Question Details

Arun and Bhaskar run a race of 3 km. First, Arun gives Bhaskar a head start of 400 m and beats him by 30 seconds. While coming back, Arun gives Bhaskar a lead of 2.5 minutes and gets beaten by 500 m. What is the difference between the times in minutes in which Arun and Bhaskar can run the race for one side separately?

Options

1.5 min

2.5 min

3 min

2 min

Show Answer

Correct Answer :

Option A

1.5 min

Solution :

Correct Answer: 1.5 min

Step-by-Step Explanation:

Let the total length of the race for one side be $D = 3 km = 3000 m$ .

Let the speed of Arun be $v_{A}$ (in m/s) and the speed of Bhaskar be $v_{B}$ (in m/s).

Let the time taken by Arun to cover the full distance of 3000 m be $T_{A}$ (in seconds) and the time taken by Bhaskar to cover 3000 m be $T_{B}$ (in seconds).

Thus, we have:

$T_{A} = \frac{3000}{v_{A}}$

$T_{B} = \frac{3000}{v_{B}}$

Case 1: First Race

Arun runs the full distance of 3000 m.

Bhaskar gets a head start of 400 m, so Bhaskar only runs a distance of:

$3000 - 400 = 2600 m$

Arun beats Bhaskar by 30 seconds. This means the time taken by Arun to cover 3000 m is 30 seconds less than the time taken by Bhaskar to cover 2600 m:

$\frac{3000}{v_{A}} = \frac{2600}{v_{B}} - 30$

Expressing this in terms of total individual times $T_{A}$ and $T_{B}$ :

$T_{A} = \frac{2600}{3000} T_{B} - 30 = \frac{13}{15} T_{B} - 30$ — (Equation 1)

Case 2: Return Race

Arun gives Bhaskar a time lead of 2.5 minutes ( $2.5 \times 60 = 150 seconds$ ).

Arun gets beaten by 500 m. This means when Bhaskar completes the full distance of 3000 m, Arun has covered only:

$3000 - 500 = 2500 m$

Since Bhaskar starts 150 seconds earlier than Arun, the running time of Bhaskar for 3000 m is 150 seconds more than the running time of Arun for 2500 m:

$T_{B} = \frac{2500}{v_{A}} + 150$

Since $\frac{2500}{v_{A}} = \frac{2500}{3000} T_{A} = \frac{5}{6} T_{A}$ , we have: