Problem Solving

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 $9,900, find the contribution made by David.

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

The OA is D.

I'm really confused with this PS question. Experts, any suggestion? I think that I can solve it using an equations system in agree of the description,

I know that the total price is $9900, the equation for it can be,
$$A+B+C+D=9900$$
Then for each one of the brothers contribution will be,
$$A=\frac{1}{2}\cdot\left(B+C+D\right)$$
$$B=\frac{1}{4}\cdot\left(A+C+D\right)$$
$$C=\frac{2}{3}\cdot\left(A+B+D\right)$$
I have 4 equations and 4 incognits, now I can get each contribution.

Thanks in advance.

Hello LUANDATO.

Let's take a look at your question.

Your answer is right. We have to set four equations with four variables. Your equations are great: $$(1) A+B+C+D=9900$$ $$(2) A=\frac{1}{2}\cdot\left(B+C+D\right)$$ $$(3) B=\frac{1}{4}\cdot\left(A+C+D\right)$$ $$(4) C=\frac{2}{3}\cdot\left(A+B+D\right)$$ Now, from equation (2) and (1) we can get $$2A=B+C+D\ \ \Leftrightarrow\ 3A=A+B+C+D=9900\ \Leftrightarrow\ A=\frac{9900}{3}=3300.$$ Now we know A. From equation (3) and (1) we can get $$4B=A+C+D\ \Leftrightarrow\ 5B=A+B+C+D=9900\ \Leftrightarrow\ B=\frac{9900}{5}=1980.$$ From the equation (4) and (1) we can get $$3C=2\left(A+B+D\right)\ \Leftrightarrow\ 5C=2\left(A+B+D\right)+2C=2\left(A+B+C+D\right)$$ $$5C=2\left(9900\right)=19800\ \Leftrightarrow\ C=\frac{19800}{5}=3960.$$ Finally, from equation (1) we can conclude: $$3300+1980+3960+D=9900\ \Leftrightarrow\ D=9900-9240=660\ \text{dollars}.$$ Hence, the correct answer is the option [spoiler]D=$660[/spoiler].

I hope this answer can help you.

Feel free to contact me again if you have a doubt.

Regards.