240 liters of mixture contains juice and water in the ratio of 5:3. If x liters of mixture is taken out and (x-4) liters of juice and (x-10) liters of water are added to the mixture, then the final quantity of juice and water is in the ratio 17:12. Find the value of X.

To solve for \( x \), let's break down the problem step by step.

### Step 1: Initial Quantities of Juice and Water
The total mixture is 240 liters, with the juice and water in the ratio 5:3.

Let:
- The initial quantity of juice be \( 5k \).
- The initial quantity of water be \( 3k \).

Since the total quantity is 240 liters, we have:
\[
5k + 3k = 240
\]
\[
8k = 240
\]
\[
k = \frac{240}{8} = 30
\]

Thus:
- The initial quantity of juice = \( 5k = 5 \times 30 = 150 \) liters.
- The initial quantity of water = \( 3k = 3 \times 30 = 90 \) liters.

### Step 2: Mixture Modification
Now, \( x \) liters of mixture is taken out. The amount of juice and water taken out is in the same ratio, 5:3. So:
- Juice taken out = \( \frac{5}{8} \times x \).
- Water taken out = \( \frac{3}{8} \times x \).

After removing \( x \) liters of the mixture, the remaining quantities are:
- Juice remaining = \( 150 - \frac{5x}{8} \).
- Water remaining = \( 90 - \frac{3x}{8} \).

Next, \( (x - 4) \) liters of juice and \( (x - 10) \) liters of water are added to the mixture. Therefore, the new quantities are:
- Juice = \( 150 - \frac{5x}{8} + (x - 4) \).
- Water = \( 90 - \frac{3x}{8} + (x - 10) \).

### Step 3: Final Ratio Condition
The final ratio of juice to water is given as 17:12. Therefore:
\[
\frac{\text{Juice}}{\text{Water}} = \frac{17}{12}
\]
Substituting the expressions:
\[
\frac{150 - \frac{5x}{8} + (x - 4)}{90 - \frac{3x}{8} + (x - 10)} = \frac{17}{12}
\]

### Step 4: Simplify and Solve for \( x \)
Let's simplify the expression:
\[
\frac{150 + \frac{3x}{8} - 4}{90 + \frac{5x}{8} - 10} = \frac{17}{12}
\]
Simplify further:
\[
\frac{146 + \frac{3x}{8}}{80 + \frac{5x}{8}} = \frac{17}{12}
\]

Now, cross-multiply to eliminate the fraction:
\[
12 \left( 146 + \frac{3x}{8} \right) = 17 \left( 80 + \frac{5x}{8} \right)
\]
Expand both sides:
\[
1752 + \frac{36x}{8} = 1360 + \frac{85x}{8}
\]
Simplify by multiplying through by 8 to clear the fractions:
\[
14016 + 36x = 10880 + 85x
\]
Bring like terms together:
\[
14016 - 10880 = 85x - 36x
\]
\[
3136 = 49x
\]
\[
x = \frac{3136}{49} = 64
\]

So, the value of \( x \) is **64 liters**.

---

Would you like more details on any of the steps or have any further questions? 

Here are five related questions for further practice:
1. What is the final quantity of juice in the mixture after the changes?
2. What is the final quantity of water in the mixture after the changes?
3. How would the solution change if the final ratio were different, say 3:2?
4. What would happen to the mixture if an additional 10 liters of juice were added instead of removing \( x \) liters initially?
5. If the ratio of juice and water in the initial mixture were 7:5, how would that affect the value of \( x \)?

**Tip:** Always check that the units and ratios are consistent when setting up equations from word problems.

Learn how to solve a mixture problem involving a 240-liter mixture with a ratio of juice to water, and how changes in the mixture affect the final ratio. Detailed step-by-step solution provided.

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