Let's solve the problem step by step.

Given:

1. A and B run a 1 km race, and A wins by 60 seconds.
2. A and C run a 1 km race, and A wins by 375 meters.
3. B and C run a 1 km race, and B wins by 30 seconds.

We need to determine the time taken by A, B, and C to complete 1 km.

Step 1: Let Variables

• Let A's time to run 1 km be x minutes.
• Let B's time be y minutes.
• Let C's time be z minutes.

From the first condition: $y = x + 1 \quad \text{(since A beats B by 60 seconds, which is 1 minute)}$

From the second condition:

• A covers 1 km in x minutes.
• C covers 625 meters in x minutes (since A wins by 375 meters, meaning C only runs 625 meters in the same time).
• Speed of C = $\frac{625}{1000} \times \frac{1}{x} = \frac{5}{8x}$.
• Since speed = distance/time, we get: $z = \frac{8}{5} x$

From the third condition: $z = y + \frac{1}{2} \quad \text{(since B beats C by 30 seconds, which is 0.5 minutes)}$

Step 2: Solve Equations

From the equation $z = \frac{8}{5} x$ and $z = y + \frac{1}{2}$, we substitute $y = x + 1$:

$\frac{8}{5} x = (x + 1) + \frac{1}{2}$

Multiply everything by 10 to clear fractions:

$16x = 10x + 10 + 5$

$16x - 10x = 15$

$6x = 15$

$x = 2.5 \quad \text{(A's time)}$

$y = x + 1 = 3.5 \quad \text{(B's time)}$

$z = \frac{8}{5} x = \frac{8}{5} \times 2.5 = 4 \quad \text{(C's time)}$

Step 3: Check Answer Choices

The closest option is:

$\mathbf{2.5 \text{ min}, 3.5 \text{ min}, 4 \text{ min}}$

which matches option 1.

Would you like a more detailed explanation on any part?

Related Questions:

1. How do you calculate relative speed in a race?
2. What happens if the distances in the problem are changed?
3. How do you derive speed from time and distance in general?
4. Can this method be used for different types of races (e.g., car races)?
5. How do you apply equations to motion problems in physics?

Tip:

In race problems, always express speeds as a ratio of distance/time before setting up equations.