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

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

 Given:

• When A covers distance \( D \), B covers \( D - 45 \), and C covers \( D - 90 \)
• When B covers \( D \), C covers \( D - 50 \)

 Step-by-Step Explanation:

From the given, we use the concept that: \[ \text{Speed} \propto \text{Distance in same time} \]

So, from the first case: \[ \frac{\text{Speed of B}}{\text{Speed of C}} = \frac{D - 45}{D - 90} \]

From the second case: \[ \frac{\text{Speed of B}}{\text{Speed of C}} = \frac{D}{D - 50} \]

Equating both expressions: \[ \frac{D - 45}{D - 90} = \frac{D}{D - 50} \]

Cross-multiplying: \[ (D - 45)(D - 50) = D(D - 90) \]

Expanding both sides: \[ D^2 - 95D + 2250 = D^2 - 90D \]

Cancelling \( D^2 \) and rearranging: \[ -95D + 2250 = -90D \Rightarrow -5D = -2250 \Rightarrow D = 450 \]

 Final Answer:

\[ \boxed{D = 450} \]

Answer 2

Let the distance of the race be \( D \) km.

 Step 1: A vs B

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

Thus, the ratio of their speeds is:

\[ \frac{\text{Speed of A}}{\text{Speed of B}} = \frac{D}{D - 45} \quad \text{...(1)} \]

 Step 2: B vs C

When B completes the race (distance \( D \)), C covers only \( D - 50 \) km.

So the ratio of their speeds is:

\[ \frac{\text{Speed of B}}{\text{Speed of C}} = \frac{D}{D - 50} \quad \text{...(2)} \]

 Step 3: A vs C

When A completes the race (distance \( D \)), C covers only \( D - 90 \) km.

Thus, the ratio of their speeds is:

\[ \frac{\text{Speed of A}}{\text{Speed of C}} = \frac{D}{D - 90} \quad \text{...(3)} \]

 Step 4: Combine All Three

Multiply equations (1) and (2) to get A:C directly:

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

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

Cancel one \( D \) from both sides:

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

 Step 5: Cross-Multiply and Solve

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

Left-hand side:

\[ D^2 - 90D \]

Right-hand side:

\[ D^2 - 45D - 50D + 2250 = D^2 - 95D + 2250 \]

Now, equate both sides:

\[ D^2 - 90D = D^2 - 95D + 2250 \]

Subtract \( D^2 \) from both sides:

\[ -90D = -95D + 2250 \]

\[ 5D = 2250 \quad \Rightarrow \quad D = 450 \]

Final Answer:

\[ \boxed{D = 450 \text{ km}} \]