Acollege charters a bus for $1300 to take a group of students to see a broadway production. when 14 more students join the trip, the cost per student decreases by $17.50. how many students were in the original group?

let x = the original no. of students
then
(x+10) = the actual no. that went on the trip
:
 = the original cost per student
and
 = the actual cost 
:
Original cost - actual cost = $12.50
 -  = 12.50
multiply equation by x(x+10)
x(x+10)* - x(x+10)* = 12.50x(x+10)
Cancel the denominators

1500(x+10) - 1500x = 12.5x(x+10)
1500x + 15000 - 1500x = 12.5x^2 + 125x
Combine on the right to form a quadratic equation
0 = 12.5x^2 + 125x - 15000
Simplify, divide equation by 12.5
x^2 + 10x - 1200 = 0
You can use the quadratic formula; a=1; b=10; c=-1200, but this will factor to
(x + 40(x - 30) = 0
The positive solution is what we want here
x = 30 students in the original group
Check this by finding the cost per student for each scenario
1500/30 = $50.00; original cost
1500/40 = $37.50; actual cost

saving: $12.50