There are two mixture comprising milk and water. Ratio of milk to water in both mixture is 4:1, 50% of mixture - B is mixed in mixture - A, then quantity of water in the resulting mixture becomes 20 liters. Then, find ratio of total quantity of mixture - A to total quantity of mixture - B if total quantity of both the mixture is 140 lit.

Options

A. A)`4:3`
B. B)`3:4`
C. C)`5:6`
D. D)`6:5`

Answer

To solve the problem step by step, we will follow the information provided and apply the concepts of ratios and mixtures.

### Step 1: Understand the Ratios
Both mixtures A and B have a ratio of milk to water of 4:1. This means that in every 5 parts of the mixture, 4 parts are milk and 1 part is water.

### Step 2: Define the Quantities
Let the total quantity of mixture A be \( A \) liters and the total quantity of mixture B be \( B \) liters. According to the problem, we know that:
\[ A + B = 140 \text{ liters} \] (Equation 1)

### Step 3: Calculate Water in Mixture B
Since 50% of mixture B is mixed with mixture A, the quantity of mixture B that is added to A is:
\[ \frac{B}{2} \text{ liters} \]

### Step 4: Calculate the Water Content
In mixture B, the amount of water can be calculated as follows:
- Total parts in mixture B = 5 (4 parts milk + 1 part water)
- Water in mixture B = \( \frac{1}{5} \times B \)

When we take 50% of mixture B, the water content from this portion is:
\[ \text{Water from B} = \frac{1}{5} \times \frac{B}{2} = \frac{B}{10} \]

### Step 5: Calculate Total Water in Resulting Mixture
In the resulting mixture (let's call it mixture C), the total amount of water is given as 20 liters. The water in mixture A can be calculated similarly:
- Water in mixture A = \( \frac{1}{5} \times A \)

Thus, the total water in mixture C is:
\[ \frac{A}{5} + \frac{B}{10} = 20 \text{ liters} \] (Equation 2)

### Step 6: Solve the Equations
Now we have two equations:
1. \( A + B = 140 \)
2. \( \frac{A}{5} + \frac{B}{10} = 20 \)

From Equation 2, we can eliminate the fractions by multiplying the entire equation by 10:
\[ 2A + B = 200 \] (Equation 3)

### Step 7: Substitute Equation 1 into Equation 3
From Equation 1, we can express \( B \) in terms of \( A \):
\[ B = 140 - A \]

Substituting this into Equation 3:
\[ 2A + (140 - A) = 200 \]
\[ 2A - A + 140 = 200 \]
\[ A + 140 = 200 \]
\[ A = 60 \text{ liters} \]

### Step 8: Find Quantity of Mixture B
Now substituting \( A \) back into Equation 1 to find \( B \):
\[ 60 + B = 140 \]
\[ B = 140 - 60 = 80 \text{ liters} \]

### Step 9: Find the Ratio of A to B
Now we can find the ratio of total quantities of mixtures A and B:
\[ A : B = 60 : 80 \]
This can be simplified by dividing both sides by 20:
\[ A : B = 3 : 4 \]

### Final Answer
The ratio of total quantity of mixture A to total quantity of mixture B is:
\[ \boxed{3 : 4} \]

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