how many quarts of pineapple juice should Theo add to a 20% pineapple drink to create 5 quarts of a 50% pineapple juice mixture?

To find out how many quarts of pineapple juice Theo should add, we need to use a proportion based on the percentages of the two mixtures.

Let's set up the proportion:

(Quarts of pineapple juice / Total quarts of mixture) = (Percentage of pineapple juice / 100)

From the given information, we know that:
- The total quarts of mixture needed is 5.
- The initial pineapple drink is 20% pineapple juice, meaning it contains 20% of pineapple juice and 80% other ingredients.
- The desired mixture should be 50% pineapple juice.

Now, let's set up the proportion with the known quantities:

(Quarts of pineapple juice / 5) = (50 / 100)

To solve for the unknown variable, we can cross-multiply and then solve for quarts of pineapple juice:

Quarts of pineapple juice = (5 * 50) / 100
Quarts of pineapple juice = 2.5

Therefore, Theo should add 2.5 quarts of pineapple juice to the 20% pineapple drink in order to create 5 quarts of a 50% pineapple juice mixture.

In 5 quarts of 50% mixture, there are 2.5 quarts of water.

The amount of 20% drink that contains 2.5 quart of water is 2.5/(1-0.20) = 3.125 quarts.

So Theo will need 1.875 quarts of pure pineapple juice, and 3.125 quarts of 20% drink to make 5 quarts of 50% mixture.