Calculating Added Milk for Ratio Adjustment

This problem involves calculating the amount of milk needed to change the ratio of milk to water in a given mixture.

Initial Mixture Analysis

We start with a total mixture volume of 90 L. The initial ratio of milk to water is given as 4:5.

• The sum of the ratio parts is $4 + 5 = 9$.
• Calculate the initial quantity of milk: $ \text{Initial Milk} = \left( \frac{4}{9} \right) \times 90 \text{ L} = 40 \text{ L} $
• Calculate the initial quantity of water: $ \text{Initial Water} = \left( \frac{5}{9} \right) \times 90 \text{ L} = 50 \text{ L} $

So, the initial mixture contains 40 L of milk and 50 L of water.

Determining the Required Milk Addition

The goal is to change the ratio of milk to water to 5:4. We are adding only milk to the mixture.

• Let the quantity of milk to be added be '$x$' liters.
• The new quantity of milk will be the initial milk plus the added milk: $ (40 + x) $ L.
• The quantity of water remains unchanged: $ 50 $ L.
• The new ratio of milk to water is given as 5:4. We can write this as an equation: $ \frac{\text{New Milk}}{\text{New Water}} = \frac{40 + x}{50} = \frac{5}{4} $

Solving the Equation

Now, we solve the equation for '$x$':

• Cross-multiply the terms: $ 4 \times (40 + x) = 5 \times 50 $
• Expand the equation: $ 160 + 4x = 250 $
• Isolate the term with '$x$': $ 4x = 250 - 160 $ $ 4x = 90 $
• Solve for '$x$': $ x = \frac{90}{4} $ $ x = 22.5 \text{ L} $

Therefore, 22.5 L of milk must be added to the mixture to achieve the desired ratio of 5:4.

Final Check

After adding 22.5 L of milk:

• New milk quantity = $40 \text{ L} + 22.5 \text{ L} = 62.5 \text{ L}$
• Water quantity = $50 \text{ L}$
• New ratio = $\frac{62.5}{50} = \frac{625}{500} = \frac{125}{100} = \frac{5}{4}$.

The calculation is correct.