Question

Evaluate the problem, and the two statements - labelled (1) and (2) - that contain certain data or information. Using the given statements, decide whether the information provided is sufficient to answer the question.
Abhay started a bike ride from his house at 10:00 AM and rode to a nearby park. He immediately turned around and rode back home, not stopping once during the journey. Did Abhay make it back home by 10:40 AM?
Statement 1: Abhay's average speed while riding back home was 10% slower than his average speed while riding to the park.
Statement 2: Abhay arrived at the park at 10:18 AM.

• A. Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient
• B. Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient.
• C. Both Statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient
• E. Statements (1) and (2) TOGETHER are NOT sufficient

Hint:

In Data Sufficiency time-speed-distance problems, focus on what variables you can solve for. Statement 1 gives a relationship between speeds (and thus times), but no actual values. Statement 2 gives an actual value for one part of the journey. Combining them allows you to use the relationship from (1) with the value from (2) to solve the entire problem.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency problem involving time, speed, and distance. The question is a "Yes/No" question. To be sufficient, the information must lead to a definite "Yes" or a definite "No" answer. The question asks if the total journey time was less than or equal to 40 minutes (from 10:00 AM to 10:40 AM).
Step 2: Key Formula or Approach:
Let \(D\) be the distance from the house to the park. Let \(T_1\) and \(S_1\) be the time taken and average speed while going to the park. Let \(T_2\) and \(S_2\) be the time taken and average speed while returning home. Total time for the journey = \(T_1 + T_2\). The question is: Is \(T_1 + T_2 \le 40\) minutes?
Step 3: Detailed Explanation:
Analyze Statement (1): Abhay's average speed while riding back home was 10% slower than his average speed while riding to the park.
This means \(S_2 = S_1 \times (1 - 0.10) = 0.9 \times S_1\). Since the distance is the same for both legs of the journey (\(D\)), and \(T = D/S\), we can establish a relationship between the times: \[ T_2 = \frac{D}{S_2} = \frac{D}{0.9 \times S_1} = \frac{1}{0.9} \times \frac{D}{S_1} = \frac{10}{9} \times T_1 \] The total time is \(T_1 + T_2 = T_1 + \frac{10}{9}T_1 = \frac{19}{9}T_1\). The question becomes: Is \(\frac{19}{9}T_1 \le 40\)?
We do not know the value of \(T_1\). If \(T_1 = 18\) minutes, total time is \(\frac{19}{9} \times 18 = 38\) minutes (Yes). If \(T_1 = 20\) minutes, total time is \(\frac{19}{9} \times 20 \approx 42.2\) minutes (No). Statement (1) alone is not sufficient.
Analyze Statement (2): Abhay arrived at the park at 10:18 AM.
Since he started at 10:00 AM, the time to get to the park, \(T_1\), is 18 minutes. The total time is \(18 + T_2\). The question is: Is \(18 + T_2 \le 40\), or is \(T_2 \le 22\)? We have no information about his speed on the return trip, so we don't know \(T_2\). It could be less than or greater than 22 minutes. Statement (2) alone is not sufficient.
Analyze Both Statements Together:
From Statement (2), we know \(T_1 = 18\) minutes. From Statement (1), we know the relationship \(T_2 = \frac{10}{9}T_1\). We can now calculate the exact value of \(T_2\): \[ T_2 = \frac{10}{9} \times 18 = 10 \times 2 = 20 \text{ minutes} \] The total journey time is \(T_1 + T_2 = 18 + 20 = 38\) minutes. Is the total time of 38 minutes less than or equal to 40 minutes? Yes. Since we have a definite "Yes" answer, the statements together are sufficient.