Solving the Milk and Water Mixture Ratio Problem

This problem involves a mixture of milk and water, where we need to change the ratio by adding only water. We start with a certain total quantity and an initial ratio, and we want to find out how much water to add to achieve a new desired ratio.

Understanding the Initial Mixture

We are given a mixture of 75 liters where the ratio of milk to water is 3:2.

The total number of ratio parts is the sum of the milk parts and the water parts:

• Milk parts: 3
• Water parts: 2
• Total parts: \(3 + 2 = 5\)

To find the initial amounts of milk and water in the 75-liter mixture, we divide the total volume by the total ratio parts and then multiply by the respective parts:

• Initial amount of Milk = \(\frac{\text{Milk parts}}{\text{Total parts}} \times \text{Total Volume}\)
• Initial amount of Milk = \(\frac{3}{5} \times 75 \text{ liters}\)
• Initial amount of Milk = \(3 \times 15 \text{ liters} = 45 \text{ liters}\)

• Initial amount of Water = \(\frac{\text{Water parts}}{\text{Total parts}} \times \text{Total Volume}\)
• Initial amount of Water = \(\frac{2}{5} \times 75 \text{ liters}\)
• Initial amount of Water = \(2 \times 15 \text{ liters} = 30 \text{ liters}\)

So, initially, the mixture contains 45 liters of milk and 30 liters of water.

Changing the Ratio by Adding Water

The problem states that we want to change the ratio of milk to water to 1:2 by adding only water. This is a crucial point: the amount of milk in the mixture will remain constant because we are only adding water.

The new ratio of milk to water is 1:2. The amount of milk is still 45 liters.

Let the new amount of water in the mixture be \(W_{\text{new}}\) liters.

According to the new ratio:

\(\frac{\text{Amount of Milk}}{\text{Amount of Water}_{\text{new}}} = \frac{1}{2}\)

Substitute the constant amount of milk (45 liters) into the equation:

\(\frac{45 \text{ liters}}{W_{\text{new}}} = \frac{1}{2}\)

Now, we can solve for \(W_{\text{new}}\) by cross-multiplication:

\(1 \times W_{\text{new}} = 45 \times 2\)

\(W_{\text{new}} = 90 \text{ liters}\)

So, to achieve a milk to water ratio of 1:2 while keeping the milk at 45 liters, the total amount of water in the final mixture must be 90 liters.

Calculating the Amount of Water Added

We know the initial amount of water was 30 liters, and the final amount of water needs to be 90 liters. The difference between the final and initial amounts of water is the amount of water that was added.

Water Added = Final amount of Water - Initial amount of Water

Water Added = \(90 \text{ liters} - 30 \text{ liters}\)

Water Added = \(60 \text{ liters}\)

Therefore, 60 liters of water should be added to the mixture to change the ratio from 3:2 to 1:2.

Let's quickly verify:

• Initial: Milk = 45L, Water = 30L. Ratio = 45:30 = 3:2. Total = 75L.
• Add 60L water.
• Final: Milk = 45L (unchanged), Water = 30L + 60L = 90L. Total = 45L + 90L = 135L.
• Final Ratio = Milk : Water = 45 : 90.

To simplify the final ratio 45:90, we can divide both numbers by their greatest common divisor, which is 45:

\(45 \div 45 = 1\)

\(90 \div 45 = 2\)

The final ratio is indeed 1:2, which matches the requirement.

The amount of water that should be added is 60 liters.

Revision Table: Mixture Ratio Calculation

Additional Information: Ratio and Proportion in Mixtures

Ratio problems involving mixtures often require understanding how adding or removing a component affects the overall ratio. Key points to remember:

• Constant Component: When only one component is added or removed, the amount of the other component(s) remains unchanged. This constant amount is crucial for setting up the equation for the new ratio.
• Ratio as a Fraction: A ratio like a:b can be written as a fraction \(\frac{a}{b}\). This allows you to use algebraic equations to solve for unknown quantities.
• Total Volume Change: Adding a substance to a mixture will increase the total volume of the mixture. The final total volume is the initial total volume plus the amount added.
• Units: Ensure all quantities are in the same units (liters, milliliters, etc.) before performing calculations.

Mixture problems can become more complex if both components are added or removed, or if a portion of the mixture is removed before a component is added. However, the principle of tracking the changes in the amounts of individual components remains fundamental.