The Royal Fruit Company produces two types of fruit drinks. The first type is 70% pure fruit juice, and the second type is 95% pure fruit juice. The company is attempting to produce a fruit drink that contains 80% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 70 pints of a mixture that is 80% pure fruit juice?

Let x pints be the required amount of 70% pure juice.

Let y pints be the required amount of 95% pure juice.

x + y = 70 pints

Therefore we can write:

y = 70 - x ... (1)

Amount of pure juice in x pints = 0.7x.

Amount of pure juice in y pints = 0.95y = 0.95 (70 - x).

Amount of pure juice in 70 pints = 0.8 x 70 = 56 pints.

Equating the amounts of pure juice, we get:

0.7x + 0.95 (70 - x) = 56 ... (2).

The solution to equation (2) is x = 42. Therefore y = 70 - 42 = 28.

The answer is: 42 pints of 70% pure fruit juice and 28 pints of 95% pure fruit juice are required.