A motorboat takes only half as much time to cover a certain distance downstream as it takes to cover the same distance upstream. What is the ratio between the rate of the current and the rate of the boat in still water?

Introduction / Context:
This question is about motion in a stream where a motorboat travels the same distance both upstream and downstream. We are told that the time taken to travel downstream is exactly half of the time taken to travel upstream. From this information, we must determine the ratio of the speed of the current to the speed of the boat in still water. This is a standard boats and streams problem that relies on the relationship between speed, distance and time.

Given Data / Assumptions:

• The boat travels the same distance downstream and upstream.
• Time taken downstream is half of the time taken upstream.
• Let the speed of the boat in still water be b km/h.
• Let the speed of the current be c km/h.
• Downstream speed is b + c and upstream speed is b - c.
• We must find the ratio c : b, that is the rate of current to the rate of boat in still water.

Concept / Approach:
For any journey, time = distance / speed. If the distance is the same in both directions, then the ratio of the times is the inverse of the ratio of the speeds. Here, the time taken downstream is half the upstream time, so the downstream speed must be twice the upstream speed. We use this idea with b + c and b - c to form an equation, solve for the relation between b and c, and then express the required ratio c : b in its simplest form.

Step-by-Step Solution:
Step 1: Let the one way distance be d km. Step 2: Time taken upstream = d / (b - c). Time taken downstream = d / (b + c). Step 3: Given that downstream time is half of upstream time. d / (b + c) = (1 / 2) * d / (b - c). Step 4: Cancel d (since d is non zero). 1 / (b + c) = (1 / 2) * 1 / (b - c). Step 5: Cross multiply to remove fractions. 2 = (b + c) / (b - c). Step 6: Multiply both sides by (b - c). 2(b - c) = b + c. 2b - 2c = b + c. Step 7: Collect like terms. 2b - b = 2c + c. b = 3c. Step 8: Rearrange to find the ratio of current speed to boat speed. c / b = 1 / 3, so c : b = 1 : 3.
Verification / Alternative check:
Assume c = 1 and b = 3 (which matches b = 3c). Then downstream speed = 3 + 1 = 4 km/h, upstream speed = 3 - 1 = 2 km/h. For a distance of 4 km, time downstream = 4 / 4 = 1 hour and time upstream = 4 / 2 = 2 hours. Downstream time is exactly half of upstream time, so the ratio works correctly.
Why Other Options Are Wrong:
Options like 3 : 2 or 2 : 3 do not satisfy the condition that downstream time is half of upstream time when used to form b and c values. If the ratio were 3 : 1 or 1 : 4, you would not get a downstream speed that is exactly double the upstream speed for any positive b, c that respect those ratios.
Common Pitfalls:
A common mistake is to set up the equation with times directly but forget that downstream time must be smaller when speed is higher, leading to inverted ratios. Another error is to confuse the ratio of times with the ratio of speeds and write 2 = (b - c) / (b + c) instead of 2 = (b + c) / (b - c).
Final Answer:
The ratio between the rate of current and the rate of the boat in still water is 1 : 3.