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

Both conditions describe B's speed in two different ways, so equating them is the natural way to solve this.

1. Dead heat with a 40 m start means B's 60 m matches A's 100 m in time, so B's speed is $\frac{60}{100}=0.6$ times A's speed.
2. With a 20 m start, B covers 80 m while A covers 100 m in $t$ seconds, but B needs $t+5$ seconds for those 80 m, since A wins by 5 seconds.
3. Using the speed ratio: B's speed $=0.6 \times \frac{100}{t}$, and this must equal $\frac{80}{t+5}$.
4. $0.6 \times \frac{100}{t} = \frac{80}{t+5} \Rightarrow \frac{60}{t} = \frac{80}{t+5} \Rightarrow 60t+300=80t \Rightarrow t=15$.
5. A's speed $=\frac{100}{15}=\frac{20}{3}$ m/s, so time for 200 m $= \frac{200}{20/3}=30$ seconds.

So the correct answer is option C, 30 seconds.