Two trains start at the same time, P from A to B and Q from B to A.If they arrive at B and A,respectively,` 2 (1)/(2)` hours and 10 hours after they passed each other, and the speed of P is 90 km/hr, then the speed of Q in km/hr is?

Options

A. 80
B. 75
C. 45
D. 60

Answer

To solve the problem, we need to find the speed of train Q given the speed of train P and the times they take to reach their destinations after passing each other.

### Step-by-Step Solution:

1. **Identify Given Information:**
- Speed of train P (from A to B) = 90 km/hr
- Time taken by train P after passing train Q = 2.5 hours (which is 2 hours and 30 minutes)
- Time taken by train Q after passing train P = 10 hours

2. **Let the Speed of Train Q be x km/hr.**

3. **Using the Rule for Speeds of Two Trains:**
The ratio of the speeds of two trains is equal to the square root of the inverse ratio of the times they take after crossing each other. Mathematically, this can be expressed as:
\[ \frac{\text{Speed of P}}{\text{Speed of Q}} = \frac{\sqrt{\text{Time taken by Q}}}{\sqrt{\text{Time taken by P}}} \]
Substituting the known values:
\[ \frac{90}{x} = \frac{\sqrt{10}}{\sqrt{2.5}} \]

4. **Simplifying the Right Side:**
We can simplify \(\sqrt{10}\) and \(\sqrt{2.5}\):
\[ \sqrt{2.5} = \sqrt{\frac{25}{10}} = \frac{5}{\sqrt{10}} \]
Thus,
\[ \frac{\sqrt{10}}{\sqrt{2.5}} = \frac{\sqrt{10}}{\frac{5}{\sqrt{10}}} = \frac{10}{5} = 2 \]

5. **Setting Up the Equation:**
Now we can set up the equation:
\[ \frac{90}{x} = 2 \]

6. **Cross-Multiplying to Solve for x:**
Cross-multiplying gives us:
\[ 90 = 2x \]

7. **Dividing by 2:**
To find x, divide both sides by 2:
\[ x = \frac{90}{2} = 45 \text{ km/hr} \]

### Final Answer:
The speed of train Q is **45 km/hr**.