A rabit takes 22 leaps for every 17 leaps of cat and 22 leaps of a rabit are equal to 17 leaps of the cat. What is the ratio of the speeds of rabit and cat?

Options

A. `1:1`
B. `484:289`
C. `17:22`
D. None of these

Answer

To find the ratio of the speeds of the rabbit and the cat, we can follow these steps:

### Step 1: Understand the relationship between leaps and distances
We know that:
- A rabbit takes 22 leaps for every 17 leaps of a cat.
- 22 leaps of the rabbit are equal to 17 leaps of the cat.

### Step 2: Define the distance per leap
Let:
- The distance covered by the rabbit in one leap be \( x \) kilometers.
- The distance covered by the cat in one leap be \( y \) kilometers.

### Step 3: Set up the equation based on the given information
From the information given:
- The total distance covered by the rabbit in 22 leaps is \( 22x \).
- The total distance covered by the cat in 17 leaps is \( 17y \).

Since 22 leaps of the rabbit are equal to 17 leaps of the cat, we can write the equation:
\[ 22x = 17y \]

### Step 4: Express the speeds
The speed of an animal can be expressed as the distance covered per unit time. Since both animals take the same amount of time to complete their respective leaps, we can express their speeds as:
- Speed of the rabbit \( = \frac{22x}{t} \)
- Speed of the cat \( = \frac{17y}{t} \)

Where \( t \) is the time taken for the leaps.

### Step 5: Find the ratio of speeds
Now, we need to find the ratio of the speeds of the rabbit to the cat:
\[ \text{Ratio of speeds} = \frac{\text{Speed of rabbit}}{\text{Speed of cat}} = \frac{\frac{22x}{t}}{\frac{17y}{t}} = \frac{22x}{17y} \]

### Step 6: Substitute \( y \) in terms of \( x \)
From the equation \( 22x = 17y \), we can express \( y \) in terms of \( x \):
\[ y = \frac{22x}{17} \]

### Step 7: Substitute \( y \) back into the speed ratio
Now, substituting \( y \) in the speed ratio:
\[ \text{Ratio of speeds} = \frac{22x}{17 \cdot \frac{22x}{17}} = \frac{22x}{\frac{374x}{17}} = \frac{22 \cdot 17}{17 \cdot 22} = 1 \]

Thus, the ratio of the speeds of the rabbit and the cat is:
\[ \text{Ratio of speeds} = 1:1 \]

### Final Answer
The ratio of the speeds of the rabbit and the cat is \( 1:1 \).

---