Mixture Ratio: Initial Quantities Calculation

The problem asks us to determine how much water needs to be added to a 50 L mixture, where the current milk to water ratio is 3:2, to achieve a new ratio of 1:1. Let's break this down step-by-step.

First, we find the initial amounts of milk and water in the 50 L mixture.

• Total volume of the mixture = 50 L.
• The ratio of milk to water is 3:2.
• The sum of the ratio parts is $3 + 2 = 5$.

Now, we calculate the quantity of each component:

• Initial quantity of milk = $ \frac{3}{5} \times 50 \text{ L} = 30 \text{ L} $.
• Initial quantity of water = $ \frac{2}{5} \times 50 \text{ L} = 20 \text{ L} $.

We can verify this: $30 \text{ L} + 20 \text{ L} = 50 \text{ L}$.

Mixture Ratio: Adjusting Water Quantity

The goal is to make the ratio of milk to water 1:1. This means the quantity of milk should be equal to the quantity of water in the final mixture. We are only adding water, so the quantity of milk remains constant.

• Quantity of milk (constant) = 30 L.
• Let the amount of water to be added be $x$ L.
• The new quantity of water = Initial water + Added water = $ (20 + x) $ L.

For the ratio to be 1:1, the final quantity of milk must equal the final quantity of water:

$ \text{Quantity of Milk} = \text{New Quantity of Water} $

$ 30 \text{ L} = (20 + x) \text{ L} $

Mixture Ratio: Final Calculation

To find the value of $x$, we solve the equation:

$ 30 = 20 + x $

Subtract 20 from both sides:

$ x = 30 - 20 $

$ x = 10 $

Therefore, 10 L of water must be added to the mixture.

Final Check:

If we add 10 L of water:

• New water quantity = $20 \text{ L} + 10 \text{ L} = 30 \text{ L}$.
• Milk quantity = 30 L.
• New ratio = Milk:Water = 30:30 = 1:1.

This matches the required ratio.