A boat travels 72 km downstream in 8 hours and 84 km upstream in 12 hours. What are the speeds (in km/h) of the boat in still water and of the water current respectively?

Introduction / Context:
Here we have a classic boats and streams problem where we are given separate downstream and upstream journeys with different distances and times. From these we can compute effective speeds in each direction and then deduce both the speed of the boat in still water and the speed of the current.

Given Data / Assumptions:
- Downstream distance = 72 km, time = 8 hours. - Upstream distance = 84 km, time = 12 hours. - Let b be the speed of the boat in still water (km/h). - Let c be the speed of the current (km/h). - Downstream speed = b + c, upstream speed = b - c.

Concept / Approach:
First we compute downstream and upstream speeds by distance / time. Then we use the standard relations b + c = downstream speed and b - c = upstream speed. Solving these two simple equations gives values for b and c. This method is central to most boats and streams aptitude questions.

Step-by-Step Solution:
Step 1: Downstream speed = 72 / 8 = 9 km/h. Step 2: Upstream speed = 84 / 12 = 7 km/h. Step 3: So, b + c = 9 and b - c = 7. Step 4: Add the equations: (b + c) + (b - c) = 9 + 7 gives 2b = 16. Step 5: Therefore b = 16 / 2 = 8 km/h. Step 6: Substitute b into b + c = 9: 8 + c = 9, so c = 1 km/h.

Verification / Alternative check:
With b = 8 km/h and c = 1 km/h, downstream speed = 8 + 1 = 9 km/h, giving time downstream = 72 / 9 = 8 hours, which matches the question. Upstream speed = 8 - 1 = 7 km/h, giving time upstream = 84 / 7 = 12 hours, also matching. The values are therefore correct and consistent with both journeys.

Why Other Options Are Wrong:
- 9 and 3 km/h give downstream and upstream speeds that do not match 9 and 7 km/h. - 6 and 7 km/h or 7 and 2 km/h would lead to huge differences between calculated and given times. - 10 and 2 km/h would produce different effective speeds than those derived from the distances and times in the problem.

Common Pitfalls:
Some learners mistakenly divide a wrong distance or misread upstream and downstream data. Others average the distances instead of speeds. Always compute the actual speeds first before forming equations, and be careful with which direction is upstream and which is downstream.

Final Answer:
The speed of the boat in still water is 8 km/h and the speed of the current is 1 km/h.