In a car race, car A beats car B by 45 km, car B beats car C by 50 km, and car A beats car C by 90 km. The distance (in km) over which the race has been conducted is

The correct answer is 450 km.

Let the total distance of the race be D km. We use the key principle: when a faster car finishes the full race distance D, the slower car has only covered a shorter distance.

Step 1 — Establish the speed ratios from the given conditions.

Since all cars race simultaneously, their speeds are constant. We express each car's speed relative to another:

A beats B by 45 km: When A completes D km, B has covered (D − 45) km.
∴ Speed ratio of A to B:

$$\frac{\mathrm{Speed\; of\; A}}{\mathrm{Speed\; of\; B}}=\frac{D}{D-45}$$

B beats C by 50 km: When B completes D km, C has covered (D − 50) km.
∴ Speed ratio of B to C:

$$\frac{\mathrm{Speed\; of\; B}}{\mathrm{Speed\; of\; C}}=\frac{D}{D-50}$$

A beats C by 90 km: When A completes D km, C has covered (D − 90) km.
∴ Speed ratio of A to C (directly):

$$\frac{\mathrm{Speed\; of\; A}}{\mathrm{Speed\; of\; C}}=\frac{D}{D-90}$$

Step 2 — Use the chain rule for speed ratios.

The ratio of A's speed to C's speed can also be expressed as the product of the two intermediate ratios:

$$\frac{\mathrm{Speed\; of\; A}}{\mathrm{Speed\; of\; C}}=\frac{\mathrm{Speed\; of\; A}}{\mathrm{Speed\; of\; B}}\times \frac{\mathrm{Speed\; of\; B}}{\mathrm{Speed\; of\; C}}$$

Substituting the expressions from Step 1:

$$\frac{D}{D-90}=\frac{D}{D-45}\times \frac{D}{D-50}$$

Step 3 — Solve for D.

$$\frac{D}{D-90}=\frac{D^2}{(D-45)(D-50)}$$

Divide both sides by D (since D ≠ 0):

$$\frac{1}{D-90}=\frac{D}{(D-45)(D-50)}$$

Cross-multiply:

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

Expand the left side:

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

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

Cancel D² from both sides:

$$-95D+2250=-90D$$

$$2250=-90D+95D=5D$$

$$D=\frac{2250}{5}=450$$

Step 4 — Verify the answer.

With D = 450 km, let's cross-check all three conditions:

• Speed ratio A:B = 450 : (450−45) = 450 : 405
• Speed ratio B:C = 450 : (450−50) = 450 : 400
• Speed ratio A:C = (450 × 450) / (405 × 400) = 202,500 / 162,000 = 450 : 360
• Direct A:C = 450 : (450−90) = 450 : 360 ✓

All three conditions are perfectly satisfied, confirming the total race distance is 450 km.