Question

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

• A. 500
• B. 475
• C. 550
• D. 450

Answer 1

The Correct Option is D

The correct answer is (D): \(450\)

Let the length of the racetrack be \(D\).

When \(A\) covers \(D\) km, \(B\) covers \(( D -45)\) km and \(C\) covers \((D-90)\) km

When \(B\) covers \(D\) km, C covers \(( D -50)\) km

The ratio of the speeds of the racers is same as the ratio of the distance travelled in a given time period.

Ratio of speed of \(B\) and \(C\) is \(\frac{\text{speed of B}}{\text{speed of C}} = \frac{D-45}{D-90} = \frac{D}{D-50}\)

\(⇒ \frac{D}{D-50} = \frac{45}{40}\)

\(⇒ 40D = 45D-50×45\)

\(⇒ D = 450\)

Answer 2

Let the distance over which the race has been conducted be \(D\) km.

When A completes the race of \(D\) km, in the same time B covers \((D-45)\) km.

Time is the same, so the ratio of speeds of A and B is equal to the ratio of the distance traveled by A and B:

\[\text{Ratio of speed of A and B} = \frac{D}{D-45}\] -----------------(1)

When B completes the race of D km, in the same time C covers \((D-50)\) km.

So, the ratio of speed of B and C is:

\[\text{Ratio of speed of B and C} = \frac{D}{D-50}\] -----------------(2)

When A completes the race of D km, in the same time C covers \((D-90)\) km.

So, the ratio of speed of A and C is:

\[\text{Ratio of speed of A and C} = \frac{D}{D-90}\]-----------------(3)

From (1), (2), and (3):

\[\left(\frac{D}{D-45}\right)\left(\frac{D}{D-50}\right) = \frac{D}{D-90}\]

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

\[D(D-90) = (D-45)(D-50)\]

\[D^2 - 90D = D^2 - 45D - 50D + 2250\]

\[95D - 90D = 2250\]

\[5D = 2250\]

\[D = \frac{2250}{5}\]

\[D = 450\]

The race has been conducted for a distance of 450 km.