correct answer is: a) ₹\;2000,\;₹\;4000\;\&\;₹\;3000

Explanation

Let,
First friend gets ₹ x.
\therefore Second friend gets ₹ \left(2\times x\right) or ₹ 2x.
And, third friend gets ₹ \large\left(\frac{x+2x}{2}\right) or ₹ \large\frac{3x}{2}.
According to the question,
₹ \left(x+2x+\large\frac{3x}{2}\right)\normalsize=₹ 9000
\rightarrow\large\left(\frac{2x+4x+3x}{2}\right)\normalsize=9000
\rightarrow\large\frac{9x}{2}\normalsize=9000
\rightarrow 9x=\left(9000\times2\right) _{[cross multiply]}
\rightarrow x=\large\left(\frac{9000\times2}{9}\right)
\rightarrow x=2000
So that,
First friend gets ₹ 2000.
Second friend get ₹ 2x or ₹ \left(2\times2000\right)=₹ 4000.
And, third friend gets ₹ \large\frac{3x}{2} or, ₹ \large\left(\frac{3\times2000}{2}\right)\normalsize=₹ 3000.
Ans: First friend will get ₹ 2000, Second friend will gets ₹ 4000 and third friend will gets ₹ 3000.

Other ways to phrase this particular math query are:

1. Three friends divide ₹ 9000 among themselves, with the second friend receiving twice the amount received by the first friend, and the third friend receiving half the total amount received by the first two friends. Determine the amount each friend receives.
2. ₹ 9000 is distributed among three friends, with the second friend receiving twice the amount received by the first friend, and the third friend receiving half the total amount received by the first two friends combined. Find the amount received by each friend.
3. A total of ₹ 9000 is split among three friends such that the second friend receives twice the amount the first friend receives, and the third friend receives half the sum of the amounts received by the other two friends. Determine the amount each friend receives.