A buket contains a mixture of two liquids A & B in the proportion `5 : 3`. If 16 litres of the mixture is replaced by 16 litres of liquid B, then the ratio of the two liquids becomes `3 : 5`. How much of the liquid B was there in the bucket?

Options

A. 25 litres
B. 15 litres
C. 18 litres
D. None of these

Answer

To solve the problem, we need to find out how much of liquid B was initially in the bucket, given that the mixture of liquids A and B is in the ratio of 5:3. Let's break down the solution step by step.

### Step 1: Define the initial quantities of liquids A and B.
Let the quantities of liquids A and B be represented as:
- Liquid A = 5x
- Liquid B = 3x

### Step 2: Calculate the total volume of the mixture.
The total volume of the mixture is:
\[ \text{Total Volume} = 5x + 3x = 8x \]

### Step 3: Determine the quantities after removing 16 liters of the mixture.
When we remove 16 liters of the mixture, we need to find out how much of liquids A and B are removed. Since the mixture is in the ratio of 5:3, the amounts of A and B removed can be calculated as follows:

- Amount of liquid A removed:
\[ \text{Liquid A removed} = \frac{5}{8} \times 16 = 10 \text{ liters} \]

- Amount of liquid B removed:
\[ \text{Liquid B removed} = \frac{3}{8} \times 16 = 6 \text{ liters} \]

### Step 4: Calculate the remaining quantities of liquids A and B.
After removing 16 liters of the mixture, the remaining quantities will be:
- Remaining liquid A:
\[ \text{Remaining A} = 5x - 10 \]

- Remaining liquid B:
\[ \text{Remaining B} = 3x - 6 \]

### Step 5: Add 16 liters of liquid B to the mixture.
Now, we replace the 16 liters removed with 16 liters of liquid B. The new quantity of liquid B becomes:
\[ \text{New quantity of B} = (3x - 6) + 16 = 3x + 10 \]

### Step 6: Set up the new ratio of liquids A and B.
According to the problem, after the replacement, the ratio of A to B becomes 3:5. Therefore, we can write:
\[ \frac{5x - 10}{3x + 10} = \frac{3}{5} \]

### Step 7: Cross-multiply to solve for x.
Cross-multiplying gives us:
\[ 5(5x - 10) = 3(3x + 10) \]
Expanding both sides:
\[ 25x - 50 = 9x + 30 \]

### Step 8: Rearrange and solve for x.
Rearranging the equation:
\[ 25x - 9x = 30 + 50 \]
\[ 16x = 80 \]
\[ x = 5 \]

### Step 9: Calculate the initial quantity of liquid B.
Now, substituting \(x\) back to find the initial quantity of liquid B:
\[ \text{Initial quantity of B} = 3x = 3 \times 5 = 15 \text{ liters} \]

### Final Answer:
The initial quantity of liquid B in the bucket was **15 liters**.

---