A tank contains 384 liters mixture of milk & water in the ratio of 15 :17. If X liters of mixture taken out and (X-14) liters of milk & (X - 34) liters of water added in-remaining mixture so ratio of milk & water becomes equal. Find quantity of milk in original mixture is what percent of total final mixture?

Options

A. `40%`
B. `35%`
C. `45%`
D. `50%`

Answer

To solve the problem step-by-step, we will follow the instructions given in the video transcript.

### Step 1: Determine the quantities of milk and water in the original mixture.

The total volume of the mixture is 384 liters, and the ratio of milk to water is 15:17.

1. **Calculate the total parts in the ratio:**
\[ \text{Total parts} = 15 + 17 = 32 \]

2. **Calculate the quantity of milk:**
\[ \text{Quantity of milk} = \frac{15}{32} \times 384 = 180 \text{ liters} \]

3. **Calculate the quantity of water:**
\[ \text{Quantity of water} = \frac{17}{32} \times 384 = 204 \text{ liters} \]

### Step 2: Set up the equation after removing and adding the mixtures.

Let \( X \) be the liters of the mixture taken out. The quantities of milk and water removed will be in the same ratio as the original mixture.

1. **Quantity of milk removed:**
\[ \text{Milk removed} = \frac{15}{32} \times X \]

2. **Quantity of water removed:**
\[ \text{Water removed} = \frac{17}{32} \times X \]

After removing \( X \) liters, the remaining quantities of milk and water are:
- Remaining milk:
\[ 180 - \frac{15}{32}X \]
- Remaining water:
\[ 204 - \frac{17}{32}X \]

### Step 3: Add the new quantities of milk and water.

We add \( X - 14 \) liters of milk and \( X - 34 \) liters of water to the remaining mixture.

1. **New quantity of milk:**
\[ \text{New milk} = \left(180 - \frac{15}{32}X\right) + (X - 14) \]

2. **New quantity of water:**
\[ \text{New water} = \left(204 - \frac{17}{32}X\right) + (X - 34) \]

### Step 4: Set the new quantities equal to each other.

Since the new ratio of milk to water becomes equal (1:1):
\[ \left(180 - \frac{15}{32}X + X - 14\right) = \left(204 - \frac{17}{32}X + X - 34\right) \]

### Step 5: Simplify and solve for \( X \).

1. **Simplify the left side:**
\[ 166 + \left(1 - \frac{15}{32}\right)X = 166 + \frac{17}{32}X \]

2. **Simplify the right side:**
\[ 170 + \left(1 - \frac{17}{32}\right)X = 170 + \frac{15}{32}X \]

3. **Setting both sides equal:**
\[ 166 + \frac{17}{32}X = 170 + \frac{15}{32}X \]

4. **Rearranging gives:**
\[ \frac{2}{32}X = 4 \implies X = 64 \text{ liters} \]

### Step 6: Calculate the final quantities of milk and water.

1. **Final quantity of milk:**
\[ \text{Final milk} = 180 - \frac{15}{32} \times 64 + (64 - 14) = 180 - 30 + 50 = 200 \text{ liters} \]

2. **Final quantity of water:**
\[ \text{Final water} = 204 - \frac{17}{32} \times 64 + (64 - 34) = 204 - 34 + 30 = 200 \text{ liters} \]

### Step 7: Calculate the total final mixture.

\[ \text{Total final mixture} = \text{Final milk} + \text{Final water} = 200 + 200 = 400 \text{ liters} \]

### Step 8: Calculate the required percentage.

1. **Percentage of milk in the final mixture:**
\[ \text{Percentage} = \left(\frac{200}{400}\right) \times 100 = 50\% \]

### Final Answer:
The quantity of milk in the original mixture is **50%** of the total final mixture.

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