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**Hint:**It is given that A can run $22.5$ meters while B runs $25$ meters at the same time, so first find the time taken by B to run $1000$ meters because B runs faster than A, he early runs 1000 meters. The resultant time is the time that A have, to run with a speed of 22.5 meters. After getting the distance traveled by A, find the difference in the distances.

**Complete step-by-step answer:**

It is given that A can run $22.5$ while B runs $25$ meters at the same time.

When $B$ runs 1000 meter, then the time that is taken by $B$is given using the formula:

${\text{Time = }}\dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}$

The distance is $1000$mt and the speed is $25$mt, so the time taken is given as:

${\text{Time = }}\dfrac{{1000}}{{25}}$

Therefore, the time taken by B to run 1000 meters is $\dfrac{{1000}}{{25}}$.

In the same time, the distance traveled by $A$ is given using the formula:

${\text{Distance}} = {\text{Speed}} \times {\text{Time}}$

The time is same as taken by $B$,${\text{Time = }}\dfrac{{1000}}{{25}}$ and the speed of $A$ is$22.5mt$.

${\text{Distance}} = {\text{22}}{\text{.5}} \times \left( {\dfrac{{1000}}{{25}}} \right)$

Simplify the above expression to find the distance traveled by $A$.

${\text{Distance}} = {\text{22}}{\text{.5}} \times \left( {40} \right) = 900$

Thus, when $B$ can run 1000 meter, then $A$ can run 900 meters.

Therefore, the difference between $B$ can run and $A$ can run is given as:

$ = \left( {1000 - 900} \right)$ meters

Thus, in the race of one kilometer, B beats A by $100$ meters.

Therefore, option A is correct.

[

**Note:**We know that change in the distance with the time is expressed as a speed. When an object is traveled x meters in t time then the speed is given as:

${\text{Speed}} = \dfrac{x}{t}$ ]

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