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

1. $1800$
2. $1900$
3. $1500$
4. $1600$

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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