Two cyclists start at the same point and travel in opposite directions. One cyclist travels 4 miles an hour slower than the other. If the two cyclists are 64 miles apart after 2 hours, what is the rate of each cyclist?

Let x be the speed of the faster cyclist in miles per hour. The slower cyclist's speed is then x - 4 miles per hour.

In 2 hours, the distance travelled by the faster cyclist is:
\text{Distance}_{\text{fast}} = x \times 2

And the distance travelled by the slower cyclist is:
\text{Distance}_{\text{slow}} = (x - 4) \times 2

The total distance between them after 2 hours is 64 miles:
2x + 2(x - 4) = 64

Simplifying the equation:
2x + 2x - 8 = 64
4x - 8 = 64
4x = 72
x = 18

The speed of the faster cyclist is:
x = 18 \, \text{mph}

The speed of the slower cyclist is:
x - 4 = 18 - 4 = 14 \, \text{mph}

Thus, the rate of each cyclist is 14 mph and 18 mph.