A vessel contains a mixture of O2 and N2. The quantity of 02 is 36% of total mixture. Some quantity of mixture taken out from it and replaced by N2. This process is repeated once more. Now the quantity of 02 is 16% of total mixture. If total quantity of mixture was 18 litres then, find the quantity of mixture which is taken out?

Options

A. 12 L
B. 4.5 L
C. 9 L
D. 6 L

Answer

To solve the problem step by step, we will follow the information provided in the question and use the concept of mixtures.

### Step 1: Determine the initial quantities of O2 and N2
The total quantity of the mixture is given as 18 liters. The quantity of O2 is 36% of the total mixture.

\[ \text{Quantity of O2} = 36\% \text{ of } 18 \text{ liters} = \frac{36}{100} \times 18 = 6.48 \text{ liters} \]

\[ \text{Quantity of N2} = 18 \text{ liters} - 6.48 \text{ liters} = 11.52 \text{ liters} \]

### Step 2: Let the quantity taken out be X liters
Let the quantity of the mixture taken out be \( X \) liters. After taking out \( X \) liters, the remaining quantity of O2 and N2 will change.

### Step 3: Calculate the remaining quantities after the first extraction
When \( X \) liters of the mixture is taken out, the quantities of O2 and N2 taken out will be proportional to their initial amounts.

\[ \text{O2 taken out} = \frac{6.48}{18} \times X = 0.36X \]
\[ \text{N2 taken out} = \frac{11.52}{18} \times X = 0.64X \]

After removing \( X \) liters, the remaining quantities will be:

\[ \text{Remaining O2} = 6.48 - 0.36X \]
\[ \text{Remaining N2} = 11.52 - 0.64X \]

### Step 4: Replace the taken out mixture with N2
After removing \( X \) liters, we add \( X \) liters of N2 back into the vessel. Therefore, the new quantity of N2 becomes:

\[ \text{New N2} = (11.52 - 0.64X) + X = 11.52 + 0.36X \]

### Step 5: Calculate the new percentages after the first replacement
The total volume remains 18 liters. The new percentages are:

\[ \text{New quantity of O2} = 6.48 - 0.36X \]

### Step 6: Repeat the process
This process is repeated once more, so we will take out \( X \) liters again. The new quantity of O2 after the second extraction will be:

\[ \text{O2 taken out again} = \frac{6.48 - 0.36X}{18} \times X \]

Let’s denote the new remaining O2 after the second extraction as:

\[ \text{Remaining O2 after second extraction} = (6.48 - 0.36X) - \left(\frac{6.48 - 0.36X}{18} \times X\right) \]

### Step 7: Set up the equation based on the final percentage of O2
According to the problem, after the second replacement, the quantity of O2 is now 16% of the total mixture:

\[ \text{Remaining O2 after second extraction} = 16\% \text{ of } 18 = \frac{16}{100} \times 18 = 2.88 \text{ liters} \]

### Step 8: Solve for X
Set the equation from the remaining O2 equal to 2.88 liters and solve for \( X \):

\[ (6.48 - 0.36X) - \left(\frac{6.48 - 0.36X}{18} \times X\right) = 2.88 \]

This equation can be solved for \( X \) to find the quantity taken out.

### Final Calculation
After simplifying and solving the equation, you will find:

\[ X = 6 \text{ liters} \]

### Conclusion
The quantity of the mixture which is taken out is **6 liters**.

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