Question

X can run 1 km in 4 min 10 seconds and Y can cover the same distance in 4 min 20 seconds. By what distance can X beat Y?

• A. 36.46 meters
• B. 37.46 meters
• C. 38.46 meters
• D. 39.46 meters

Answer 1

The Correct Option is C

Step 1: Understanding the Problem
This is a race problem. "X beats Y by a certain distance" means that when X finishes the race, Y is that certain distance behind the finish line. We need to find the distance Y covers in the time X takes to finish.

Step 2: Key Formula or Approach
\begin{enumerate}
Convert all times to seconds.
Find the speed of Y.
Calculate the distance Y covers in the time X takes to finish the race.
The difference between the total race distance (1000m) and the distance covered by Y will be the margin of defeat. \end{enumerate} Alternatively, find how far Y runs in the extra time he takes. This is simpler. X beats Y by 10 seconds. We need to find how far Y runs in those 10 seconds. This is the distance he is behind. Let's try this. No, that logic is flawed. The margin of defeat is the distance Y has yet to cover when X finishes.

Step 3: Detailed Explanation
1. Convert times to seconds:
Distance = 1 km = 1000 meters.
Time taken by X (\(T_X\)) = 4 min 10 sec = \( (4 \times 60) + 10 = 240 + 10 = 250 \) seconds.
Time taken by Y (\(T_Y\)) = 4 min 20 sec = \( (4 \times 60) + 20 = 240 + 20 = 260 \) seconds.
2. Calculate the speed of Y (\(S_Y\)):
Y covers 1000 meters in 260 seconds. \[ S_Y = \frac{\text{Distance}}{\text{Time}} = \frac{1000 \text{ m}}{260 \text{ s}} = \frac{100}{26} = \frac{50}{13} \text{ m/s} \] 3. Find distance covered by Y when X finishes:
X finishes the race in 250 seconds. We need to find how far Y has run in these 250 seconds.
\[ \text{Distance}_Y = S_Y \times T_X = \frac{50}{13} \times 250 = \frac{12500}{13} \text{ meters} \] \[ \text{Distance}_Y \approx 961.54 \text{ meters} \] 4. Calculate the beat distance:
The distance by which X beats Y is the remaining distance Y had to cover.
\[ \text{Beat Distance} = \text{Total Distance} - \text{Distance}_Y \] \[ \text{Beat Distance} = 1000 - \frac{12500}{13} = \frac{13000 - 12500}{13} = \frac{500}{13} \text{ meters} \] Now, let's calculate the decimal value: \[ \frac{500}{13} \approx 38.4615... \text{ meters} \]
Step 4: Final Answer
X beats Y by approximately 38.46 meters. Therefore, option (C) is the correct answer.