Two trains whose length difference is 170 m, crosses each other in 16 sec when moves in opposite direction but crosses each other in 36 sec when move in the same direction. Find the speed of faster train if speed of slower train is 35 km/hr.

Options

A. 98 km/hr
B. 70 km/hr
C. 91 km/hr
D. 65 km/hr

Answer

To solve the problem, we need to find the speed of the faster train given the speed of the slower train and the time taken for both trains to cross each other in different scenarios.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the speed of the slower train be \( S_1 = 35 \) km/hr.
- Let the speed of the faster train be \( S_2 \) km/hr.
- Let the length of the first train be \( L_1 \) meters.
- The length of the second train will then be \( L_1 + 170 \) meters.

2. **Convert Speeds to m/s:**
- To convert km/hr to m/s, we use the conversion factor \( \frac{5}{18} \).
- Thus, \( S_1 = 35 \times \frac{5}{18} = \frac{175}{18} \) m/s.

3. **Crossing in Same Direction:**
- When the trains cross each other in the same direction, the relative speed is \( S_2 - S_1 \).
- The time taken to cross each other is 36 seconds.
- The total distance covered when they cross each other is \( L_1 + (L_1 + 170) = 2L_1 + 170 \).
- Using the formula for speed, we have:
\[ \frac{(S_2 - S_1) \cdot 36}{1} = 2L_1 + 170 \]
- Substituting \( S_1 \):
\[ (S_2 - 35) \cdot 36 = 2L_1 + 170 \quad \text{(Equation 1)} \]

4. **Crossing in Opposite Direction:**
- When the trains cross each other in opposite directions, the relative speed is \( S_2 + S_1 \).
- The time taken to cross each other is 16 seconds.
- The total distance is still \( 2L_1 + 170 \).
- Using the speed formula again, we have:
\[ \frac{(S_2 + S_1) \cdot 16}{1} = 2L_1 + 170 \]
- Substituting \( S_1 \):
\[ (S_2 + 35) \cdot 16 = 2L_1 + 170 \quad \text{(Equation 2)} \]

5. **Setting Up the Equations:**
- From Equation 1:
\[ 36(S_2 - 35) = 2L_1 + 170 \]
- From Equation 2:
\[ 16(S_2 + 35) = 2L_1 + 170 \]

6. **Equating the Two Equations:**
- Set the right-hand sides equal to each other:
\[ 36(S_2 - 35) = 16(S_2 + 35) \]
- Expanding both sides:
\[ 36S_2 - 1260 = 16S_2 + 560 \]
- Rearranging gives:
\[ 36S_2 - 16S_2 = 560 + 1260 \]
\[ 20S_2 = 1820 \]
\[ S_2 = \frac{1820}{20} = 91 \text{ km/hr} \]

### Final Answer:
The speed of the faster train is **91 km/hr**.