A boat takes a total of 19 hours to travel downstream from point A to point B and then return upstream to point C, where C is the midpoint of AB. If the speed of the stream is 4 km/h and the speed of the boat in still water is 14 km/h, what is the distance between A and B (in km)?

Introduction / Context:
This problem combines downstream and upstream travel over different portions of a route in a river with a known current. You are given the total time for a two-leg journey and the speeds in still water and in the stream, and you are asked to find the distance between two points on the river.

Given Data / Assumptions:

Let the distance from A to B be D km.
Point C is the midpoint of AB, so AC = CB = D/2.
The boat travels downstream from A to B, then upstream from B to C.
Speed of boat in still water = 14 km/h.
Speed of stream = 4 km/h.
Downstream speed = 14 + 4 = 18 km/h.
Upstream speed = 14 - 4 = 10 km/h.
Total time for A → B (downstream) and B → C (upstream) = 19 hours.

Concept / Approach:
We compute the time for each leg using time = distance / speed. The downstream leg covers the full distance D, while the upstream leg covers only half the distance (from B to midpoint C, which is D/2). The sum of these two times is given as 19 hours. This gives a linear equation in D, which we can solve directly.

Step-by-Step Solution:
Step 1: Express the time taken downstream from A to B. Time downstream, t₁ = D / 18 hours. Step 2: Express the time taken upstream from B to C. Distance B to C = D/2. Upstream speed = 10 km/h. Time upstream, t₂ = (D/2) / 10 = D / 20 hours. Step 3: Set up the total time equation. t₁ + t₂ = 19. So D/18 + D/20 = 19. Step 4: Solve for D. Take LCM of 18 and 20, which is 180. (10D + 9D) / 180 = 19. 19D / 180 = 19 ⇒ D = 180 km.

Verification / Alternative check:
Check the times with D = 180 km: t₁ = 180 / 18 = 10 hours (downstream), t₂ = (180/2) / 10 = 90 / 10 = 9 hours (upstream). Total = 10 + 9 = 19 hours, which matches the given total time exactly, confirming that D = 180 km is correct.

Why Other Options Are Wrong:
For D = 160 km, total time would be 160/18 + 80/10 ≈ 8.89 + 8 = 16.89 h, not 19 h.
For D = 140 km or 120 km, the total times would be even smaller, farther from the required 19 hours. Hence they cannot satisfy the given condition.

Common Pitfalls:
A common mistake is to assume the boat returns the entire distance from B back to A, instead of only to midpoint C, which changes the upstream distance from D/2 to D and gives the wrong equation. Another error is miscomputing the LCM or algebraic simplification when combining fractions. Keeping track of distances on each leg and carefully setting up the time equation avoids these problems.

Final Answer:
The distance between A and B is 180 km.