A hare sees a hound 100 m away from her and runs off in the opposite direction at a speed of 12 KM an hour. A minute later the hound perceives her and gives a chase at a speed of 16 KM an hour. The distance at which the hound catches the hare (in meters) is________

Answer

To solve the problem step by step, we will analyze the situation and use the given speeds and distances to find out how far the hound catches the hare.

### Step 1: Convert the speeds from km/h to m/s
- The speed of the hare is 12 km/h.
\[ \text{Speed of hare in m/s} = 12 \times \frac{5}{18} = \frac{60}{18} = \frac{10}{3} \text{ m/s} \]

- The speed of the hound is 16 km/h.
\[ \text{Speed of hound in m/s} = 16 \times \frac{5}{18} = \frac{80}{18} = \frac{40}{9} \text{ m/s} \]

### Step 2: Determine the distance traveled by the hare in the first minute
- The hare runs for 1 minute (60 seconds) before the hound starts chasing.
\[ \text{Distance traveled by hare in 1 minute} = \text{Speed of hare} \times \text{Time} = \frac{10}{3} \times 60 = 200 \text{ meters} \]

### Step 3: Calculate the total distance between the hare and the hound when the hound starts chasing
- Initially, the hare is 100 meters away from the hound. After running for 1 minute, the hare has traveled an additional 200 meters.
\[ \text{Total distance between hare and hound} = 100 + 200 = 300 \text{ meters} \]

### Step 4: Set up the equation for the time taken by the hound to catch the hare
- Let \( t \) be the time (in seconds) taken by the hound to catch the hare after it starts chasing.
- In this time, the distance covered by the hound is:
\[ \text{Distance by hound} = \text{Speed of hound} \times t = \frac{40}{9} t \]

- The distance covered by the hare during this time is:
\[ \text{Distance by hare} = \text{Speed of hare} \times t = \frac{10}{3} t \]

### Step 5: Write the equation based on the distances
- The distance the hound covers must equal the distance the hare covers plus the initial distance (300 meters):
\[ \frac{40}{9} t = \frac{10}{3} t + 300 \]

### Step 6: Solve the equation
- Rearranging gives:
\[ \frac{40}{9} t - \frac{10}{3} t = 300 \]

- To combine the fractions, find a common denominator (which is 9):
\[ \frac{40}{9} t - \frac{30}{9} t = 300 \]
\[ \frac{10}{9} t = 300 \]

- Multiply both sides by 9:
\[ 10t = 2700 \]

- Divide by 10:
\[ t = 270 \text{ seconds} \]

### Step 7: Calculate the distance the hound travels during this time
- The distance covered by the hound in 270 seconds is:
\[ \text{Distance} = \text{Speed of hound} \times \text{Time} = \frac{40}{9} \times 270 \]
\[ = \frac{40 \times 270}{9} = 1200 \text{ meters} \]

### Final Answer:
The distance at which the hound catches the hare is **1200 meters**.

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