In a racing over a distance 'x' at a uniform speed, A can beat B by 20 meters, B can beat C by 10 meters and A can beat C by 28 meters, then the value of x will be:

100 m

The correct answer is Option (4) → 100 m

$\text{Let total race distance} = x$

$\text{A beats B by 20 m} \Rightarrow \frac{v_A}{v_B} = \frac{x}{x-20}$

$\text{B beats C by 10 m} \Rightarrow \frac{v_B}{v_C} = \frac{x}{x-10}$

$\text{A beats C by 28 m} \Rightarrow \frac{v_A}{v_C} = \frac{x}{x-28}$

$\frac{v_A}{v_C} = \frac{v_A}{v_B} \cdot \frac{v_B}{v_C}$

$\frac{x}{x-28} = \frac{x}{x-20} \cdot \frac{x}{x-10}$

$\frac{x}{x-28} = \frac{x^2}{(x-20)(x-10)}$

$\frac{1}{x-28} = \frac{x}{(x-20)(x-10)}$

$(x-20)(x-10) = x(x-28)$

$x^2 - 30x + 200 = x^2 - 28x$

$-30x + 200 = -28x$

$200 = 2x$

$x = 100$

The value of $x$ is $100$ meters.