx + c = 0 by subtracting 15360 from both sides of this equation to get:
x^2 - 8x - 15360 = 0
-----
a = 1
b = -8
c = -15360
-----
possible factors to get a c factor of 15360 are:
15360 * 1
7680 * 2
3840 * 4
1920 * 8
960 * 16
480 * 32
240 * 64
120 * 128
-----
of these 120 * 128 is the most promising because 128 minus 120 = 8 which is our b factor.
-----
our possible factors for this quadratic equation are:
(x-128) * (x+120) = 0
we multiply these factors out to get:
x*x = x^2
x*120 = 120x
-128*x = -128x
-128*120 = -15360
-----
We add these together and combine like terms to get:
x^2 - 8x - 15360 = 0
-----
since this is identical to our quadratic equation, we have the right factors.
x = 128 or x = -120
-----
x can't be -120 because then the number of persons would be negative.
x has to be 128.
-----
the original number in the party is 128.
the cost per person would have been 42240/128 = $330.
-----
the new number in the party is 128 - 8 = 120.
the new cost per person is 42240/120 = $352.
-----
$352 - $330 = $22 which is the increas in cost per person.
-----
Your answer is:
128 people originally intended to take the trip.
-----