BTGmoderatorLU wrote:Currently, y percent of the members on the finance committee are women and next month, z percent of the men on the finance committee will resign. If no other personnel changes occur, then after the resignations next month, the men who remain on the finance committee will represent what percent of the total finance committee members?
$$A.\ \frac{100\left(100-z\right)\left(100-y\right)}{100^2-z\left(100-y\right)}$$
$$B.\ \frac{\left(100-z\right)\left(100-y\right)}{100}$$
$$C.\ \left(100-z\right)\left(100-y\right)$$
$$D.\ \frac{zy}{100}-z$$
$$E.\ \frac{z\left(100-y\right)}{100}$$
We can let n = the total number of members currently on the committee. Since y percent of members are women, then (100 - y) percent of members are men. Therefore, ny/100 are women and n(100 - y)/100 are men. Since z percent of men will resign next month, the number of men left will be n(100 - y)/100 * (100 - z)/100. Therefore, the percent of members that will be men after the resignations next month is:
[n(100 - y)/100 * (100 - z)/100] / [n(100 - y)/100 * (100 - z)/100 + ny/100] * 100
[(100 - y)(100 - z)/10,000] / [(100 - y)(100 - z)/10,000 + y/100] * 100
[(100 - y)(100 - z)] / [(100 - y)(100 - z) + 100y] * 100
[(100 - y)(100 - z)] / [10,000 - 100z - 100y + yz + 100y] * 00
[(100 - y)(100 - z)] / [10,000 - 100z + yz] * 100
[100(100 - y)(100 - z)] / [100^2 - z(100 - y)]
Alternate Solution:
Let's suppose that there are 100 people on the committee, y = 50 percent of them are women and z = 50 percent of the men will resign.
According to this information, there are 50 women and 50 men on the committee and 25 men will resign. After resignations, the percentage of the men on the committee will be 25/75 = 1/3 = 33 1/3 %.
Let us see which of the answer choices produce 33 1/3 when y = 50 and z = 50:
Answer Choice A: [100(100 - z)(100 - y)]/[100^2 - z(100 - y)]
(100*50*50)/(10000 - 50*50) = 250000/(10000 - 2500) = 250000/7500 = 33 1/3
We see that answer choice A is possible.
Answer Choice B:
[(100 - z)(100 - y)]/100
(50*50)/100 = 2500/100 = 25
Answer Choice B is not possible
Answer Choice C:
(100 - z)(100 -y)
50 * 50 = 2500
Answer Choice C is not possible
Answer Choice D:
(zy/100) - z
[(50*50)/100] - 50 = 2500/100 - 50 = -25
Answer Choice D is not possible
Answer Choice E:
[z(100 - y)]/100
(50*50)/100 = 2500/100 = 25
Answer Choice E is not possible
Therefore, A is the only possible answer choice.
Answer: A
