The Royal Fruit Company produces two types of fruit drinks. The first type is 20% pure fruit juice, and the second type is 100% pure fruit juice. The company is attempting to produce a fruit drink that contains 85% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 160 pints of a mixture that is 85% pure fruit juice?

To solve this problem, we'll set up a system of linear equations to find the quantities of each type of drink needed to produce the desired mixture.

Let's denote:
- \( x \) as the amount of the first type of drink (20% pure fruit juice).
- \( y \) as the amount of the second type of drink (100% pure fruit juice).

We have the following information:
1. The total amount of the mixture should be 160 pints.
2. The concentration of pure fruit juice in the final mixture should be 85%.

From this information, we can set up the following equations:

1. \( x + y = 160 \)
2. \( 0.20x + 1.00y = 0.85 \times 160 \)

Now, let's solve these equations step by step.

### Step 1: Simplify the equations

First, rewrite the second equation to eliminate the percentages:
\[ 0.20x + y = 136 \]

### Step 2: Solve the system of equations

From the first equation:
\[ y = 160 - x \]

Substitute \( y \) in the second equation:
\[ 0.20x + (160 - x) = 136 \]

Simplify and solve for \( x \):
\[ 0.20x + 160 - x = 136 \]
\[ 0.20x - x = 136 - 160 \]
\[ -0.80x = -24 \]
\[ x = \frac{-24}{-0.80} \]
\[ x = 30 \]

So, the amount of the first type of drink (20% pure fruit juice) is 30 pints.

Now, substitute \( x \) back into the first equation to find \( y \):
\[ y = 160 - 30 \]
\[ y = 130 \]

So, the amount of the second type of drink (100% pure fruit juice) is 130 pints.

### Conclusion

The Royal Fruit Company should use:
- 30 pints of the first type of drink (20% pure fruit juice)
- 130 pints of the second type of drink (100% pure fruit juice)

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:
1. What if the desired concentration was 90% instead of 85%?
2. How would the mixture change if the total desired volume was 200 pints?
3. What if the company only had 100 pints of the second type of drink available?
4. How can we verify that our solution is correct?
5. What would be the concentration if equal amounts of both types of drinks were used?
6. How does changing the percentages in each type of drink affect the mixture?
7. Can we solve this problem using a graphical method?
8. How would the amounts change if there was a third type of drink with a different concentration?

**Tip:** Always ensure that the units of measurement are consistent throughout the problem to avoid errors in calculations.

Learn how to solve the Royal Fruit Company's mixture problem to create a drink with 85% pure fruit juice. Understand the use of linear equations and percentage calculations in this practical example. Suitable for students in Grades 7-9.

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