If P, Q, and R invested in 1 : 2 : 3 and after 6 months P withdrew all of his capital, Q doubled his capital and R halved his capital. After one year out of a total profit of Rs. 13800, what is the 33.33% of R's profit?

1800

Investment of P = 1R

Investment of Q = 2R

Investment of Q' = 4R                     (after doing double)

Investment of R = 3R

Investment of R' = 1.5R                   (after doing half)

Profit Ratio = Investment ratio × time ratio

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So, ATQ

                         P   :   Q   :   Q'  :   R   :   R'

Investment ⇒   1   :    2   :   4   :   3   :   1.5

Time           ⇒    6   :    6   :   6   :   6   :    6

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Profit          ⇒    6  :   12   :  24 :  18   :   9

So,

                           (P)    :   (Q + Q')  :  (R + R')

Profit          ⇒       6      :       36      :      27

Now, total profit = 6 + 36 + 27 = 69R = 13800

⇒ 1R = 200

⇒ 33.33% of R's profit = \(\frac{1}{3}\) × 27R ⇒ 9R = 9 × 200 = 1800