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!