The ratio of length of two trains is 6 : 5 and theratio of their speed is 3 : 2 The ratio of time taken by them to cross a pole is:

Options

A. `3:5`
B. `4:5`
C. `5:6`
D. `5:8`

Answer

To solve the problem step by step, we will find the ratio of the time taken by two trains to cross a pole based on the given ratios of their lengths and speeds.

### Step 1: Define the variables
Let:
- Length of Train 1 = L1
- Length of Train 2 = L2
- Speed of Train 1 = S1
- Speed of Train 2 = S2

### Step 2: Use the given ratios
From the problem, we have:
- The ratio of lengths of the two trains:
\[ \frac{L1}{L2} = \frac{6}{5} \]
- The ratio of speeds of the two trains:
\[ \frac{S1}{S2} = \frac{3}{2} \]

### Step 3: Write the formula for time
The time taken by each train to cross a pole is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Thus, the time taken by Train 1 (T1) and Train 2 (T2) can be expressed as:
\[ T1 = \frac{L1}{S1} \quad \text{and} \quad T2 = \frac{L2}{S2} \]

### Step 4: Find the ratio of times
To find the ratio of the times taken by the two trains, we can write:
\[ \frac{T1}{T2} = \frac{L1/S1}{L2/S2} = \frac{L1}{L2} \times \frac{S2}{S1} \]

### Step 5: Substitute the known ratios
Substituting the known ratios into the equation:
\[ \frac{T1}{T2} = \frac{L1}{L2} \times \frac{S2}{S1} = \frac{6/5} \times \frac{2/3} \]

### Step 6: Simplify the expression
Now, we simplify the expression:
\[ \frac{T1}{T2} = \frac{6}{5} \times \frac{2}{3} = \frac{6 \times 2}{5 \times 3} = \frac{12}{15} \]
This can be further simplified:
\[ \frac{12}{15} = \frac{4}{5} \]

### Conclusion
Thus, the ratio of the time taken by the two trains to cross a pole is:
\[ \boxed{4 : 5} \]