In a vessel two medicine A and B are in the ratio `4 : 1. 10` ltof mixture is taken out and replaced with 10 ltof medicine B and the ratio of A and B thus becomes` 2 : 3`. Find the initial quantity of medicine A.

Answer

To solve the problem step by step, let's break down the information provided and calculate the initial quantity of medicine A.

### Step 1: Define the initial quantities
Let the initial quantities of medicines A and B be represented as:
- Medicine A = 4x
- Medicine B = x

### Step 2: Calculate the quantities after removing 10 liters of the mixture
The total quantity of the mixture is:
\[ 4x + x = 5x \]

When 10 liters of the mixture is removed, the quantity of A and B removed can be calculated based on their ratio in the mixture.

- The fraction of medicine A in the mixture:
\[ \frac{4}{5} \]
- The fraction of medicine B in the mixture:
\[ \frac{1}{5} \]

Now, calculate the amount of A and B removed:
- Amount of A removed:
\[ \text{Amount of A removed} = \frac{4}{5} \times 10 = 8 \text{ liters} \]
- Amount of B removed:
\[ \text{Amount of B removed} = \frac{1}{5} \times 10 = 2 \text{ liters} \]

After removing 10 liters:
- Remaining quantity of A:
\[ 4x - 8 \]
- Remaining quantity of B:
\[ x - 2 \]

### Step 3: Add 10 liters of medicine B
After removing the 10 liters, we add 10 liters of medicine B:
- New quantity of B:
\[ (x - 2) + 10 = x + 8 \]

### Step 4: Set up the new ratio
According to the problem, the new ratio of A to B is 2:3:
\[ \frac{4x - 8}{x + 8} = \frac{2}{3} \]

### Step 5: Cross-multiply to solve for x
Cross-multiplying gives:
\[ 3(4x - 8) = 2(x + 8) \]
Expanding both sides:
\[ 12x - 24 = 2x + 16 \]

### Step 6: Rearranging the equation
Rearranging the equation to isolate x:
\[ 12x - 2x = 16 + 24 \]
\[ 10x = 40 \]
\[ x = 4 \]

### Step 7: Calculate the initial quantity of medicine A
Now that we have the value of x, we can find the initial quantity of medicine A:
\[ \text{Initial quantity of A} = 4x = 4 \times 4 = 16 \text{ liters} \]

### Final Answer
The initial quantity of medicine A is **16 liters**.

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