A boat covers equal distances upstream and downstream in a total of 5 hours. The sum of its upstream and downstream speeds is 40 km/h. The speed of the boat in still water is 600% more than the speed of the stream. What is the approximate distance (in km) covered by the boat in downstream direction?

Introduction / Context:
This question combines several pieces of information about a boat moving in a river: total time for equal upstream and downstream journeys, the sum of the two effective speeds, and a percentage relationship between the boat speed in still water and the stream speed. Our goal is to use these conditions to determine the approximate downstream distance covered by the boat. This involves working with ratios, percentages and time expressions together.

Given Data / Assumptions:

• The boat covers equal distances upstream and downstream in a total of 5 hours.
• Let the one way distance be s km.
• Sum of upstream and downstream speeds = 40 km/h.
• Let b be the speed of the boat in still water (km/h).
• Let c be the speed of the stream (km/h).
• Upstream speed = b - c; downstream speed = b + c.
• b is 600% more than c, meaning b = c + 6c = 7c.
• We must find s, the downstream distance, approximately.

Concept / Approach:
First we translate the 600% condition into the relation b = 7c. Then we express upstream and downstream speeds in terms of c. Using the given sum of speeds, we solve for c and hence b. Next, we write the equation for total time as the sum of the upstream and downstream times for distance s, equated to 5 hours. Solving this time equation yields s in terms of c, which we then evaluate numerically. Finally, we compare the result to the given options, picking the closest integer value.

Step-by-Step Solution:
Step 1: Use the 600% condition. b is 600% more than c, so b = c + 6c = 7c. Step 2: Express upstream and downstream speeds in terms of c. Upstream speed u = b - c = 7c - c = 6c. Downstream speed d = b + c = 7c + c = 8c. Step 3: Use the sum of speeds. u + d = 6c + 8c = 14c. Given that u + d = 40 km/h, we have 14c = 40. c = 40 / 14 = 20 / 7 km/h. Then b = 7c = 7 * (20 / 7) = 20 km/h. Thus u = 6c = 6 * (20 / 7) = 120 / 7 km/h. And d = 8c = 8 * (20 / 7) = 160 / 7 km/h. Step 4: Write the total time equation with equal distances. Time upstream = s / u = s / (120 / 7) = 7s / 120. Time downstream = s / d = s / (160 / 7) = 7s / 160. Total time = 7s / 120 + 7s / 160 = 5 hours. Step 5: Solve for s. Take 7s common: 7s (1 / 120 + 1 / 160) = 5. Compute 1 / 120 + 1 / 160 = (160 + 120) / (120 * 160) = 280 / 19200. Simplify 280 / 19200 = 28 / 1920 = 7 / 480. So 7s * (7 / 480) = 5, meaning 49s / 480 = 5. Therefore s = 5 * 480 / 49 = 2400 / 49 km. Step 6: Approximate the value of s. 49 * 49 = 2401, so 2400 / 49 is just under 49. Thus s ≈ 48.98 km, which is approximately 49 km.
Verification / Alternative check:
With s ≈ 49 km, upstream time = 49 / (120 / 7) ≈ 49 * 7 / 120 ≈ 343 / 120 ≈ 2.86 hours. Downstream time = 49 / (160 / 7) ≈ 49 * 7 / 160 ≈ 343 / 160 ≈ 2.14 hours. Total ≈ 2.86 + 2.14 = 5 hours, matching the given total time.
Why Other Options Are Wrong:
Distances like 45, 55 or 60 km would produce total times significantly different from 5 hours when combined with the calculated upstream and downstream speeds. A distance of 50 km is closest to the computed value of about 49 km, whereas 40 km would be too small to account for 5 hours of travel at the given speeds.
Common Pitfalls:
One common misconception is to interpret 600% more as 6 times instead of 7 times the original speed, leading to b = 6c instead of b = 7c. Another frequent error is mishandling fractions when adding 1 / 120 and 1 / 160, which can lead to an incorrect total time expression and hence a wrong distance.
Final Answer:
The approximate distance covered downstream is 50 km.