A mixture of milk and water in a jar comprises 12 litres of milk. If 8 liters of pure milk and 3 liters of pure water were added to this jar, the percentage of water in the new mixture would be `20%`. What was the initial quantity of water in the jar? (in litre)

Options

A. 5
B. 4
C. 2
D. 6

Answer

To solve the problem, we need to find the initial quantity of water in the jar. Let's break it down step by step.

### Step 1: Define the Variables
Let the initial quantity of water in the jar be \( x \) liters. We know that the initial quantity of milk is 12 liters.

### Step 2: Determine the New Quantities After Adding Milk and Water
After adding 8 liters of pure milk and 3 liters of pure water, the new quantities will be:
- New quantity of milk = \( 12 + 8 = 20 \) liters
- New quantity of water = \( x + 3 \) liters

### Step 3: Calculate the Total Quantity of the New Mixture
The total quantity of the new mixture is:
\[ \text{Total mixture} = \text{New milk} + \text{New water} = 20 + (x + 3) = 23 + x \text{ liters} \]

### Step 4: Set Up the Percentage Equation
According to the problem, the percentage of water in the new mixture is 20%. Therefore, we can set up the equation:
\[ \frac{\text{Quantity of water}}{\text{Total mixture}} \times 100 = 20 \]
Substituting the known values:
\[ \frac{x + 3}{23 + x} \times 100 = 20 \]

### Step 5: Simplify the Equation
To eliminate the percentage, we can rewrite the equation:
\[ \frac{x + 3}{23 + x} = \frac{20}{100} = \frac{1}{5} \]

### Step 6: Cross Multiply
Cross multiplying gives us:
\[ 5(x + 3) = 1(23 + x) \]
Expanding both sides:
\[ 5x + 15 = 23 + x \]

### Step 7: Rearrange the Equation
Now, we rearrange the equation to isolate \( x \):
\[ 5x - x = 23 - 15 \]
This simplifies to:
\[ 4x = 8 \]

### Step 8: Solve for \( x \)
Dividing both sides by 4:
\[ x = \frac{8}{4} = 2 \]

### Conclusion
The initial quantity of water in the jar was \( 2 \) liters.

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