In a 1000 m race, the ratio of the speeds of two contestants P and Q is 4: 5. P has a start of 280 m. Then, P wins by?

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120 m

100 m

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90 m

Given that the ratio of the speeds of contestants \( P \) and \( Q \) is \( 4:5 \), and \( P \) has a start of 280 meters in a 1000-meter race, we need to find out by how many meters \( P \) wins the race.

### Step 1: Calculate the distance covered by Q when P finishes the race.
Let's assume the speed of \( P \) is \( 4x \) and the speed of \( Q \) is \( 5x \).

Since \( P \) starts with a 280-meter advantage, \( P \) only needs to cover \( 1000 - 280 = 720 \) meters to complete the race.

### Step 2: Time taken by P to cover 720 meters
The time taken by \( P \) to cover 720 meters is:

\[
\text{Time taken by } P = \frac{\text{Distance}}{\text{Speed}} = \frac{720}{4x} = \frac{180}{x} \text{ seconds}
\]

### Step 3: Distance covered by Q in the same time
In this time, \( Q \) will cover a distance of:

\[
\text{Distance covered by } Q = \text{Speed} \times \text{Time} = 5x \times \frac{180}{x} = 900 \text{ meters}
\]

### Step 4: Determine how much P wins by
Since \( Q \) covers 900 meters when \( P \) finishes the race, \( P \) wins by:

\[
1000 - 900 = 100 \text{ meters}
\]

Thus, \( P \) wins by **100 meters**.

\(\boxed{100 \text{ meters}}\)

Would you like further details on this solution? Here are five related questions you might find useful:

1. How does the start advantage affect the outcome of a race?
2. What happens if the speed ratio changes to something other than \( 4:5 \)?
3. How can you calculate the time difference between two contestants in a race?
4. If the race was longer (e.g., 1500 meters), by how much would \( P \) win?
5. What if \( P \) had no start advantage—how would the race outcome change?

**Tip:** When dealing with races and ratios, breaking down the problem into smaller steps—like calculating time and distance separately—can simplify the solution process.

Explore how to calculate the outcome of a 1000m race between contestants P and Q with a speed ratio of 4:5. P starts with a 280m advantage. Understand the math behind P's victory by 100 meters.

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