Members of the Computer Club were assessed equal amounts to raise $1200 to buy some software. When 8 new members joined, the per-member assessment was reduced by $7.50. What was the new size of the club?
Let's assume the original number of club members is 'x'.
We know that the members were assessed equal amounts to raise $1200, so the original per-member assessment can be calculated as:
Original per-member assessment = Total amount raised / Number of members
Original per-member assessment = $1200 / x
After 8 new members joined, the per-member assessment was reduced by $7.50. This means the new per-member assessment is equal to the original per-member assessment minus $7.50.
New per-member assessment = Original per-member assessment - $7.50
New per-member assessment = ($1200 / x) - $7.50
We also know that the new per-member assessment is equal to the original per-member assessment divided by (x + 8) since the number of members increased by 8.
New per-member assessment = Original per-member assessment / (x + 8)
New per-member assessment = $1200 / (x + 8)
Now we can equate the two expressions for the new per-member assessment and solve for x:
($1200 / x) - $7.50 = $1200 / (x + 8)
To simplify the equation, let's get rid of the fractions by multiplying both sides by x(x + 8):
($1200 * (x + 8)) - ($7.50 * x * (x + 8)) = $1200 * x
Expanding the equation and combining like terms:
1200x + 9600 - 7.50x^2 - 60x = 1200x
Rearranging the terms:
7.50x^2 + 60x + 1200x - 1200x - 9600 = 0
Combining like terms:
7.50x^2 + 60x - 9600 = 0
Now we can solve this quadratic equation. We can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case:
a = 7.50
b = 60
c = -9600
x = (-(60) ± sqrt((60)^2 - 4(7.50)(-9600))) / 2(7.50)
Simplifying:
x = (-60 ± sqrt(3600 + 288000)) / 15
x = (-60 ± sqrt(291600)) / 15
x = (-60 ± 540) / 15
If we take the positive value of x:
x = (-60 + 540) / 15
x = 480 / 15
x = 32
Therefore, the original number of club members was 32.
Since 8 new members joined, the new size of the club will be:
New size of the club = Original number of members + 8
New size of the club = 32 + 8
New size of the club = 40
So, the new size of the club is 40.
To solve this problem, we need to set up an equation. Let's denote the original number of members as "x" and the original per-member assessment as "y".
According to the given information, the members were assessed equal amounts to raise $1200. Therefore, the equation can be written as:
xy = 1200 -- (Equation 1)
We are also told that when 8 new members joined, the per-member assessment was reduced by $7.50. This means the new per-member assessment is (y - 7.50). The new assessment must still raise a total of $1200, but now based on the new number of members (x + 8). Therefore, we can write a second equation:
(x + 8)(y - 7.50) = 1200 -- (Equation 2)
To find the new size of the club, we need to solve for x in Equation 2.
Expanding Equation 2, we get:
xy - 7.50x + 8y - 60 = 1200
Rearranging the equation:
xy + 8y - 7.50x - 60 - 1200 = 0
xy + 8y - 7.50x - 1260 = 0
xy - 7.50x + 8y - 1260 = 0 -- (Equation 3)
Now we have two equations: Equation 1 and Equation 3. We can solve these two equations simultaneously to find the new number of members (x + 8).
Subtracting Equation 1 from Equation 3, we get:
xy - xy - 7.50x + 7.50x + 8y - 8y - 1260 - 1200 = 0 - 0
- 7.50x + 7.50x - 1260 - 1200 = 0
- 2460 = 0
This equation is not possible because the left side is a constant value and the right side is zero. This suggests that there might be an error in the problem statement.
10
number of original members ---x
share per member = 1200/x
number of members now --- x+8
share per member = 1200/(x+8)
solve 1200/(x+8) = 7.5