Question

If Runner A followed Runner B down a portion of track that is \(\frac{1}{3}\) mile long, how many seconds did it take Runner A to run the track?
(1) Runner A ran onto the track 10 seconds after Runner B ran onto the track and ran off the track 8 seconds after Runner B ran off the track.
(2) Runner B ran the track at a constant speed of 9 miles per hour.

• 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 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 problems involving relative motion or time, clearly define variables for each entity. Pay close attention to units (e.g., hours vs. seconds) and perform conversions when necessary. When combining statements, check if the information from one statement can be plugged into the relationships established by the other.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks for the time it took Runner A to run a specific distance. Let's denote the time for Runner A as \(T_A\) and for Runner B as \(T_B\).
Step 2: Detailed Explanation:
Analyzing Statement (1):
This statement gives a relationship between the start and end times of the two runners.
Let's say Runner B starts at time \(t=0\). Runner B finishes at time \(T_B\).
Runner A starts 10 seconds later, at \(t=10\).
Runner A finishes 8 seconds after Runner B, so at time \(T_B + 8\).
The total time taken by Runner A is the difference between A's finish time and A's start time:
\[ T_A = (\text{A's finish time}) - (\text{A's start time}) = (T_B + 8) - 10 = T_B - 2 \text{ seconds} \]
This statement gives a relationship between \(T_A\) and \(T_B\), but we cannot find the value of \(T_A\) without knowing \(T_B\). Thus, statement (1) alone is not sufficient.
Analyzing Statement (2):
This statement provides the speed of Runner B.
Speed of B, \(V_B = 9\) miles per hour.
Distance, \(D = \frac{1}{3}\) mile.
We can calculate the time taken by Runner B using the formula Time = Distance / Speed.
\[ T_B = \frac{D}{V_B} = \frac{1/3 \text{ mile}}{9 \text{ miles/hour}} = \frac{1}{27} \text{ hours} \]
This statement gives us the time for Runner B, but provides no information about Runner A. Thus, statement (2) alone is not sufficient.
Combining Statements (1) and (2):
From statement (2), we found \(T_B = \frac{1}{27}\) hours. The question asks for the time in seconds, so let's convert the units.
\[ T_B = \frac{1}{27} \text{ hours} \times \frac{3600 \text{ seconds}}{1 \text{ hour}} = \frac{3600}{27} = \frac{400}{3} \text{ seconds} \]
From statement (1), we have the relation \(T_A = T_B - 2\).
Now we can substitute the value of \(T_B\) to find \(T_A\).
\[ T_A = \frac{400}{3} - 2 = \frac{400}{3} - \frac{6}{3} = \frac{394}{3} \text{ seconds} \]
Since we can find a unique value for \(T_A\), the two statements together are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but combining them allows us to find the time taken by Runner A. Therefore, the correct option is (C).