A container contains milk and water in ratio 13 : 7. When 30 lit mixture is taken out of x lit mixture and 2.5 lit of milk is added then quantity of milk becomes `66 2/3`% of total mixture. Find 'x'.

Options

A. 80 lit
B. 70 lit
C. 90 lit
D. 100 lit

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the initial ratio of milk and water
The initial ratio of milk to water in the container is 13:7. This means that for every 20 parts of the mixture, 13 parts are milk and 7 parts are water.

### Step 2: Define the total volume of the mixture
Let the total volume of the mixture be \( x \) liters. Therefore, the volume of milk in the mixture is:
\[ \text{Volume of milk} = \frac{13}{20}x \]
And the volume of water in the mixture is:
\[ \text{Volume of water} = \frac{7}{20}x \]

### Step 3: Calculate the remaining mixture after taking out 30 liters
When 30 liters of the mixture is taken out, the remaining volume of the mixture is:
\[ \text{Remaining mixture} = x - 30 \text{ liters} \]

### Step 4: Calculate the volumes of milk and water in the remaining mixture
The volume of milk in the remaining mixture can be calculated as:
\[ \text{Remaining milk} = \left( x - 30 \right) \times \frac{13}{20} \]
The volume of water in the remaining mixture can be calculated as:
\[ \text{Remaining water} = \left( x - 30 \right) \times \frac{7}{20} \]

### Step 5: Add 2.5 liters of milk to the remaining mixture
After removing 30 liters, 2.5 liters of milk is added. Therefore, the new volume of milk becomes:
\[ \text{New volume of milk} = \left( \left( x - 30 \right) \times \frac{13}{20} \right) + 2.5 \]

### Step 6: Calculate the total volume of the new mixture
The total volume of the new mixture is:
\[ \text{New total mixture} = (x - 30) + 2.5 = x - 27.5 \]

### Step 7: Set up the equation based on the percentage of milk
According to the problem, the new quantity of milk is \( 66 \frac{2}{3} \% \) of the total mixture. Converting \( 66 \frac{2}{3} \% \) to a fraction gives:
\[ 66 \frac{2}{3} \% = \frac{2}{3} \]
Thus, we can write the equation:
\[ \left( \left( x - 30 \right) \times \frac{13}{20} + 2.5 \right) = \frac{2}{3} \times (x - 27.5) \]

### Step 8: Solve the equation
Now we can solve the equation:
1. Multiply both sides by 3 to eliminate the fraction:
\[ 3 \left( \left( x - 30 \right) \times \frac{13}{20} + 2.5 \right) = 2(x - 27.5) \]
2. Distributing on both sides:
\[ \frac{39}{20}(x - 30) + 7.5 = 2x - 55 \]
3. Multiply through by 20 to eliminate the denominator:
\[ 39(x - 30) + 150 = 40x - 1100 \]
4. Expand and simplify:
\[ 39x - 1170 + 150 = 40x - 1100 \]
\[ 39x - 1020 = 40x - 1100 \]
5. Rearranging gives:
\[ 1100 - 1020 = 40x - 39x \]
\[ 80 = x \]

### Step 9: Conclusion
Thus, the value of \( x \) is 80 liters.