Three cats are roaming in a zoo n such a way that when cat A takes 5 steps, B takes 6 steps and C takes 7 steps.But the 6 steps of A are equal to the 7 steps of B and 8 steps of C. what is the ratio of their speeds?

Options

A. a. `140:144:147`
B. b. `40:44:47`
C. c. `15:21:28`
D. d. `252:245:240`

Answer

To solve the problem, we need to find the ratio of the speeds of the three cats A, B, and C based on the steps they take and the relationships between their steps.

### Step-by-Step Solution:

1. **Identify the Steps Taken by Each Cat**:
- Cat A takes 5 steps.
- Cat B takes 6 steps.
- Cat C takes 7 steps.

2. **Establish the Relationships Between Steps**:
- We know that 6 steps of A are equal to 7 steps of B.
- We also know that 6 steps of A are equal to 8 steps of C.

3. **Express the Length of Each Step**:
- Let the length of one step of Cat A be \( a \).
- Let the length of one step of Cat B be \( b \).
- Let the length of one step of Cat C be \( c \).

From the relationships given:
- \( 6a = 7b \) (1)
- \( 6a = 8c \) (2)

4. **Express \( b \) and \( c \) in terms of \( a \)**:
- From equation (1):
\[ b = \frac{6a}{7} \]
- From equation (2):
\[ c = \frac{6a}{8} = \frac{3a}{4} \]

5. **Calculate the Total Distance Covered by Each Cat**:
- Distance covered by Cat A in 5 steps:
\[ D_A = 5a \]
- Distance covered by Cat B in 6 steps:
\[ D_B = 6b = 6 \left(\frac{6a}{7}\right) = \frac{36a}{7} \]
- Distance covered by Cat C in 7 steps:
\[ D_C = 7c = 7 \left(\frac{3a}{4}\right) = \frac{21a}{4} \]

6. **Find the Ratio of Their Speeds**:
- Speed is defined as distance per unit time. Since we are comparing speeds, we can take the distances calculated above and express them in a common form.
- The speeds of the cats can be expressed as:
\[ \text{Speed of A} = \frac{D_A}{t} = \frac{5a}{t} \]
\[ \text{Speed of B} = \frac{D_B}{t} = \frac{36a/7}{t} = \frac{36a}{7t} \]
\[ \text{Speed of C} = \frac{D_C}{t} = \frac{21a/4}{t} = \frac{21a}{4t} \]

7. **Form the Ratio of the Speeds**:
- The ratio of the speeds of A, B, and C is:
\[ \text{Speed Ratio} = 5a : \frac{36a}{7} : \frac{21a}{4} \]

8. **Eliminate \( a \) and \( t \)**:
- The ratio simplifies to:
\[ 5 : \frac{36}{7} : \frac{21}{4} \]
- To eliminate the fractions, we can multiply through by 28 (the least common multiple of 7 and 4):
\[ 5 \times 28 : 36 \times 4 : 21 \times 7 \]
- This gives us:
\[ 140 : 144 : 147 \]

9. **Simplify the Ratio**:
- The final ratio can be simplified by dividing each term by their greatest common divisor (GCD). The GCD of 140, 144, and 147 is 1, so the ratio remains:
\[ 140 : 144 : 147 \]

### Final Ratio of Their Speeds:
The ratio of the speeds of cats A, B, and C is:
\[ \text{Speed Ratio} = 140 : 144 : 147 \]