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We are given two vessels containing milk and water in different proportions:

- The first vessel contains **24 L milk** and **6 L water**.
- The second vessel contains **15 L milk** and **10 L water**.

We need to determine how much mixture from each vessel should be transferred to a third vessel such that the third vessel ends up with **25 L milk** and **10 L water**.

Let’s define the amounts to be taken from each vessel as follows:
- From the first vessel, let \( x \) liters of mixture be taken.
- From the second vessel, let \( y \) liters of mixture be taken.

The goal is to find \( x \) and \( y \) such that:
- The total amount of milk in the third vessel is 25 L.
- The total amount of water in the third vessel is 10 L.

### Step 1: Ratio of milk and water in each vessel
- In the first vessel, the ratio of milk to total mixture is \( \frac{24}{24+6} = \frac{24}{30} = \frac{4}{5} \), meaning 4/5 of the mixture is milk.
- In the second vessel, the ratio of milk to total mixture is \( \frac{15}{15+10} = \frac{15}{25} = \frac{3}{5} \), meaning 3/5 of the mixture is milk.

### Step 2: Set up equations for milk and water
Let’s set up the following system of equations based on the conditions for the third vessel:
1. Milk equation:  
   \[
   \frac{4}{5}x + \frac{3}{5}y = 25
   \]
2. Water equation:  
   \[
   \frac{1}{5}x + \frac{2}{5}y = 10
   \]

### Step 3: Solve the system of equations

We can now solve these two equations to find \( x \) and \( y \). I will calculate this now.The solution to the system of equations is:
- \( x = 20 \) L of mixture from the first vessel,
- \( y = 15 \) L of mixture from the second vessel.

Thus, **20 L of mixture should be taken from the first vessel** and **15 L from the second vessel** to achieve a mixture of 25 L of milk and 10 L of water in the third vessel.

Would you like more details on the steps? Here are five related questions to explore further:

1. How are mixture ratios calculated for problems like this?
2. What happens if the target milk and water amounts change?
3. Could we solve this using a different method, like substitution?
4. How would the problem change if the total mixture in each vessel were different?
5. What if one of the vessels contained no water or no milk?

**Tip:** When solving mixture problems, always start by finding the proportions of each substance in the original vessels.

A vessel contains a mixture of 24 L milk and 6 L water, and a second vessel contains a mixture of 15 L milk and 10 L water. How much mixture of milk and water should be taken from the first and the second vessel separately and kept in a third vessel so that the third vessel may contain a mixture of 25 L milk and 10 L water?

Learn how to solve a mixture problem where two vessels contain different volumes of milk and water. This example involves finding how much mixture from each vessel must be transferred to a third vessel to achieve a desired mixture. Suitable for students in grades 9-12.

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