Question

The problem below consists of a question and two statements numbered 1 & 2. You have to decide whether the data provided in the statements are sufficient to answer the question. Rahim is riding upstream on a boat, from point A to B, at a constant speed. The distance from A to B is 30 km. One minute after Rahim leaves from point A, a speedboat starts from point A to go to point B. It crosses Rahim’s boat after 4 minutes. If the speed of the speedboat is constant from A to B, what is Rahim’s speed in still water? 1. The speed of the speedboat in still water is 30 km/hour. 2. Rahim takes three hours to reach point B from point A.

• A. Statement 1 alone is sufficient to answer the question, but statement 2 alone is not sufficient
• B. Statement 2 alone is sufficient to answer the question, but statement 1 alone is not sufficient
• C. Each statement alone is sufficient
• D. Both statements together are sufficient, but neither of them alone is sufficient
• E. Statements 1 & 2 together are not sufficient

Answer 1

The Correct Option is D

The problem consists of determining Rahim's speed in still water, given two statements. Let's evaluate each statement one by one to determine if it provides sufficient information to solve the problem.

Statement 1: The speed of the speedboat in still water is 30 km/hour.

This statement gives us information about the speedboat's speed in still water, but it does not provide any information about Rahim's speed or how it influences the time to intersect with the speedboat. Without knowing how the river's current affects both boats or how this relates to Rahim's speed, this statement alone is not sufficient to determine Rahim’s speed in still water.

Statement 2: Rahim takes three hours to reach point B from point A.

This statement provides the time Rahim takes to travel 30 km upstream. However, without knowing the river's current speed or any information about the relative speed of the speedboat, we cannot derive Rahim's speed in still water from this information alone.

Now, let's consider both statements together:

• From Statement 1, we know the speedboat crosses Rahim's boat 4 minutes after it starts. Knowing its speed (30 km/hour), we can calculate the distance covered by the speedboat in that time.
• The speedboat traveled for 5 minutes (since it started 1 minute after Rahim), i.e., \(\frac{1}{12} \text{ hours}\). Therefore, it covers: \(\text{distance} = \text{speed} \times \text{time} = 30 \times \frac{1}{12} = 2.5 \text{ km}\)
• Simultaneously, Rahim also covers 2.5 km in 5 minutes. Thus, Rahim's speed relative to the still water, considering the current is consistent with the speedboat's forward journey from the remaining part of the answer, can now be determined with both sets of given information.

Using both statements, we can deduce Rahim's speed in still water because we now have the relative speeds set against a constant distance where both the current and Rahim's travel time are known. Thus, the correct answer is:

Both statements together are sufficient, but neither of them alone is sufficient.

Answer 2

Let the speeds be: \[ \text{Rahim’s speed} = R \ \text{kmph}, \quad \text{Speedboat (still water)} = S \ \text{kmph}, \quad \text{Stream speed} = W \ \text{kmph}. \] They cross each other at point \(X\).

From given conditions: \[ \frac{AX}{R - W} = \frac{5}{60}, \quad \frac{AX}{S - W} = \frac{4}{60}. \] Eliminating \(AX\): \[ W = 5R - 4S \quad \ldots (1) \]

Statement I: \(S = 30 \ \text{kmph}\). Since \(W\) or \(AX\) is not known, Statement I alone is not sufficient.

Statement II: \[ \frac{BA}{R + W} = \frac{3}{60} \ \text{hours}, \quad \text{or simply speed } (R+W)=3 \ \text{km/hr}. \] Since \(S\) or \(AX\) is not known, Statement II alone is not sufficient.

Combining I and II:
Substituting \(S = 30\) in (1): \[ W = 5R - 4(30) = 5R - 120 \] From Statement II: \[ R + W = 10 \quad \Rightarrow \quad W = 10 - R. \] Equating: \[ 10 - R = 5R - 120 \] \[ 6R = 130 \quad \Rightarrow \quad R = \tfrac{65}{3}. \]

Final Answer:
Both statements together are sufficient. Hence, the correct option is: \[ \boxed{D} \]