A motorboat running upstream takes 8 hours 48 minutes to cover a certain distance and takes 4 hours to cover the same distance downstream. What is the ratio of the speed of the motorboat in still water to the speed of the water current?

Introduction / Context:
This question involves comparing upstream and downstream times for the same distance to find the ratio of the speed of the motorboat in still water to the speed of the current. It is a good example of how relative speed in water is affected by current and how timing data can be converted into speed ratios.

Given Data / Assumptions:
- Time taken upstream = 8 hours 48 minutes. - Time taken downstream = 4 hours. - 8 hours 48 minutes = 8 + 48/60 = 8.8 hours. - Let the distance be D km. - Let b be the speed of the boat in still water (km/h). - Let c be the speed of the current (km/h). - Upstream speed = b - c, downstream speed = b + c.

Concept / Approach:
Speed equals distance divided by time. Using the same distance D, we can express upstream and downstream speeds in terms of D and times. These two speeds must match b - c and b + c. We then add and subtract the resulting equations to isolate b and c, and finally form the ratio b : c. The distance D cancels out, so its specific value is not needed.

Step-by-Step Solution:
Step 1: Upstream speed = D / 8.8 km/h. Step 2: Downstream speed = D / 4 km/h. Step 3: So b - c = D / 8.8 and b + c = D / 4. Step 4: Write 8.8 as 44/5 so that 1 / 8.8 = 5 / 44. Step 5: Hence b - c = D * 5 / 44 and b + c = D / 4. Step 6: Add equations: (b - c) + (b + c) = D * 5 / 44 + D / 4. Step 7: This gives 2b = D * (5 / 44 + 11 / 44) = D * 16 / 44 = D * 4 / 11. Step 8: So b = D * 2 / 11. Step 9: Subtract equations: (b + c) - (b - c) = D / 4 - D * 5 / 44. Step 10: This gives 2c = D * (11 / 44 - 5 / 44) = D * 6 / 44 = D * 3 / 22. Step 11: Therefore c = D * 3 / 44. Step 12: Ratio b : c = (D * 2 / 11) : (D * 3 / 44). Step 13: Cancel D and simplify: (2 / 11) : (3 / 44) = (2 / 11) * (44 / 3) = 88 / 33 = 8 / 3.

Verification / Alternative check:
The ratio 8 : 3 indicates that the boat speed in still water is more than twice the current speed, which is reasonable given that upstream time is more than double downstream time. Any alternative ratio must be tested against the time data, and 8 : 3 is the only one consistent with the observed 8.8 versus 4 hours for the same distance.

Why Other Options Are Wrong:
- Ratios 3 : 7, 7 : 9, 5 : 4, and 4 : 1 do not match the precise relationship between upstream and downstream times shown by the 8.8 and 4 hour durations. - Using these incorrect ratios would give very different predicted times for either upstream or downstream travel.

Common Pitfalls:
Mistakes often occur when converting 8 hours 48 minutes to hours, or when trying to directly write ratios from times without going through speeds. Another common error is to try averaging times instead of speeds. Always reduce mixed times to hours, express speeds as D divided by time, and then work systematically with equations.

Final Answer:
The ratio of the speed of the motorboat in still water to the speed of the current is 8 : 3.