Two vessels A and B filled with mixture of milk and water in ratio 1:4 and 2:3 both mixtures are poured in vessel C of capacity 70 l vessel C becomes full if water is 150% more than milk in C find quantity of milk in vessel A

Options

A. 8
B. 6
C. 12
D. 5

Answer

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

### Step 1: Define Variables for Vessel A
Vessel A has a mixture of milk and water in the ratio of 1:4.
- Let the quantity of milk in vessel A be \( x \) liters.
- Then the quantity of water in vessel A will be \( 4x \) liters.

### Step 2: Define Variables for Vessel B
Vessel B has a mixture of milk and water in the ratio of 2:3.
- Let the quantity of milk in vessel B be \( 2y \) liters.
- Then the quantity of water in vessel B will be \( 3y \) liters.

### Step 3: Total Volume in Vessel C
When both mixtures are poured into vessel C, the total volume of milk and water in vessel C will be:
- Total milk: \( x + 2y \)
- Total water: \( 4x + 3y \)

The capacity of vessel C is 70 liters, so we have:
\[ x + 2y + 4x + 3y = 70 \]
Combining like terms gives:
\[ 5x + 5y = 70 \]
Dividing the entire equation by 5:
\[ x + y = 14 \quad \text{(Equation 1)} \]

### Step 4: Relationship Between Milk and Water in Vessel C
According to the problem, the amount of water in vessel C is 150% more than the amount of milk. This means:
- If the amount of milk is \( M \), then the amount of water \( W \) is given by:
\[ W = M + 1.5M = 2.5M \]
Thus, we can express this in terms of the total volume:
\[ M + W = 70 \]
Substituting \( W \):
\[ M + 2.5M = 70 \]
This simplifies to:
\[ 3.5M = 70 \]
Solving for \( M \):
\[ M = \frac{70}{3.5} = 20 \]

### Step 5: Finding the Total Milk in Vessel C
From the previous steps, we found that the total milk in vessel C is \( 20 \) liters. Therefore:
\[ x + 2y = 20 \quad \text{(Equation 2)} \]

### Step 6: Solve the System of Equations
Now we have two equations:
1. \( x + y = 14 \) (Equation 1)
2. \( x + 2y = 20 \) (Equation 2)

We can solve these equations simultaneously.

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 14 - x \]

Substituting \( y \) into Equation 2:
\[ x + 2(14 - x) = 20 \]
This simplifies to:
\[ x + 28 - 2x = 20 \]
Combining like terms:
\[ -x + 28 = 20 \]
Solving for \( x \):
\[ -x = 20 - 28 \]
\[ -x = -8 \implies x = 8 \]

### Step 7: Conclusion
The quantity of milk in vessel A is \( x = 8 \) liters.