In a race of 2500m, A beats B by 500 m and in a race of 2000m, B beats C by 800 m. By what distance A gives start to C so that they will end up at same time in 4 km race.

Answer

To solve the problem step by step, we will analyze the information given and calculate the required distance A gives a start to C in a 4 km race.

### Step 1: Understand the races between A, B, and C
1. In a race of 2500 m, A beats B by 500 m. This means when A finishes 2500 m, B has run only 2000 m.
2. In a race of 2000 m, B beats C by 800 m. This means when B finishes 2000 m, C has run only 1200 m.

### Step 2: Calculate the speeds of A, B, and C
Let’s denote the speeds of A, B, and C as \( v_A \), \( v_B \), and \( v_C \) respectively.

From the first race:
- When A runs 2500 m, B runs 2000 m.
- The ratio of their speeds can be expressed as:
\[ \frac{v_A}{v_B} = \frac{2500}{2000} = \frac{5}{4} \]

From the second race:
- When B runs 2000 m, C runs 1200 m.
- The ratio of their speeds can be expressed as:
\[ \frac{v_B}{v_C} = \frac{2000}{1200} = \frac{5}{3} \]

### Step 3: Find the ratio of A to C's speed
To find the ratio of A's speed to C's speed, we can multiply the two ratios we have:
\[ \frac{v_A}{v_C} = \frac{v_A}{v_B} \times \frac{v_B}{v_C} = \frac{5}{4} \times \frac{5}{3} = \frac{25}{12} \]

### Step 4: Set up the equation for the 4 km race
In a 4 km (4000 m) race, let’s denote the distance A gives a start to C as \( d \). Therefore, C will run \( 4000 - d \) meters while A runs 4000 meters.

Since they finish at the same time, we can set up the equation based on their speeds:
\[ \frac{4000}{v_A} = \frac{4000 - d}{v_C} \]

### Step 5: Substitute the speed ratio into the equation
Using the ratio \( \frac{v_A}{v_C} = \frac{25}{12} \), we can express \( v_C \) in terms of \( v_A \):
\[ v_C = \frac{12}{25} v_A \]

Now substituting \( v_C \) into the equation:
\[ \frac{4000}{v_A} = \frac{4000 - d}{\frac{12}{25} v_A} \]

### Step 6: Simplify the equation
Cancelling \( v_A \) from both sides:
\[ 4000 = \frac{25(4000 - d)}{12} \]

### Step 7: Solve for \( d \)
Multiplying both sides by 12:
\[ 48000 = 25(4000 - d) \]
Dividing by 25:
\[ 1920 = 4000 - d \]
Rearranging gives:
\[ d = 4000 - 1920 = 2080 \]

### Final Answer
Thus, A gives C a start of **2080 meters**.

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