P,Q and R entered into a business by making investment in the ratio of `3:4:6` respectively .After eight months Q and R withdrew Rs. 2000 and Rs . 4000 respectively .if after 15 months ratio of profit share of P,Q and R is `45:53:76` , then find initial investment of R ?

Options

A. A)18000 Rs
B. B)12000 Rs
C. C)6000 Rs
D. D)24000 Rs

Answer

To solve the problem step by step, we will analyze the investments and the profit ratios of P, Q, and R.

### Step 1: Define the Investments
Let the initial investments of P, Q, and R be represented as:
- P's investment = 3X
- Q's investment = 4X
- R's investment = 6X

### Step 2: Calculate the Time of Investment
P, Q, and R invested their amounts for 15 months. However, Q and R withdrew their investments after 8 months. Therefore:
- P's investment duration = 15 months
- Q's investment duration = 8 months (before withdrawal) + 7 months (after withdrawal)
- R's investment duration = 8 months (before withdrawal) + 7 months (after withdrawal)

### Step 3: Calculate the Effective Investments
The effective investments for profit calculation will be:
- P's effective investment = 3X * 15 = 45X
- Q's effective investment = 4X * 8 + (4X - 2000) * 7
- Q's investment after withdrawal = 4X - 2000
- Q's effective investment = 32X + (4X - 2000) * 7 = 32X + 28X - 14000 = 60X - 14000
- R's effective investment = 6X * 8 + (6X - 4000) * 7
- R's investment after withdrawal = 6X - 4000
- R's effective investment = 48X + (6X - 4000) * 7 = 48X + 42X - 28000 = 90X - 28000

### Step 4: Set Up the Profit Ratio
According to the problem, the profit ratio of P, Q, and R is given as 45:53:76. Therefore, we can set up the following ratios:
- P's profit = 45X
- Q's profit = 60X - 14000
- R's profit = 90X - 28000

### Step 5: Set Up the Equation
From the profit ratios of Q and R, we can set up the equation:
\[ \frac{60X - 14000}{90X - 28000} = \frac{53}{76} \]

### Step 6: Cross-Multiply and Solve for X
Cross-multiplying gives us:
\[ 76(60X - 14000) = 53(90X - 28000) \]
Expanding both sides:
\[ 4560X - 1064000 = 4770X - 1484000 \]
Rearranging the equation:
\[ 4770X - 4560X = 1484000 - 1064000 \]
\[ 210X = 420000 \]
Dividing both sides by 210:
\[ X = 2000 \]

### Step 7: Calculate R's Initial Investment
Now that we have the value of X, we can find R's initial investment:
\[ R's \, investment = 6X = 6 \times 2000 = 12000 \]

### Final Answer
The initial investment of R is **Rs. 12000**.

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