The ratio of lengh of two trains is `5:3` and the ratio of their speed is `6:5`. The ratio of time taken by them to cross a pole is

Options

A. `5:6`
B. `11:8`
C. `25:18`
D. `27:16`

Answer

To find the ratio of the time taken by two trains to cross a pole, we can follow these steps:

### Step 1: Understand the relationship between distance, speed, and time
We know that:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
In this case, the distance for each train to cross a pole is equal to the length of the train.

### Step 2: Define the lengths and speeds of the trains
Let the length of the first train be \(5x\) and the length of the second train be \(3x\) (since the ratio of their lengths is \(5:3\)).
Let the speed of the first train be \(6y\) and the speed of the second train be \(5y\) (since the ratio of their speeds is \(6:5\)).

### Step 3: Calculate the time taken by each train to cross the pole
For the first train:
\[ \text{Time}_1 = \frac{\text{Length of Train 1}}{\text{Speed of Train 1}} = \frac{5x}{6y} \]

For the second train:
\[ \text{Time}_2 = \frac{\text{Length of Train 2}}{\text{Speed of Train 2}} = \frac{3x}{5y} \]

### Step 4: Find the ratio of the times taken by the two trains
Now, we need to find the ratio of \(\text{Time}_1\) to \(\text{Time}_2\):
\[ \text{Ratio} = \frac{\text{Time}_1}{\text{Time}_2} = \frac{\frac{5x}{6y}}{\frac{3x}{5y}} \]

### Step 5: Simplify the ratio
To simplify this, we can multiply by the reciprocal:
\[ \text{Ratio} = \frac{5x}{6y} \times \frac{5y}{3x} = \frac{5 \times 5}{6 \times 3} = \frac{25}{18} \]

### Conclusion
The ratio of the time taken by the two trains to cross a pole is \(25:18\).

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