Two vessels A and B containing a mixture of milk and water. The ratio of milk and water in the vessel A is 4 : 5 and in vessel B is 5:8. If x lit of mixture from vessel A and 39 lit of mixture from vessel B is taken out and mix into another vessel C then ratio of milk to water become 2:3 in vessel C. Find the value of x.

Options

A. A)16 lit
B. B)12.8 lit
C. C)14.5 lit
D. D)13.5 lit

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Understand the Ratios in Each Vessel
- Vessel A has a milk to water ratio of 4:5.
- Vessel B has a milk to water ratio of 5:8.

### Step 2: Calculate the Amount of Milk and Water Taken from Vessel B
- From Vessel B, we take 39 liters of mixture.
- The total parts in Vessel B = 5 (milk) + 8 (water) = 13 parts.
- Milk in 39 liters from Vessel B = (5/13) * 39 = 15 liters.
- Water in 39 liters from Vessel B = (8/13) * 39 = 24 liters.

### Step 3: Express the Amount of Milk and Water Taken from Vessel A
- Let x liters be taken from Vessel A.
- The total parts in Vessel A = 4 (milk) + 5 (water) = 9 parts.
- Milk in x liters from Vessel A = (4/9) * x liters.
- Water in x liters from Vessel A = (5/9) * x liters.

### Step 4: Set Up the Equation for Vessel C
- In Vessel C, the total milk = Milk from A + Milk from B = (4/9)x + 15 liters.
- In Vessel C, the total water = Water from A + Water from B = (5/9)x + 24 liters.
- The ratio of milk to water in Vessel C is given as 2:3.

### Step 5: Write the Ratio Equation
- According to the ratio, we can set up the equation:
\[ \frac{(4/9)x + 15}{(5/9)x + 24} = \frac{2}{3} \]

### Step 6: Cross-Multiply to Solve for x
- Cross-multiplying gives us:
\[ 3((4/9)x + 15) = 2((5/9)x + 24) \]
- Expanding both sides:
\[ \frac{12}{9}x + 45 = \frac{10}{9}x + 48 \]

### Step 7: Simplify the Equation
- Rearranging gives:
\[ \frac{12}{9}x - \frac{10}{9}x = 48 - 45 \]
- This simplifies to:
\[ \frac{2}{9}x = 3 \]

### Step 8: Solve for x
- Multiplying both sides by 9 gives:
\[ 2x = 27 \]
- Dividing by 2 gives:
\[ x = 13.5 \]

### Final Answer
The value of x is **13.5 liters**.

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