Two vessels A and B contain milk and water mixed in the ratio 4: 3 and 2: 3. The ratio in which these mixtures be mixed to form a new mixture containing half milk and half water is

Options

A. `7:5`
B. `6:5`
C. `5:6`
D. `4:3`

Answer

To solve the problem of mixing two vessels A and B containing milk and water in specified ratios to form a new mixture with equal parts milk and water, we can follow these steps:

### Step 1: Determine the milk ratio in each vessel
- For vessel A, the ratio of milk to water is 4:3.
- The fraction of milk in vessel A = Milk / (Milk + Water) = 4 / (4 + 3) = 4 / 7.

- For vessel B, the ratio of milk to water is 2:3.
- The fraction of milk in vessel B = Milk / (Milk + Water) = 2 / (2 + 3) = 2 / 5.

### Step 2: Determine the milk ratio in the desired mixture
- The desired mixture is to have equal parts of milk and water, which means:
- The fraction of milk in the mixture = 1 / (1 + 1) = 1 / 2.

### Step 3: Set up the alligation formula
- We will use the alligation method to find the ratio in which the two mixtures should be combined.
- Place the fractions of milk in vessels A and B and the desired fraction of milk in the mixture in a format suitable for alligation:

```
A: 4/7
M: 1/2
B: 2/5
```

### Step 4: Calculate the differences
- Calculate the difference between the fraction of milk in the mixtures and the desired fraction:
- Difference for A: (1/2) - (2/5)
- Difference for B: (4/7) - (1/2)

### Step 5: Solve the differences
- For A:
- Convert to a common denominator:
- (1/2) = 5/10 and (2/5) = 4/10, so (1/2) - (2/5) = 5/10 - 4/10 = 1/10.

- For B:
- Convert to a common denominator:
- (4/7) = 28/49 and (1/2) = 24.5/49, so (4/7) - (1/2) = 28/49 - 24.5/49 = 3.5/49 = 1/14.

### Step 6: Form the ratio
- The ratio in which the mixtures should be mixed is the inverse of the differences calculated:
- Ratio of A to B = (1/10) : (1/14) = 14 : 10.

### Step 7: Simplify the ratio
- Simplifying 14:10 gives us 7:5.

### Final Answer
The required ratio in which the mixtures from vessels A and B should be mixed is **7:5**.

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