A group of choir members decided to raise 3600/= to buy a guitar. Each member was to contribute an equal amount. In the preparation process, five members transferred to another church, which meant the remaining contributors had to pay more to achieve the target.
a. Show that the increase in the contribution per member was: Sh. 18,000 / (n(n-5)) if n is the initial number of members.
b. If the increase in the contribution per member was Sh. 24, what was the original contribution before the other members left?
c. Calculate the percentage increase in the contribution after the others left.
Answer (1)
Let's address each part of the question step by step.
a. Show that the increase in the contribution per member was: n ( n − 5 ) 18000 if n is the initial number of members.
Initially, the total amount required was 3600, and n members agreed to contribute equally. Therefore, each member's initial contribution is given by: C i = n 3600
When 5 members leave, the number of contributors becomes n − 5 . The new contribution per member is then: C n e w = n − 5 3600
The increase in contribution per member then follows from: C in cre a se = C n e w − C i = n − 5 3600 − n 3600
Combining these fractions over a common denominator gives: C in cre a se = n ( n − 5 ) 3600 n − 3600 ( n − 5 ) = n ( n − 5 ) 3600 × 5 = n ( n − 5 ) 18000
This proves the increase is indeed n ( n − 5 ) 18000 .
b. If the increase in the contribution per member was 24 , what was the original contribution before the other members left?
We know: n ( n − 5 ) 18000 = 24
Solving for n :
18000 = 24 n ( n − 5 ) n ( n − 5 ) = 750
We need to find integer solutions for n . Since n ( n − 5 ) = 750 , consider the quadratic equation: n 2 − 5 n − 750 = 0
Using the quadratic formula for roots n = 2 a − b ± b 2 − 4 a c , where a = 1 , b = − 5 , c = − 750 :
n = 2 5 ± 25 + 3000 = 2 5 ± 55
Calculating the roots:
n = 2 60 = 30
n = 2 − 50 = − 25 (not feasible as n must be positive)
Thus, n = 30 members initially.
The original contribution per member was: C i = 30 3600 = 120
c. Calculate the percentage increase in the contribution after the others left.
The increased contribution per member after 5 members left is: C n e w = 30 − 5 3600 = 25 3600 = 144
The increase is 144 − 120 = 24 . So, the percentage increase is: Percentage Increase = ( 120 24 ) × 100% = 20%
Thus, the contribution per member increased by 20% after 5 members left.
