This problem asks us to determine how much water needs to be added to a 90 litre solution of acid and water to change their ratio. We start with an initial ratio of acid to water as 2 ∶ 1 and aim to achieve a new ratio of 1 ∶ 2.

Initial Solution Composition

Let's first find the individual quantities of acid and water in the initial 90 litre solution. The acid and water are in the ratio 2 ∶ 1.

• Total parts in the initial ratio = 2 (acid parts) + 1 (water part) = 3 parts.
• Since the total volume of the solution is 90 litres, each part represents: \[\frac{90 \text{ litres}}{3 \text{ parts}} = 30 \text{ litres per part}\]

Now, we can calculate the initial quantity of acid and water:

• Quantity of Acid: \(2 \text{ parts} \times 30 \text{ litres/part} = 60 \text{ litres}\)
• Quantity of Water: \(1 \text{ part} \times 30 \text{ litres/part} = 30 \text{ litres}\)

To double-check, \(60 \text{ litres (acid)} + 30 \text{ litres (water)} = 90 \text{ litres}\), which matches the given total volume of the 90 litre solution.

Acid Quantity Remains Constant

The problem states that only water is added to the solution to change the ratio of acid and water. This means the quantity of acid in the solution will remain the same as the initial quantity.

• The quantity of acid in the new solution will still be 60 litres.

The desired new ratio of acid to water is 1 ∶ 2. Let \(W_{\text{new}}\) be the new quantity of water in the solution. We can set up a proportion based on the new ratio:

\[\frac{\text{Acid}}{\text{Water}_{\text{new}}} = \frac{1}{2}\]

Substitute the constant acid quantity (60 litres) into the proportion:

\[\frac{60 \text{ litres}}{W_{\text{new}}} = \frac{1}{2}\]

To solve for \(W_{\text{new}}\), cross-multiply the terms:

\[1 \times W_{\text{new}} = 60 \text{ litres} \times 2\] \[W_{\text{new}} = 120 \text{ litres}\]

So, for the ratio of acid and water to become 1 ∶ 2, the solution must contain 120 litres of water.

Calculating Water Added

To find out how many litres of water should be added to the solution, we subtract the initial quantity of water from the new required quantity of water.

• New quantity of water required = 120 litres
• Initial quantity of water = 30 litres

Quantity of water added = New quantity of water - Initial quantity of water

Quantity of water added = \(120 \text{ litres} - 30 \text{ litres}\)

Quantity of water added = \(90 \text{ litres}\)

Therefore, 90 litres of water should be added to the 90 litre solution to change the ratio of acid and water from 2 ∶ 1 to 1 ∶ 2.

Summary of Solution Composition

The amount of water added is the difference between the final and initial water quantities, which is 120 litres - 30 litres = 90 litres.