In a car race, car $A$ beats car $B$ by $45 \mathrm{~km}$ , car $B$ beats car $C$ by $50$ km, and car $A$ beats $C$ by $90 \mathrm{~km}$ . the distance (in km) over which the race has been conducted is

$550$

$500$

$475$

$450$

✅ Correct Option: 4

We need to set up relationships between the cars' performances and use the concept of relative speeds.

Let us define the race distance as x km.

When the race ends (i.e., when car A finishes):

Car A travels = x km (full race distance)

Car B travels = (x - 45) km (since A beats B by 45 km)

Car C travels = (x - 90) km (since A beats C by 90 km)

This means when car B completes the full race distance of x km, car C has only covered (x - 50) km.

Here's the key insight: The speed ratio between any two cars remains constant throughout the race.

From the first condition, the speed ratio of B to C is:

$\dfrac{\text{Distance covered by B}}{\text{Distance covered by C}} = \dfrac{x - 45}{x - 90}$

From the second condition, when B covers the full race distance x, C covers (x - 50):

$\dfrac{\text{Distance covered by B}}{\text{Distance covered by C}} = \dfrac{x}{x - 50}$

Since the speed ratio is constant:

$\dfrac{x - 45}{x - 90} = \dfrac{x}{x - 50}$

$(x - 45)(x - 50) = x(x - 90)$

$x^2 - 50x - 45x + 2250 = x^2 - 90x$

$x^2 - 95x + 2250 = x^2 - 90x$

$-95x + 2250 = -90x$

$2250 = 5x$

Therefore: x = 450 km

Answer: The race distance is 450 km.

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In a car race, car $A$ beats car $B$ by $45 \mathrm{~km}$ , car $B$ beats car $C$ by $50$ km, and car $A$ beats $C$ by $90 \mathrm{~km}$ . the distance (in km) over which the race has been conducted is