M07-17

Official Solution:

Metropolis Corporation has four shareholders: Fritz, Luis, Alfred, and Werner. The number of shares Fritz owns is \(\frac{2}{3}\) of the total number of shares owned by the other three shareholders combined. Similarly, Luis owns \(\frac{3}{7}\) of the total number of shares owned by the other three shareholders, while Alfred owns \(\frac{4}{11}\) of the total number of shares owned by the other three shareholders. If the company distributed $3,600,000 in dividends among the four shareholders, how much of this amount was received by Werner?

A. $90,000
B. $100,000
C. $120,000
D. $180,000
E. $240,000


Fritz owns \(\frac{2}{3}\) of the shares owned by the other three shareholders. In other words, Fritz owns \(\frac{2}{2+3}=\frac{2}{5}\) of all the shares. To put it more simply, Fritz owns 2 parts, while the other three shareholders together own 3 parts. Thus, Fritz owns 2 parts out of a total of 5 parts.

Luis owns \(\frac{3}{7}\) of the shares owned by the other three shareholders. In other words, Luis owns \(\frac{3}{3+7}=\frac{3}{10}\) of all the shares.

Alfred owns \(\frac{4}{11}\) of the shares owned by the other three shareholders. In other words, Alfred owns \(\frac{4}{4+11}=\frac{4}{15}\) of all the shares.

Combined, these three shareholders own \(\frac{2}{5}+\frac{3}{10}+\frac{4}{15}=\frac{29}{30}\) of all the shares. This implies that Werner owns the remaining shares, which amounts to \(1-\frac{29}{30}=\frac{1}{30}\). Therefore, from the total of $3,600,000, Werner receives \($3,600,000 * \frac{1}{30} = $120,000\).


Answer: C
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f=\(\frac{2}{3}\)(a+l+w) - this is what the stem provides to us

treat this as a fraction - in a simpler form it is: \(\frac{f}{1}\)=\(\frac{2 (a+l+w)}{3}\)

hence \(\frac{f}{(a+l+w)}\)=\(\frac{2}{3}\) - it means there are total 5 parts of a whole, of which f gains 2, hence

f=\(\frac{2}{5}\) of total

For anyone still having doubts,

Remember: The money will be divided as per each person's individual share of the whole. Thus, you need to find W's share

Consider total share = 1

Denote share of each guy as:- F,L,A and W. Thus, F+L+A+W = 1

F has \(\frac{2}{3}\) of the rest of the 3's share combined. "Rest of the 3 means" (1-F), so we can write:-

F = \(\frac{2(1-F)}{3} Therefore, F = \frac{2}{5}\)

Now, for the rest:-

\(L = \frac{3(1-L)}{7} Therefore, L = \frac{3}{9}\)

\(A = \frac{4(1-A)}{11} Therefore, A = \frac{4}{15}\)

\(So, W's share = 1 - (\frac{2}{5}+\frac{3}{9}+\frac{4}{15})\)
Thus, W has \(\frac{1}{30}\) of the whole or\(\frac{1*3,600,000}{30}\) = $120,000

For those who are still confused.
F=2/3 of (1-F)
F=2/3-2F/3
3F=2-2F
5F=2
hence, F=2/5
This is a longer method and not advised during the exam, just for clarity

It's quite simple: A has $2 and B has 3$ --> A has 2/3rd of B's amount and also 2/(2+3)=2/5th of total amount of $5.

Hope it's clear.
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A simple and clear explanation for those who couldn't understand the fraction part -

Given - F = 2/3 * ( L+A+W)

Simplifying : 3/2 * F = L+A+W

Adding "F" to both sides : F + 3/2 * F= F + L + A + W

5/2* F = F + L + A + W = Total (T)

Therefore : F = 2/5 * T

I hope this helps.

I'm not sure if i am interpreting this question correctly, but for those who are still confused;

"F = 2/3 of the other three shareholders" basically says "F's ratio to the other three is 2:3". Hence "total part" = 5.

Other solution:
I think you can treat this problem as a ratio problem:

Number of shares that Fritz owns is \(\frac{2}{3}\) of number of the shares of the other three shareholders.
That means: F=\(\frac{2}{3}\)(L+A+W). We can re-write this as \(\frac{F}{L+A+W}\)=\(\frac{2}{3}\).
So, the ratio of F to (L+A+W) to Total (=F+L+A+W) is 2:3:5. That means \(\frac{F}{Total}\)=\(\frac{2}{5}\) <=> F=\(\frac{2}{5}\)Total

The same for the others
Please correct me if i'm wrong.

How do you get the fraction 2/5th's of the total out of 2/3rd's of the shares of the other 3? Like why do you do 2/2+3? I can't seem to figure out how that computes.... thank you

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.
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