Two vessels A and B contain mixtures of ethanol and water, having these in the…

To solve this problem, we can use the method of allegation, which is ideal for finding the ratio in which two mixtures must be combined.

Step-by-Step Analysis
Analyze the Ratios:

Vessel A: Ethanol to Water = 5:2. The fraction of ethanol is 5 / (5 + 2) = 5/7.

Vessel B: Ethanol to Water = 7:6. The fraction of ethanol is 7 / (7 + 6) = 7/13.

Target Mixture: Ethanol to Water = 8:5. The fraction of ethanol is 8 / (8 + 5) = 8/13.

Apply Allegation:
We want to find the ratio of Vessel A to Vessel B. Let's look at the difference between the ethanol fractions:

Difference between Vessel B and Target: |7/13 - 8/13| = 1/13

Difference between Vessel A and Target: |5/7 - 8/13|
To calculate |5/7 - 8/13|, find a common denominator (7 * 13 = 91):
|65/91 - 56/91| = 9/91

Find the Ratio:
The ratio of Vessel A to Vessel B is the ratio of these differences:
Ratio = (1/13) : (9/91)
Multiply both sides by 91:
(1/13 * 91) : (9/91 * 91) = 7 : 9

The correct ratio in which the mixtures from vessels A and B should be mixed is 7:9, which corresponds to Option 3.