Question 349557: the members of the student catholic action decided to raise Php 4 000 to help the local charity clinic. The amount to be raised was divided equally among the members. Due to an increase of 20 members, the amount to be raised by each member went down by Php 10. How many members were originally in the club? (show your solution involving quadratic equation)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you want to raise 4000.
let x = original number of members.
let y = amount required from each member.
you get:
y = 4000 / x
with an increase of 20 members, the amount each member needed to raise went down by 10.
you get:
y-10 = 4000 / (x + 20)
you have 2 equations that need to be solved simultaneously.
since y = 4000 / x, you can substitute for y in the second equation to get:
(4000 / x) - 10 = 4000 / (x + 20)
multiply both sides of this equation by (x + 20) * x to get:
4000 * (x + 20) - (10 * (x + 20) * x) = 4000 * x
Simplify by removing parentheses to get:
4000*x + 80,000 - 10*x^2 - 200*x = 4000 * x
subtract 4000 * x from both sides of this equation to get:
80,000 - 10*x^2 - 200*x = 0
re=order the terms to get:
-10*x^2 - 200*x + 80,000 = 0
multiply both sides of this equation by -1 to get:
10x^2 + 200*x - 80,000 = 0
you don't have to do that, but it's easier to factor the equation if the coefficient of the x^2 term is positive.
divide both sides of the equation by 10 to get:
x^2 + 20*x - 8000 = 0
factor the equation to get:
(x + 100) * (x - 80) = 0
your values of x can be either -100 or +80.
since x has to be positive, then your answer is x = 80.
substitute in your original equation to confirm this is a good solution.
y = 4000 / x becomes y = 4000 / 80 becomes y = 50
with the original number of members, each member is required to provide 50.
y - 10 would then equal 40.
your second equation is:
y-10 = 4000 / (x + 20)
y - 10 = 40
x + 20 = 100
you get:
40 = 4000 / 100 which becomes:
40 = 40 which is true.
when you add 20 members, each member now has to pay 40.
solution is confirmed as good.
there were 80 original members in the club.
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