In a vessel ratio of milk and water is `7:3`. When x ltr milk is added then ratio of milk and water becomes `17:6` Now in this mixture 3.5 ltr milk and 3 ltr water is added then ratio of milk and water becomes `5:2`. Find the value of `x`.

Options

A. 4ltr
B. 6ltr
C. 8ltr
D. 2ltr

Answer

To solve the problem step by step, we will follow the given information and equations derived from the ratios of milk and water.

### Step 1: Define the initial quantities of milk and water
Let the initial quantities of milk and water be:
- Milk = 7a liters
- Water = 3a liters

### Step 2: Set up the equation after adding x liters of milk
When x liters of milk is added, the new quantities become:
- New Milk = \( 7a + x \)
- Water remains the same = \( 3a \)

According to the problem, the new ratio of milk to water becomes \( 17:6 \). Therefore, we can set up the equation:
\[ \frac{7a + x}{3a} = \frac{17}{6} \]

### Step 3: Cross-multiply to solve for x
Cross-multiplying gives:
\[ 6(7a + x) = 17(3a) \]
Expanding both sides:
\[ 42a + 6x = 51a \]
Rearranging the equation to isolate x:
\[ 6x = 51a - 42a \]
\[ 6x = 9a \]
\[ x = \frac{9a}{6} = \frac{3a}{2} \]

### Step 4: Set up the equation after adding 3.5 liters of milk and 3 liters of water
Now, we add 3.5 liters of milk and 3 liters of water:
- New Milk = \( 7a + x + 3.5 \)
- New Water = \( 3a + 3 \)

The new ratio of milk to water becomes \( 5:2 \). Therefore, we can set up the equation:
\[ \frac{7a + x + 3.5}{3a + 3} = \frac{5}{2} \]

### Step 5: Cross-multiply to solve for a
Cross-multiplying gives:
\[ 2(7a + x + 3.5) = 5(3a + 3) \]
Expanding both sides:
\[ 14a + 2x + 7 = 15a + 15 \]
Rearranging the equation:
\[ 2x + 7 = 15a - 14a + 15 \]
\[ 2x + 7 = a + 15 \]
\[ 2x = a + 15 - 7 \]
\[ 2x = a + 8 \]

### Step 6: Substitute the value of x from Step 3
From Step 3, we have \( x = \frac{3a}{2} \). Substitute this into the equation:
\[ 2\left(\frac{3a}{2}\right) = a + 8 \]
This simplifies to:
\[ 3a = a + 8 \]
Now, isolate a:
\[ 3a - a = 8 \]
\[ 2a = 8 \]
\[ a = 4 \]

### Step 7: Find the value of x
Now substitute \( a = 4 \) back into the equation for x:
\[ x = \frac{3a}{2} = \frac{3 \times 4}{2} = \frac{12}{2} = 6 \]

### Final Answer
The value of \( x \) is **6 liters**.

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