Solving the Milk and Water Mixture Ratio Problem

This problem involves changing the ratio of milk and water in a mixture by adding only water. We start with a known total volume and initial ratio, and we want to find the amount of water needed to achieve a new ratio.

Initial Mixture Composition Calculation

The total volume of the mixture is 60 litres. The initial ratio of milk to water is 2 : 1. This means for every 2 parts of milk, there is 1 part of water.

The total number of parts in the ratio is \(2 + 1 = 3\) parts.

The value of one part is the total volume divided by the total number of parts:

\( \text{Value of one part} = \frac{\text{Total volume}}{\text{Total parts}} = \frac{60 \text{ litres}}{3} = 20 \text{ litres/part} \)

Now, we can find the initial quantities of milk and water:

• Initial quantity of Milk: \(2 \text{ parts} \times 20 \text{ litres/part} = 40 \text{ litres}\)
• Initial quantity of Water: \(1 \text{ part} \times 20 \text{ litres/part} = 20 \text{ litres}\)

Check: \(40 \text{ litres (Milk)} + 20 \text{ litres (Water)} = 60 \text{ litres (Total)}\). The initial ratio is \(40 : 20\), which simplifies to \(2 : 1\), matching the problem statement.

Adding Water to Change the Ratio

We want to change the ratio of milk to water to 1 : 2 by adding only water. This means the quantity of milk will remain constant, while the quantity of water will increase.

Let \(x\) be the amount of water added in litres.

• New quantity of Milk: 40 litres (remains unchanged)
• New quantity of Water: Initial water + added water = \(20 + x\) litres

The new desired ratio of milk to water is 1 : 2. We can set up an equation using the new quantities and the new ratio:

\( \frac{\text{New quantity of Milk}}{\text{New quantity of Water}} = \frac{1}{2} \)

\( \frac{40}{20 + x} = \frac{1}{2} \)

Calculating the Amount of Water Added

Now, we solve the equation for \(x\) by cross-multiplying:

\( 40 \times 2 = 1 \times (20 + x) \)

\( 80 = 20 + x \)

Subtract 20 from both sides of the equation:

\( 80 - 20 = x \)

\( x = 60 \)

So, 60 litres of water must be added to the mixture.

Verification of the New Mixture

After adding 60 litres of water:

• Milk: 40 litres
• Water: \(20 + 60 = 80\) litres
• New Ratio: \(40 : 80\), which simplifies to \(1 : 2\). This matches the desired ratio.

The total volume of the new mixture will be \(40 + 80 = 120\) litres.

The amount of water that must be added is 60 litres.

Revision Table: Mixture Calculation Summary

Additional Information: Ratio and Mixture Concepts

Ratio problems involving mixtures are common in quantitative aptitude tests. Understanding how adding or removing a component affects the ratio is key.

• Ratio: A ratio is a comparison of two or more quantities of the same unit. It can be written as \(a:b\), \(a/b\), or "a to b".
• Mixture: A mixture is a substance containing two or more components not chemically combined.
• Changing Ratios: When one component is added to or removed from a mixture, the proportion (and thus the ratio) of the components changes. The quantity of the component that is NOT added or removed remains constant, which is crucial for setting up equations to solve these problems.
• Setting up Equations: If you know the quantity of the component that remains constant and the desired new ratio, you can set up a proportional equation to find the unknown quantity (the amount added or removed of the other component).

In this problem, milk remained constant (40 litres). In the new ratio \(1:2\), milk represents 1 part. This means 1 part corresponds to 40 litres. Since water represents 2 parts in the new ratio, 2 parts correspond to \(2 \times 40 = 80\) litres. The new water quantity must be 80 litres. Since we started with 20 litres of water, we need to add \(80 - 20 = 60\) litres.