P, Q and R invested in a business in the ratio 1 : 2 : 3. After 6 months, P withdrew all his capital, Q doubled his capital and R halved his capital. Out of a total profit of Rs 13,800 at the end of one year, what is \(33\frac{1}{3}\%\) of R’s profit (in Rs)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1800
Profit is shared in the ratio of capital × months, adding the part before the change and the part after it. \(33\frac{1}{3}\%\) is the same as one-third.
Let the capitals be \(x\), \(2x\) and \(3x\)
P: \(x\times6=6x\) (nothing after 6 months)
Q: \(2x\times6+4x\times6=36x\)
R: \(3x\times6+\frac{3}{2}x\times6=18x+9x=27x\)
P : Q : R \(=6:36:27\), total \(69\) parts
R’s profit \(\displaystyle =\frac{27}{69}\times13800=27\times200=\) Rs 5,400
\(33\frac{1}{3}\%\) of 5400 \(\displaystyle =\frac{5400}{3}=1800\)
The required amount is Rs 1,800.
Why the other options are incorrect
Option (2) 5400 — Incorrect. This is R’s whole profit; the question asks for one-third of it.
Option (3) 5900 — Incorrect. It is more than R’s whole profit of Rs 5,400, so it cannot be a third of it.
Option (4) 1600 — Incorrect. Three times 1600 is 4800, which is not R’s profit of Rs 5,400.