Train Distance Calculation: Karnal to Kurukshetra

This problem involves calculating the distance between two locations given the speeds for the journey in both directions and the total time taken.

Key Formulas

The fundamental relationship is Distance = Speed $\times$ Time ($d = v \times t$). Rearranging this gives Time = Distance / Speed ($t = d / v$).

For this problem:

• Let $d$ represent the distance between Karnal and Kurukshetra (in Km).
• Speed from Karnal to Kurukshetra ($v_1$) = $90$ km/hr.
• Speed from Kurukshetra to Karnal ($v_2$) = $92$ km/hr.
• Total travel time ($T$) = $182$ hours.

Distance Calculation Steps

1. Calculate the time taken for the outward journey ($t_1$):

   $t_1 = \frac{d}{v_1} = \frac{d}{90} \text{ hours}$

2. Calculate the time taken for the return journey ($t_2$):

   $t_2 = \frac{d}{v_2} = \frac{d}{92} \text{ hours}$

3. Use the total time ($T$) to form an equation:

   $T = t_1 + t_2$

   $182 = \frac{d}{90} + \frac{d}{92}$

4. Solve the equation for $d$:

   Factor out $d$: $182 = d \left( \frac{1}{90} + \frac{1}{92} \right)$.

   Combine the fractions inside the parentheses using a common denominator ($90 \times 92 = 8280$):

   $ \frac{1}{90} + \frac{1}{92} = \frac{92}{8280} + \frac{90}{8280} = \frac{92 + 90}{8280} = \frac{182}{8280} $.

   Substitute this back into the equation:

   $182 = d \left( \frac{182}{8280} \right)$.

   Isolate $d$ by multiplying both sides by $\frac{8280}{182}$:

   $d = 182 \times \frac{8280}{182}$

   $d = 8280$

Conclusion

The distance between Karnal and Kurukshetra is 8280 Km.