Two vessels A and B contain milik and water in the ratios 7: 5 and 17:7 respectively. In what ratio mixture from two vessels should be mixed to get a new mixture containing milk and water in the ratio 5:3 ?

Options

A. `1:3`
B. `2:3`
C. `2:1`
D. `3:2`

Answer

To solve the problem of mixing two vessels containing milk and water in specific ratios to achieve a desired ratio, we can follow these steps:

### Step 1: Understand the Ratios
- Vessel A contains milk and water in the ratio of 7:5.
- Vessel B contains milk and water in the ratio of 17:7.
- We want to mix these two to achieve a new mixture with a milk to water ratio of 5:3.

### Step 2: Calculate the Fraction of Milk in Each Vessel
- For Vessel A:
- Total parts = 7 (milk) + 5 (water) = 12 parts.
- Fraction of milk in A = 7/12.

- For Vessel B:
- Total parts = 17 (milk) + 7 (water) = 24 parts.
- Fraction of milk in B = 17/24.

- For the desired mixture:
- Total parts = 5 (milk) + 3 (water) = 8 parts.
- Fraction of milk in the new mixture = 5/8.

### Step 3: Set Up the Alligation
We can use the alligation method to find the ratio in which the two mixtures should be combined.

- Place the fractions of milk in a line:
```
Milk in A: 7/12
Milk in B: 17/24
Milk in New Mixture: 5/8
```

### Step 4: Calculate the Differences
- Difference between the fraction of milk in A and the new mixture:
- \( \frac{7}{12} - \frac{5}{8} \)
- To calculate this, find a common denominator (24):
- \( \frac{7 \times 2}{12 \times 2} = \frac{14}{24} \)
- \( \frac{5 \times 3}{8 \times 3} = \frac{15}{24} \)
- Difference = \( \frac{14}{24} - \frac{15}{24} = -\frac{1}{24} \) (take the absolute value, so it's \( \frac{1}{24} \))

- Difference between the fraction of milk in B and the new mixture:
- \( \frac{17}{24} - \frac{5}{8} \)
- Again, find a common denominator (24):
- \( \frac{5 \times 3}{8 \times 3} = \frac{15}{24} \)
- Difference = \( \frac{17}{24} - \frac{15}{24} = \frac{2}{24} = \frac{1}{12} \)

### Step 5: Determine the Ratio
- The ratio of the quantities from vessels A and B:
- Ratio = Difference from B : Difference from A
- Ratio = \( \frac{1}{12} : \frac{1}{24} \)
- To get rid of the fractions, multiply both sides by 24:
- Ratio = \( 2 : 1 \)

### Final Answer
The mixture from vessels A and B should be mixed in the ratio of **2:1**.

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