The ratio of milk and water in a vessel is `13: 11`. If 48 ltr of mixture is taken out and 81 ltr water is added then ratio of milk and water becomes `7:8`. Find quantity of milk in initial mixture.

Options

A. 301 ltr
B. 321 ltr
C. 295 ltr
D. 299 ltr

Answer

To solve the problem step by step, we will follow the given information and perform the necessary calculations.

### Step 1: Define the initial quantities of milk and water
Let the initial quantities of milk and water be represented as:
- Milk = 13x liters
- Water = 11x liters

### Step 2: Calculate the total initial mixture
The total initial mixture is:
\[ \text{Total mixture} = \text{Milk} + \text{Water} = 13x + 11x = 24x \text{ liters} \]

### Step 3: Remove 48 liters of the mixture
When 48 liters of the mixture is taken out, the quantities of milk and water removed can be calculated using the ratio of milk to water:
- Milk removed = \( \frac{13}{24} \times 48 = 26 \text{ liters} \)
- Water removed = \( \frac{11}{24} \times 48 = 22 \text{ liters} \)

### Step 4: Calculate the remaining quantities of milk and water
After removing 48 liters:
- Remaining Milk = \( 13x - 26 \text{ liters} \)
- Remaining Water = \( 11x - 22 \text{ liters} \)

### Step 5: Add 81 liters of water
Now, we add 81 liters of water to the remaining water:
- New quantity of Water = \( (11x - 22) + 81 = 11x + 59 \text{ liters} \)

### Step 6: Set up the new ratio of milk to water
According to the problem, the new ratio of milk to water becomes \( 7:8 \). Therefore, we can set up the equation:
\[ \frac{13x - 26}{11x + 59} = \frac{7}{8} \]

### Step 7: Cross-multiply to solve for x
Cross-multiplying gives us:
\[ 8(13x - 26) = 7(11x + 59) \]
Expanding both sides:
\[ 104x - 208 = 77x + 413 \]

### Step 8: Rearranging the equation
Rearranging the equation to isolate x:
\[ 104x - 77x = 413 + 208 \]
\[ 27x = 621 \]
\[ x = \frac{621}{27} = 23 \]

### Step 9: Calculate the quantity of milk in the initial mixture
Now, we can find the quantity of milk in the initial mixture:
\[ \text{Quantity of Milk} = 13x = 13 \times 23 = 299 \text{ liters} \]

### Final Answer
The quantity of milk in the initial mixture is **299 liters**.

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