The Royal Fruit Company produces two types of fruit drinks. The first type is 55% pure fruit juice, and the second type is 80% pure fruit juice. The company is attempting to produce a fruit drink that contains 65% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 180 pints of a mixture that is 65% pure fruit juice?

Let x be the number of pints of the first type (55% pure fruit juice) and y be the number of pints of the second type (80% pure fruit juice) needed to make a 180-pint mixture that is 65% pure fruit juice.

We can set up a system of two equations based on the given information:
1. The total number of pints: x + y = 180
2. The pure fruit juice content: 0.55x + 0.80y = 0.65 \times 180

Solving the system of equations:
1. From the first equation, we have y = 180 - x
2. Substitute the value of y into the second equation:
0.55x + 0.80(180 - x) = 0.65 \times 180
0.55x + 144 - 0.80x = 117
0.25x = 27
x = 108

Therefore, x = 108 pints of the first type and y = 180 - 108 = 72 pints of the second type.

\boxed{x = 108, y = 72}