Question

In a 100 m race, if A gives B a start of 20 metres, then A wins the race by 5 seconds. Alternatively, if A gives B a start of 40 metres, the race ends in a dead heat. How long does A take to run 200 m?

• A. 10 seconds
• B. 20 seconds
• C. 30 seconds
• D. 40 seconds

Hint:

The dead-heat case gives B's speed as a fixed fraction of A's speed; combine that with the 5-second-win case to solve for A's 100 m time.

Answer 1

The Correct Option is C

Step 1: Set up variables.
Let A take t seconds to run 100 m, so A's speed is \(\dfrac{100}{t}\) m/s.

Step 2: Use the dead heat condition.
With a 40 m start, B only needs to cover 60 m, and this dead heat means B covers 60 m in the same time t that A covers 100 m. So B's speed \(=\dfrac{60}{t}\) m/s.

Step 3: Use the 20 m start condition.
With a 20 m start, B covers 80 m, and A wins by 5 seconds, meaning B takes \(t+5\) seconds to cover 80 m. So B's speed \(=\dfrac{80}{t+5}\) m/s.

Step 4: Equate the two expressions for B's speed.
\(\dfrac{60}{t}=\dfrac{80}{t+5}\Rightarrow60(t+5)=80t\Rightarrow60t+300=80t\Rightarrow20t=300\Rightarrow t=15\).

Step 5: Find A's speed and the time for 200 m.
A's speed \(=\dfrac{100}{15}=\dfrac{20}{3}\) m/s. Time for 200 m \(=\dfrac{200}{20/3}=200\times\dfrac{3}{20}=30\) seconds.

Step 6: Final Answer.
A takes 30 seconds to run 200 m, so option C is correct.