A cat takes 5 leaps for every 4 leaps of a dog but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speeds of the cat to that of the dog?

Options

A. `11:15`
B. `15 : 11`
C. `16 : 15 `
D. `15 : 16`

Answer

To find the ratio of the speeds of the cat to that of the dog, we can follow these steps:

### Step 1: Understand the relationship between the leaps of the cat and the dog.
According to the problem, a cat takes 5 leaps for every 4 leaps of a dog. This can be expressed as:
- For every 5 leaps of the cat, the dog takes 4 leaps.

### Step 2: Establish the equivalence of leaps.
We also know that 3 leaps of the dog are equal to 4 leaps of the cat. This means:
- 3 Dog Leaps = 4 Cat Leaps
From this, we can find the length of each leap in terms of a common unit.

### Step 3: Express the leaps in terms of distance.
Let’s denote the distance of one leap of the dog as \( d \) and the distance of one leap of the cat as \( c \). From the equivalence:
- \( 3d = 4c \)
This implies:
- \( d = \frac{4}{3}c \)

### Step 4: Calculate the total distance covered in a given time.
Now, let’s calculate the distance covered by both the cat and the dog in the same time frame.
- In the time the cat takes 5 leaps, the dog takes 4 leaps.
- The distance covered by the cat in 5 leaps is \( 5c \).
- The distance covered by the dog in 4 leaps is \( 4d = 4 \times \frac{4}{3}c = \frac{16}{3}c \).

### Step 5: Find the ratio of the distances covered.
Now we can find the ratio of the distances covered by the cat and the dog:
- Distance covered by the cat: \( 5c \)
- Distance covered by the dog: \( \frac{16}{3}c \)

The ratio of the distances is:
\[ \text{Ratio} = \frac{5c}{\frac{16}{3}c} = \frac{5}{\frac{16}{3}} = 5 \times \frac{3}{16} = \frac{15}{16} \]

### Step 6: Determine the ratio of speeds.
Since speed is defined as distance per unit time, the ratio of speeds (Speed of Cat : Speed of Dog) is the same as the ratio of the distances they cover in the same time:
\[ \text{Speed Ratio} = \frac{15}{16} \]

### Final Answer:
The ratio of the speeds of the cat to that of the dog is \( \frac{15}{16} \).

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