Four brothers Adam, Bill, Charles and David together contributed certain amount of money and purchased a car. The contribution made by Adam is half that of the total contribution made by the other three brothers, contribution made by Bill is one-fourth of the total contribution made by the other three brothers and the contribution made by Charles is two-third of the total contribution made by the other three brothers. If the total price of the car is $9900, find the contribution made by David.

A. $540
B. $580
C. $600
D. $660
E. $680
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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

how did you get the contributed amounts? for example A contributed 1/3, B contributed 1/5th, etc?
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there is another simple way ----

a+b+c+d=t

a= .5(t-a)
b= .25(t-b)
c=2/3(t-c)

so t = 9900
so a=3300 , b= 1980 , c= 3960
so d= 660

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