Four brothers Adam, Bill, Charles and David together

(1) A = (B + C + D)/2
(2) B = (A + C + D)/4
(3) C = 2(B + D + A)/3
(4) A + B + C + D = 9900

Combine (1) and (4)
A = (9900 - A)/2 ==> A=3300

Combine (2) and (4)
B = (9900 - B)/4==>B=1980

Combine (3) and (4)
C = 2(9900-C)/3==>C=3960

D = 9900 - (3300 + 1980 + 3960)
D = 660

Answer: 660

how did you get the contributed amounts? for example A contributed 1/3, B contributed 1/5th, etc?

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We can let the contribution made by Adam, Bill, Charles and David be A, B, C, and D respectively.

Since Adam contributes A dollars, the other three boys contribute 9900 - A dollars; so we have:

A = (1/2)(9900 - A)

2A = 9900 - A

3A = 9900

A = 3,300


Similarly, since Bill contributes B dollars, the other three boys contribute 9900 - B dollars and we have:

B = 1/4(9900 - B)

4B = 9900 - B

5B = 9900

B = 1,980

Finally, since Charles contributed C dollars, the other three boys contributed 9900 - C dollars and thus:

C = 2/3(9900 - C)

3C = 2(9900) - 2C

5C = 2(9900)

C = 3,960

Now that we know A = 3,300, B = 1,980 and C = 3,960, we can easily find D by the following equation:

3,300 + 1,980 + 3,960 + D = 9,900

9,240 + D = 9,900

D = 660

Answer: D

Hmm, if you write it down you get literally:

\(A = \frac{1}{2}(B + C + D)\)

\(B = \frac{1}{4}(A + C + D)\)

\(C = \frac{2}{3}(A + B + D)\)

\(A + B + C + D = 9900\)

Therefore you have a system of 4 unknowns and 4 equations, so it's solvable (too bad it's not a DS problem we could just stop here ). Now I guess the easiest way to solve it is to use Gauss' Pivot Method? Remember you are only interested in D, so no need to calculate A, B and C, that would be a waste of time

Hi, I don't understand where you get the 1/3 from?

Add the portions of the ratio.

1:2 => 3 parts total, A is 1/3
1:4 => 5 parts total, B is 1/5
2:3 => 5 parts total, C is 2/5

Similar question to practice: baker-s-dozen-128782-20.html#p1057509

Nice method to solve this kind of problems, when only 1 variable value is required

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