Rational expression word problem

a french club collected the same amount from each student going on a trip. When 6 students could not go, each of the remaining students was charged an extra $3. if the total cost was $540, how many students went on the trip?

10 points for best answer

let the original number = x , and the amount from each student = a

then ax is the original amount and equals 540

so a = 540/x ---- (1)

when 6 students could't go then they become x - 6 , and the new amount would be (a+3)(x-6)

so (a + 3)(x - 6) = 540

=> (540/x + 3)(x - 6) = 540

mult by x u get :

(540 + 3x)(x - 6) = 540x

=> 540x - 3240 + 3x^2 - 18x = 540x

540x cancels out and u get :

3x^2 - 18x - 3240 = 0

divide by 3 :

x^2 - 6x - 1080 = 0

(x - 36)(x + 30) = 0

so x = 36 students

when 6 didn't go , so 30 students remain

so 30 students went on the trip <<<<<<<<<<<<<<<<<

x number of students need to spend y dollars, where x * y = 540. So y = 540/x
(x - 6) students need to spend (y + 3) dollars

(x - 6)(y + 3) = 540
(x - 6)(540/x + 3) = 540

540 + 3x - 3240/x - 18 = 540
3x - 3240/x - 18 = 0
3x^2 - 3240 - 18x = 0
3x^2 - 18x - 3240 = 0
x^2 - 6x - 1080 = 0
(x - 36)(x + 30) = 0

x = 36 or x = -30

Obviously x = 36.

So 36 students were supposed to go at a cost of $15 per student.
However, what actually happened was that 30 students went at a cost of $18 each.