A jar contains a mixture of milk and water in the ratio of 3 :1. Now, `1/25` of the mixture is taken out and 24 litres water is added to it. If the resultant ratio of milk to water in the jar was 2:1, what was the initial quantity of mixture in the jar? (in litres)

Options

A. 160
B. 180
C. 200
D. 250

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Initial Ratio
The initial ratio of milk to water in the jar is given as 3:1. This means that for every 3 parts of milk, there is 1 part of water.

### Step 2: Define the Total Quantity
Let the initial total quantity of the mixture be \(4x\) liters, where \(x\) is a common factor. Thus, the quantity of milk is \(3x\) liters, and the quantity of water is \(x\) liters.

### Step 3: Calculate the Quantity Taken Out
We take out \( \frac{1}{25} \) of the mixture. Therefore, the quantity taken out is:
\[ \text{Quantity taken out} = \frac{1}{25} \times 4x = \frac{4x}{25} \text{ liters} \]

### Step 4: Calculate Milk and Water Taken Out
Now, we need to find out how much milk and water is taken out from the mixture:
- Milk taken out:
\[ \text{Milk taken out} = \frac{3x}{4x} \times \frac{4x}{25} = \frac{3x}{25} \text{ liters} \]
- Water taken out:
\[ \text{Water taken out} = \frac{x}{4x} \times \frac{4x}{25} = \frac{x}{25} \text{ liters} \]

### Step 5: Calculate Remaining Quantities
After taking out the mixture, the remaining quantities of milk and water in the jar are:
- Remaining milk:
\[ \text{Remaining milk} = 3x - \frac{3x}{25} = \frac{75x - 3x}{25} = \frac{72x}{25} \text{ liters} \]
- Remaining water:
\[ \text{Remaining water} = x - \frac{x}{25} = \frac{25x - x}{25} = \frac{24x}{25} \text{ liters} \]

### Step 6: Add Water to the Mixture
Next, we add 24 liters of water to the remaining water:
\[ \text{Final quantity of water} = \frac{24x}{25} + 24 \]

### Step 7: Set Up the Ratio Equation
The problem states that the resultant ratio of milk to water is 2:1. Therefore, we can set up the equation:
\[ \frac{\frac{72x}{25}}{\frac{24x}{25} + 24} = \frac{2}{1} \]

### Step 8: Cross Multiply and Simplify
Cross-multiplying gives us:
\[ 72x = 2 \left( \frac{24x}{25} + 24 \right) \]
Expanding the right side:
\[ 72x = \frac{48x}{25} + 48 \]

### Step 9: Eliminate the Fraction
To eliminate the fraction, multiply through by 25:
\[ 1800x = 48x + 1200 \]

### Step 10: Solve for \(x\)
Rearranging gives:
\[ 1800x - 48x = 1200 \]
\[ 1752x = 1200 \]
\[ x = \frac{1200}{1752} = \frac{100}{146} = \frac{50}{73} \]

### Step 11: Calculate the Initial Quantity
Now, substituting \(x\) back to find the initial quantity of the mixture:
\[ \text{Initial quantity} = 4x = 4 \times \frac{50}{73} = \frac{200}{73} \text{ liters} \]

### Conclusion
The initial quantity of the mixture in the jar is \(200\) liters.