1. **State the problem:** A river cruise ship sails 80 miles downstream in 4 hours and returns upstream in 5 hours. We need to find the rate of the ship in still water ($s$) and the rate of the current ($c$). 2. **Define variables and formulas:** Let $s$ = speed of the ship in still water (miles per hour), and $c$ = speed of the current (miles per hour). Downstream speed = $s + c$ Upstream speed = $s - c$ Distance = Speed \times Time 3. **Write equations from the problem:** Downstream: $$80 = (s + c) \times 4$$ Upstream: $$80 = (s - c) \times 5$$ 4. **Simplify equations:** $$s + c = \frac{80}{4} = 20$$ $$s - c = \frac{80}{5} = 16$$ 5. **Solve the system of equations:** Add the two equations: $$ (s + c) + (s - c) = 20 + 16 $$ $$ 2s = 36 $$ $$ s = \frac{\cancel{2}s}{\cancel{2}} = 18 $$ Substitute $s=18$ into $s + c = 20$: $$ 18 + c = 20 $$ $$ c = 20 - 18 = 2 $$ 6. **Final answer:** The rate of the ship in still water is $18$ miles per hour. The rate of the current is $2$ miles per hour.