x liter is taken out from a vessel full of kerosene and substituted by pure petrol. This process is repeated two more times. Finally the ratio of petrol and kerosene in the mixture becomes `1701:27`. Find the value of x if volume of the original solution is 18 liter?

Options

A. 13.5 ltr
B. 9 ltr
C. 12 ltr
D. 10 ltr

Answer

To solve the problem step by step, we will follow the process of calculating the remaining kerosene after each replacement and find the value of \( x \).

### Step 1: Understand the Problem
We have a vessel containing 18 liters of kerosene. We take out \( x \) liters of kerosene and replace it with \( x \) liters of petrol. This process is repeated two more times. Finally, the ratio of petrol to kerosene is given as \( 1701:27 \).

### Step 2: Calculate Total Parts in the Final Ratio
The total parts in the final ratio of petrol to kerosene is:
\[ 1701 + 27 = 1728 \]

### Step 3: Determine the Remaining Kerosene
Let’s denote the initial amount of kerosene as \( K_0 = 18 \) liters. After each replacement, the amount of kerosene remaining can be expressed as:
\[ K_n = K_0 \left(1 - \frac{x}{18}\right)^n \]
where \( n \) is the number of times the process is repeated.

### Step 4: Set Up the Equation for Remaining Kerosene
Since the process is repeated 3 times, we have:
\[ K_3 = 18 \left(1 - \frac{x}{18}\right)^3 \]
The final amount of kerosene can also be calculated from the ratio:
\[ \frac{K_3}{K_3 + P_3} = \frac{27}{1728} \]
where \( P_3 \) is the amount of petrol after 3 replacements. Since \( P_3 = 18 - K_3 \), we can substitute this into the equation.

### Step 5: Substitute and Solve
From the ratio, we can express \( K_3 \):
\[ \frac{K_3}{18} = \frac{27}{1728} \]
This implies:
\[ K_3 = 18 \cdot \frac{27}{1728} = \frac{486}{1728} = \frac{27}{96} = \frac{9}{32} \text{ liters} \]

### Step 6: Set Up the Equation for Kerosene
Now we can set up the equation:
\[ 18 \left(1 - \frac{x}{18}\right)^3 = \frac{9}{32} \]

### Step 7: Simplify the Equation
Dividing both sides by 18 gives:
\[ \left(1 - \frac{x}{18}\right)^3 = \frac{9}{32 \cdot 18} = \frac{9}{576} = \frac{3}{192} = \frac{1}{64} \]

### Step 8: Take the Cube Root
Taking the cube root of both sides:
\[ 1 - \frac{x}{18} = \frac{1}{4} \]

### Step 9: Solve for \( x \)
Rearranging gives:
\[ \frac{x}{18} = 1 - \frac{1}{4} = \frac{3}{4} \]
Thus:
\[ x = 18 \cdot \frac{3}{4} = \frac{54}{4} = 13.5 \text{ liters} \]

### Final Answer
The value of \( x \) is \( 13.5 \) liters.

---