P, Q, and R started a business by investing their amounts in the ratio 4:6:5, respectively. After 6 months, P increased his investment by Rs 400, while Q reduced his investment by Rs 400. If the annual profit share of P, out of a total profit of Rs 30,000, is Rs 9,000, find the initial investment(in Rs) made by R.

The correct option is 2000.

Step-by-Step Solution:

1. Express initial investments using the given ratio:
The initial investments of P, Q, and R are in the ratio 4 : 6 : 5.
Let:
Initial investment of P = 4x
Initial investment of Q = 6x
Initial investment of R = 5x

2. Calculate effective investments over 12 months (in terms of amount × time):
For P:
For the first 6 months, investment = 4x
For the next 6 months, investment = 4x + 400
Total equivalent investment of P = (4x × 6) + [(4x + 400) × 6]

$\text{P's total investment}=24x+24x+2400=48x+2400$

For Q:
For the first 6 months, investment = 6x
For the next 6 months, investment = 6x - 400
Total equivalent investment of Q = (6x × 6) + [(6x - 400) × 6]

$\text{Q's total investment}=36x+36x-2400=72x-2400$

For R:
R makes no changes for the entire 12 months.
Total equivalent investment of R = 5x × 12

$\text{R's total investment}=60x$

3. Find the Total Investment Units:

$\text{Total Investment}=(48x+2400)+(72x-2400)+60x$

$\text{Total Investment}=180x$

4. Use P's profit share to solve for x:
Given that P's share out of a total profit of Rs 30,000 is Rs 9,000:

$\frac{\text{P's Investment}}{\text{Total Investment}}=\frac{\text{P's Profit Share}}{\text{Total Profit}}$

$\frac{48x+2400}{180x}=\frac{9000}{30000}$

$\frac{48x+2400}{180x}=\frac{3}{10}$

Cross-multiplying to solve for x:

$10\times (48x+2400)=3\times 180x$

$480x+24000=540x$

$540x-480x=24000$

$60x=24000$

$x=400$

5. Calculate the initial investment made by R:

$\text{Initial Investment of R}=5x=5\times 400=\text{Rs 2000}$