A boat takes 19 hours to travel downstream from point A to point B and then return upstream to point C, where point 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?

Introduction / Context:
This boats and streams question involves a trip where a boat goes downstream from one point to another and then returns only halfway upstream. You must carefully interpret the distances and use upstream and downstream speeds to form an equation based on the total time. The challenge is to correctly express the total journey time in terms of the unknown distance between the two points.

Given Data / Assumptions:

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

Concept / Approach:
We express the total time as sum of downstream and upstream times:

• Time downstream A → B = distance / downstream speed = D / 18.
• Time upstream B → C = distance / upstream speed = (D/2) / 10 = D / 20.

We are told this total time equals 19 hours, so we form an equation D/18 + D/20 = 19 and solve for D.

Step-by-Step Solution:
Step 1: Time from A to B downstream = D / 18 hours.Step 2: Time from B to C upstream = (D/2) / 10 = D / 20 hours.Step 3: Total time = D / 18 + D / 20 = 19.Step 4: Factor D: D(1/18 + 1/20) = 19.Step 5: Compute 1/18 + 1/20 = (20 + 18) / 360 = 38 / 360 = 19 / 180.Step 6: So D * (19 / 180) = 19.Step 7: Divide both sides by 19: D / 180 = 1 ⇒ D = 180.Step 8: Therefore, the distance between A and B is 180 km.

Verification / Alternative check:
Using D = 180, time downstream from A to B is 180 / 18 = 10 hours. Time upstream from B to C (90 km at 10 km/h) is 90 / 10 = 9 hours. Total time = 10 + 9 = 19 hours, which matches the given condition, confirming that D = 180 km is correct.

Why Other Options Are Wrong:
Distances like 160 km or 140 km would give totals of time that are not equal to 19 hours when substituted into the equation D / 18 + D / 20. For example, with D = 160, the total time is less than 19 hours. Distances like 120 or 200 km similarly fail to satisfy the given condition.

Common Pitfalls:

• Misinterpreting the journey and assuming the boat returns all the way from B back to A rather than just to midpoint C.
• Using wrong upstream or downstream speeds (for example, adding instead of subtracting for upstream).
• Forgetting that the upstream distance on the return leg is only D/2, not D.
• Making arithmetic errors when adding fractions 1/18 and 1/20.

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