A group of people is sharing a hotel room. The total price of the room is $495. After two members of the group leave without contributing, the cost per person for each of the remaining members increases by $10. How many people were in the group originally?
A) 9
B) 10
C) 11
D) 13
E) 25
Let x = the
original number of people in the group
So, x - 2 = number of people in the group
after 2 leave
So, 495/x = original cost per person
And 495/(x-2) = cost per person after 2 leave
After two members of the group leave without contributing, the cost per person for each of the remaining members increases by $10.
In other words, (cost per person after 2 leave) - (original cost per person) = 10
So, 495/(x-2) - 495/x = 10
IMPORTANT: At this point, it might actually be fastest to plug in the answer choices to see which one works. Or, we can solve the equation for x . . .
495/(x-2) - 495/x = 10
Multiply both sides by (x)(x-2) to get: 495x - 495(x-2) = 10(x)(x-2)
Expand: 495x - 495x + 990 = 10x^2 - 20x
Simplify: 10x^2 - 20x - 990 = 0
Divide both sides by 10: x^2 - 2x - 99 = 0
Factor: (x - 11)(x + 9) = 0
So, x = 11 or x = -9
We can eliminate x = -9, so x must equal
11
Cheers,
Brent
