Let the total distance of the race is $d$ km .
$X$ beats $Y$ by 36 Km which means when $X$ completed the $d$ km distance $Y$ was at distance $d-36$ km.
Let $X$ completed the race in $t$ sec.
So, Speed of $X$ is $\large S_{X}=\frac{d}{t}$
Speed of $Y$ is $\large S_{Y}=\frac{d-36}{t}$
So, ratio of $\large \frac{S_{X}}{S_{Y}}=\frac{d}{d-36}$……………………..(1)
$Y$ beats $Z$ by 72 Km which means when $Y$ completed the $d$ km distance $Z$ was at distance $d-72$ km.
Let $Y$ completed the race in $t_{1}$ sec.
So, Speed of $Y$ is $\large S_{X}=\frac{d}{t_{1}}$
Speed of $Z$ is $\large S_{Y}=\frac{d-72}{t_{1}}$
So, ratio of $\large \frac{S_{Y}}{S_{Z}}=\frac{d}{d-72}$……………………..(2)
$X$ beats $Z$ by 90 Km which means when $X$ completed the $d$ km distance $Z$ was at distance $d-90$ km.
Let $Y$ completed the race in $t_{2}$ sec.
So, Speed of $Y$ is $\large S_{X}=\frac{d}{t_{2}}$
Speed of $Z$ is $\large S_{Y}=\frac{d-90}{t_{2}}$
So, ratio of $\large \frac{S_{X}}{S_{Z}}=\frac{d}{d-90}$……………………..(3)
Dividing (3) by (1) we get ,
$\large \frac{\frac{S_{X}}{S_{Z}}}{\frac{S_{X}}{S_{Y}}}=\frac{\frac{d}{d-90}}{\frac{d}{d-36}}$
=>$\large \frac{S_{Y}}{S_{Z}}=\frac{d-36}{d-90}$ ….…….……...(4)
Now (4) and (2) are equal so equating them,
$\large \frac{d}{d-72}=\frac{d-36}{d-90}$
=>$\large d(d-90)=(d-36)(d-72)$
=>$\large d=144$km.
So correct answer is $144000$ $m$.