A hare pursued by a grey-hound, is 20 of her own leaps ahead of him. While the hare takes 4 leaps the grey-hound takes 3 leaps. 2 leaps of grey-hound is equal to 3 leaps of hare. In how many leaps will the grey-hound overtake the hare?

Answer

To solve the problem of how many leaps the greyhound will take to overtake the hare, we can follow these steps:

### Step 1: Understand the Problem
The hare is 20 leaps ahead of the greyhound. While the hare takes 4 leaps, the greyhound takes 3 leaps. We also know that 2 leaps of the greyhound are equal to 3 leaps of the hare.

### Step 2: Establish the Relationship Between Leaps and Distance
Let’s denote:
- The length of one leap of the hare as \( h \).
- The length of one leap of the greyhound as \( g \).

From the problem, we have:
\[ 2g = 3h \]
This implies:
\[ g = \frac{3}{2}h \]

### Step 3: Calculate the Distance Ahead
The hare is 20 leaps ahead:
\[ \text{Distance ahead} = 20h \]

### Step 4: Determine the Speed of Each Animal
In terms of distance per leap:
- The hare covers \( 4h \) in the time it takes to make 4 leaps.
- The greyhound covers \( 3g \) in the same time.

Substituting \( g \) from the earlier equation:
\[ 3g = 3 \times \frac{3}{2}h = \frac{9}{2}h \]

### Step 5: Calculate the Effective Speeds
- Speed of the hare: \( \text{Speed}_{hare} = \frac{4h}{t} \)
- Speed of the greyhound: \( \text{Speed}_{greyhound} = \frac{9h/2}{t} \)

### Step 6: Find the Relative Speed
The relative speed of the greyhound with respect to the hare is:
\[ \text{Relative Speed} = \text{Speed}_{greyhound} - \text{Speed}_{hare} \]
\[ = \frac{9h/2}{t} - \frac{4h}{t} \]
\[ = \left( \frac{9h}{2} - 4h \right) \frac{1}{t} = \left( \frac{9h - 8h}{2} \right) \frac{1}{t} = \frac{h}{2t} \]

### Step 7: Calculate Time to Overtake
To find the time \( T \) it takes for the greyhound to cover the distance of \( 20h \):
\[ 20h = \text{Relative Speed} \times T \]
\[ 20h = \frac{h}{2t} \times T \]
Cancelling \( h \) from both sides:
\[ 20 = \frac{T}{2t} \]
\[ T = 40t \]

### Step 8: Calculate Number of Leaps Taken by the Greyhound
In the time \( T \), the greyhound takes:
\[ \text{Number of leaps} = \text{leaps per unit time} \times T \]
\[ = 3 \times T \]
Substituting \( T = 40t \):
\[ = 3 \times 40 = 120 \text{ leaps} \]

### Final Answer
The greyhound will overtake the hare in **120 leaps**.

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