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The question asks: 

**How much of a 90% orange juice drink must be mixed with 13 gallons of a 10% orange juice drink to obtain a mixture that is 50% orange juice?**

### Solution:

Let the amount of the 90% orange juice drink be **x** gallons.

We know:
- The 10% orange juice drink has 13 gallons, and the amount of orange juice in this is \( 0.1 \times 13 = 1.3 \) gallons.
- The 90% orange juice drink has \( 0.9 \times x \) gallons of pure orange juice.
- The final mixture should have \( 0.5 \times (x + 13) \) gallons of orange juice, where \( x + 13 \) is the total number of gallons of the mixture.

### Set up the equation:

The amount of pure orange juice from both drinks should equal the amount of pure orange juice in the final mixture:
\[
0.9x + 1.3 = 0.5(x + 13)
\]

### Solving the equation:

1. Expand the right side:
   \[
   0.9x + 1.3 = 0.5x + 6.5
   \]

2. Subtract \( 0.5x \) from both sides:
   \[
   0.4x + 1.3 = 6.5
   \]

3. Subtract 1.3 from both sides:
   \[
   0.4x = 5.2
   \]

4. Divide by 0.4:
   \[
   x = \frac{5.2}{0.4} = 13
   \]

Thus, **13 gallons** of the 90% orange juice drink must be mixed with the 13 gallons of the 10% orange juice drink to get a 50% orange juice mixture.

---

Would you like further details or have any questions?

### Related Questions:
1. How do you adjust this method for different percentages?
2. What if the final mixture is supposed to be 40% instead of 50%?
3. How can you derive similar mixtures using algebra?
4. What happens if you mix equal parts of the two drinks? What will be the percentage of orange juice?
5. How does the total volume of the mixture change as the initial amounts change?

### Tip:
Always double-check the total volume and percentage in mixture problems to ensure that your final equation balances both parts of the problem correctly.

How much of a 90% orange juice drink must be mixed with 13 gallons of a 10% orange juice drink to obtain a mixture that is 50% orange juice?

Learn how to calculate the amount of 90% orange juice needed to mix with 13 gallons of 10% orange juice to get a 50% concentration. This solution covers key algebraic concepts and mixture problems. Ideal for students in Grades 8-10.

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