Question

The distance from A to B is 60 km. Partha and Narayan start from A at the same time and move towards B. Partha takes four hours more than Narayan to reach B. Moreover, Partha reaches the mid-point of A and B two hours before Narayan reaches B. The speed of Partha, in km per hour, is

• A. 4
• B. 6
• C. 5
• D. 3

Answer 1

The Correct Option is C

Let the speed of Partha be \( x \) km/h. The time taken by Partha to cover 60 km is: \[ \frac{60}{x} \ \text{hours.} \] Let the speed of Narayan be \( y \) km/h. The time taken by Narayan to cover 60 km is: \[ \frac{60}{y} \ \text{hours.} \]

Step 1: Formulating the Equations 

From the problem: \[ \frac{60}{x} = \frac{60}{y} + 4 \] Additionally, Partha reaches the midpoint (30 km) two hours before Narayan reaches B: \[ \frac{30}{x} + 2 = \frac{60}{y} \]

Step 2: Rearranging

From the second equation: \[ \frac{60}{y} - \frac{30}{x} = 2 \] We now have the system: \[ \begin{cases} \frac{60}{x} - \frac{60}{y} = 4 \\ \frac{60}{y} - \frac{30}{x} = 2 \end{cases} \]

 

Step 3: Solving

Adding the two equations: \[ \left(\frac{60}{x} - \frac{60}{y}\right) + \left(\frac{60}{y} - \frac{30}{x}\right) = 4 + 2 \] \[ \frac{60}{x} - \frac{30}{x} = 6 \] \[ \frac{30}{x} = 6 \quad \Rightarrow \quad x = 5 \]

 

Step 4: Finding \(y\)

Substitute \( x = 5 \) into: \[ \frac{30}{5} + 2 = \frac{60}{y} \] \[ 6 + 2 = \frac{60}{y} \quad \Rightarrow \quad \frac{60}{y} = 8 \quad \Rightarrow \quad y = 7.5 \]

 

Step 5: Verification

• Partha's time for 60 km: \( \frac{60}{5} = 12 \) hours
• Narayan's time for 60 km: \( \frac{60}{7.5} = 8 \) hours
• Partha takes \( 12 - 8 = 4 \) hours more ✅
• Midpoint check: Partha takes 6 hours to reach midpoint; Narayan still needs 2 hours to finish ✅

Final Answer: The speed of Partha is 5 km/h.