Question

A race course is 400 metres long. A and B run a race and A wins by 5 metres. B and C run over the same course and B wins by 4 metres. C and D run over it and D wins by 16 metres. If A and D run over it, then who would win and by how much?

• A. A by 8.4 metres
• B. D by 8.4 metres
• C. D by 7.3 metres
• D. A by 7.3 metres

Answer 1

The Correct Option is C

Step 1: Translate “wins by” into speed ratios (track length $L=400$ m).
A beats B by $5$ m $\Rightarrow$ when A runs $400$, B runs $395$ $\Rightarrow \dfrac{v_A}{v_B}=\dfrac{400}{395}=\dfrac{80}{79}$.
B beats C by $4$ m $\Rightarrow$ when B runs $400$, C runs $396$ $\Rightarrow \dfrac{v_B}{v_C}=\dfrac{400}{396}=\dfrac{100}{99}$.
D beats C by $16$ m $\Rightarrow$ when D runs $400$, C runs $384$ $\Rightarrow \dfrac{v_D}{v_C}=\dfrac{400}{384}=\dfrac{25}{24}$, hence $\dfrac{v_C}{v_D}=\dfrac{24}{25}$.

Step 2: Relate A and D.
\[ \dfrac{v_A}{v_D} = \left(\dfrac{v_A}{v_B}\right)\!\left(\dfrac{v_B}{v_C}\right)\!\left(\dfrac{v_C}{v_D}\right) = \dfrac{80}{79}\cdot\dfrac{100}{99}\cdot\dfrac{24}{25} = \dfrac{192000}{195525}\approx 0.9820. \] Since $\dfrac{v_A}{v_D}<1$, D is faster than A.

Step 3: Convert the speed advantage to a distance lead over 400 m.
When D finishes $400$ m, A covers $400 \times \dfrac{v_A}{v_D} \approx 400 \times 0.9820 \approx 392.8$ m.
Therefore, D’s winning margin $\Rightarrow 400 - 392.8 \approx 7.2 \text{ m} \approx \boxed{7.3 \text{ metres}}.$