A 480 L mixture of milk and water in the ratio 5: 3. If 'X' L of the mixture is replaced by water and again '(X - 24)' L of the mixture is replaced by water. If the amount of water in the final mixture is 300 L, then find the value of X + 15'

We are given a mixture of 480 L of milk and water with an initial ratio of 5:3. This means the initial volumes of milk and water can be calculated as follows:

- Let the initial amount of milk be \( M \) and the initial amount of water be \( W \).
- The ratio of milk to water is 5:3, so:
  \[
  M = \frac{5}{8} \times 480 = 300 \text{ L of milk},
  \]
  \[
  W = \frac{3}{8} \times 480 = 180 \text{ L of water}.
  \]

Next, we are told that \( X \) L of the mixture is replaced by water. After this, \( X - 24 \) L of the new mixture is replaced by water. Finally, the amount of water in the mixture is given as 300 L, and we are tasked to find the value of \( X + 15 \).

Let's work through this step by step:

### Step 1: First Replacement
When \( X \) L of the mixture is replaced by water, the new composition of the mixture changes. Since \( X \) L of the mixture is taken out and replaced with water, both the milk and water content in the mixture decrease proportionally.

- The fraction of milk remaining after removing \( X \) L of the mixture is \( \left( \frac{300}{480} \right) \).
- The fraction of water remaining after removing \( X \) L of the mixture is \( \left( \frac{180}{480} \right) \).

So after removing \( X \) L and replacing it with water, the amounts of milk and water in the mixture are:
- Milk left = \( 300 \times \left( \frac{480 - X}{480} \right) \),
- Water left = \( 180 \times \left( \frac{480 - X}{480} \right) + X \).

### Step 2: Second Replacement
Next, \( X - 24 \) L of the new mixture is replaced by water. Again, the amounts of milk and water are reduced proportionally.

- The new amount of milk becomes: \( 300 \times \left( \frac{480 - X}{480} \right) \times \left( \frac{480 - X + 24}{480} \right) \).
- The new amount of water becomes: \( \left( 180 \times \left( \frac{480 - X}{480} \right) + X \right) \times \left( \frac{480 - X + 24}{480} \right) + (X - 24) \).

After the second replacement, we are told that the final amount of water in the mixture is 300 L. Therefore, we set up the following equation for the water content:
\[
\left( 180 \times \left( \frac{480 - X}{480} \right) + X \right) \times \left( \frac{480 - X + 24}{480} \right) + (X - 24) = 300.
\]

### Step 3: Solve for \( X \)
Now we solve this equation to find \( X \). Once \( X \) is determined, we can find \( X + 15 \).

Let me calculate this for you.The possible solutions for \( X \) are \( X = 120 \) and \( X = 864 \).

Given that the total volume of the mixture is 480 L, \( X \) must be less than 480 L. Thus, the valid solution is \( X = 120 \).

### Step 4: Find \( X + 15 \)
Now, we calculate:
\[
X + 15 = 120 + 15 = 135.
\]

Thus, the value of \( X + 15 \) is \( 135 \).

Do you want further details or have any questions? Here are some related questions to expand your understanding:

1. What happens to the ratio of milk and water after each replacement?
2. How do proportions change when a part of a mixture is replaced with another substance?
3. How can this method be applied to other mixture problems?
4. Can we generalize this method to solve any mixture-replacement problem?
5. What would happen if the amount replaced each time was different?

**Tip:** Always pay attention to constraints like total volume when solving mixture problems!

Learn how to solve a complex mixture problem involving milk and water ratios. Understand the steps to calculate the amount of mixture replaced and the resulting water content in the final solution. Suitable for high school students studying mixtures and proportions.

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