A vessel contains a solution of two liquids A and B in the ratio 5 : 3. When 10 litres of the solution is taken out and replaced by the same quantity of B, the ratio of A and B in the vessel becomes 10 : 11. The quantity (in litres) of the solution, in the vessel was ........

Options

A. 42
B. 48
C. 52
D. 44

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the initial ratio of liquids A and B
The initial ratio of liquids A and B is given as 5:3. This means that for every 5 parts of liquid A, there are 3 parts of liquid B.

### Step 2: Set up the initial quantities
Let the total quantity of the solution in the vessel be \( x \) liters.
- The quantity of liquid A = \( \frac{5}{8}x \) (since 5 parts out of a total of 8 parts)
- The quantity of liquid B = \( \frac{3}{8}x \) (since 3 parts out of a total of 8 parts)

### Step 3: Calculate the quantities after removing 10 liters
When 10 liters of the solution is taken out, the quantities of A and B removed can be calculated based on their initial proportions:
- Quantity of A removed = \( \frac{5}{8} \times 10 = \frac{50}{8} = 6.25 \) liters
- Quantity of B removed = \( \frac{3}{8} \times 10 = \frac{30}{8} = 3.75 \) liters

After removing 10 liters, the remaining quantities are:
- Remaining A = \( \frac{5}{8}x - 6.25 \)
- Remaining B = \( \frac{3}{8}x - 3.75 \)

### Step 4: Add 10 liters of liquid B
After removing the 10 liters, we add back 10 liters of liquid B. Therefore, the new quantity of B becomes:
- New quantity of B = \( \left(\frac{3}{8}x - 3.75\right) + 10 = \frac{3}{8}x + 6.25 \)

### Step 5: Set up the new ratio
According to the problem, after this operation, the new ratio of A to B becomes 10:11. This gives us the equation:
\[ \frac{\frac{5}{8}x - 6.25}{\frac{3}{8}x + 6.25} = \frac{10}{11} \]

### Step 6: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 11\left(\frac{5}{8}x - 6.25\right) = 10\left(\frac{3}{8}x + 6.25\right) \]
Expanding both sides:
\[ \frac{55}{8}x - 68.75 = \frac{30}{8}x + 62.5 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ \frac{55}{8}x - \frac{30}{8}x = 68.75 + 62.5 \]
\[ \frac{25}{8}x = 131.25 \]

### Step 8: Solve for \( x \)
Multiplying both sides by \( \frac{8}{25} \):
\[ x = 131.25 \times \frac{8}{25} = 42 \]

### Conclusion
The total quantity of the solution in the vessel was **42 liters**.