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In a $10$ km race, $A$, $B$ and $C$, each running at uniform speed, get the gold, silver, and bronze medals, respectively. If $A$ beats $B$ by $1$ km and $B$ beats $C$ by $1$ km, then by how many metres does $A$ beat $C$?

  1. $1800$
  2. $1900$
  3. $1500$
  4. $1600$
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Let's draw the diagram for a better understanding.



We know that,  $\text{Speed} = \dfrac{\text{Distance}}{\text{Time}} $

If time is constant, then  $\boxed{\text{Speed} \propto \text{Distance}}$

The ratio of the speed of $A$ and $B:$

  • $S_{A}:S_{B} =10:9$

The ratio of the speed of $B$ and $C:$

  • $S_{B}:S_{C} =10:9$

Now, we can combine the ratios.

  • $S_{A}:S_{B} =(10:9)\times 10=100:90$
  • $S_{B}:S_{C} =(10:9)\times 9=90:81$

$\Rightarrow S_{A}:S_{B}:S_{C} = 100:90:81$       

We can write the ratio of Distance, $D_{A}:D_{B}: D_{C} = 10000 : 9000 : 8100.$

$\therefore \; A$ would beat $C$ by $1900$ meters.

Correct Answer $:\text{B}$

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