A vessel contains 18 litres mixture of milk and water in the ratio 5 : 1. If 3 litres of milk is added to the vessel, then how much water (in litres) should be added to the vessel to have milk and water in the ratio 9 : 2?

The correct answer is 1.

Let us break down the problem step-by-step:

Step 1: Find the initial quantities of milk and water in the mixture.
The total volume of the mixture is 18 litres, and the ratio of milk to water is 5 : 1.
The sum of the ratio parts is:
5 + 1 = 6 parts.
Each part corresponds to:
$\frac{18}{6}=3$ litres.
Therefore, the initial quantities are:
Initial volume of milk = $5\times 3=15$ litres.
Initial volume of water = $1\times 3=3$ litres.

Step 2: Calculate the new quantity of milk after adding 3 litres.
If 3 litres of milk is added, the new volume of milk becomes:
15 + 3 = 18 litres.

Step 3: Determine the required amount of water to achieve the target ratio.
Let $x$ be the quantity of water (in litres) that needs to be added.
The new volume of water will be $3+x$ litres.
We want the new ratio of milk to water to be 9 : 2. We can set up the proportion:
$\frac{18}{3+x}=\frac{9}{2}$

Step 4: Solve for $x$.
Cross-multiplying the equation:
$18\times 2=9\times (3+x)$
$36=27+9x$
Subtract 27 from both sides:
$9=9x$
Divide by 9:
$x=1$

Thus, 1 litre of water should be added to the vessel.