Question

If A and B run a race, then A wins by 60 seconds. If B and C run the same race, then B wins by 30 seconds. Assuming that C maintains a uniform speed what is the time taken by C to finish the race?

A. A and C run the same race and A wins by 375 metres.

B. The length of the race is 1 km.

• A. The questions can be answered by one of the statements, but not by the other
• B. The question can be answered using either of the two statements alone.
• C. The questions can be answered using both the statements together, but cannot be answered using either statement alone.
• D. The question cannot be answered even by using both the statements together.

Hint:

Add the two time gaps to see how long C needs after A has finished. Then ask what each statement adds, a speed or a length.

Answer 1

The Correct Option is C

Step 1: Turn the race facts into times.
"A wins by 60 seconds" means B reaches the finish line 60 seconds after A. "B wins by 30 seconds" means C reaches it 30 seconds after B.
Let A take \( t \) seconds. Then B takes \( t + 60 \) and C takes \( t + 90 \). The same race is run each time, so the course length is the same.
So when A finishes, C still needs exactly 90 seconds to reach the end.

Step 2: Read statement A alone.
A wins by 375 m over C. When A finishes, C is 375 m short of the line, and C needs 90 seconds to cover those 375 m.
So the speed of C is \( 375/90 = 25/6 \) m/s. But the time for the whole race is length divided by speed, and the length is not known. So A alone is not enough.

Step 3: Read statement B alone.
B gives the length 1000 m. The speed of C and the time of A are both unknown. The 60 s and 30 s margins alone do not fix the time of C, since a longer or shorter race with different speeds can give the same margins. So B alone is not enough.

Step 4: Combine A and B.
From A, the speed of C is \( 25/6 \) m/s. From B, the race length is 1000 m. So the time of C is:
\[ t_C = \frac{1000}{25/6} = \frac{1000 \times 6}{25} = 240 \text{ s} \]
Check: C takes 240 s, so A takes \( 240 - 90 = 150 \) s and B takes 210 s. In 150 s, C covers \( (25/6) \times 150 = 625 \) m, so A leads C by \( 1000 - 625 = 375 \) m, as stated.

Step 5: Match with the options.
Options (1) and (2) are wrong because each statement alone leaves the answer open. Option (4) is wrong because together they give 240 s. So option (3) is correct.

Final Answer:
Both statements are needed. Together they give C a finishing time of 240 seconds, so the answer is option 3. \[ \boxed{3} \]