This problem involves a race between Atul and Vishu, where Atul gives Vishu certain advantages but still manages to beat him.

Let's break down the problem:

1. Scenario 1 (900m race):

   • Vishu gets a 20-second head-start, but Atul beats him by 135 meters in the 900m race.

2. Scenario 2 (Same race, different condition):

   • In another race of the same length, Atul gives Vishu a head start of 189 meters and beats him by 8 seconds.

Objective:

We need to calculate how much time Vishu would take to complete the full 900m race.

Step-by-step analysis:

Scenario 1:

• Let the speed of Vishu be $v_V$ and the speed of Atul be $v_A$.
• Time taken by Vishu to complete the 900m race (with a 20-second head-start) = $\frac{900}{v_V} - 20$.
• Time taken by Atul to run the same 900m race = $\frac{900}{v_A}$.
• The distance Atul covers in the same time that Vishu runs 900 - 135 = 765 meters, so:
  $\frac{765}{v_V} = \frac{900}{v_A}$ --> (Equation 1)

Scenario 2:

• In this scenario, Vishu gets a head-start of 189 meters and Atul beats him by 8 seconds.
• Time taken by Vishu to run 900 - 189 = 711 meters = $\frac{711}{v_V}$.
• Time taken by Atul to run 900 meters is $\frac{900}{v_A}$.
• Given that Atul beats Vishu by 8 seconds:
  $\frac{900}{v_A} = \frac{711}{v_V} + 8$ --> (Equation 2)

Solving these two equations will give us the value of $v_V$, which we can use to find Vishu's time for the full 900m race.

Let me calculate this for you.The solution to the system of equations gives us:

• Vishu's speed $v_V = 27/4 = 6.75$ meters per second.
• Atul's speed $v_A = 135/17 \approx 7.94$ meters per second.

Now, to calculate the time it takes for Vishu to complete the full 900m race:

$\text{Time for Vishu} = \frac{900}{v_V} = \frac{900}{6.75}$

Let me calculate that for you.Vishu will take approximately 2 minutes and 13 seconds to complete the full 900-meter race.

Would you like more details or have any further questions?

Here are 5 related questions you might find interesting:

1. How much faster is Atul compared to Vishu based on their speeds?
2. If the race were longer (e.g., 1500 meters), how would the time comparison change?
3. How does the head-start impact the race time, and how could it be calculated for different distances?
4. What if Atul gave Vishu a head-start in terms of time rather than distance?
5. How could we calculate the relative distance Atul would lead by in other race scenarios?

Tip: In races or time-distance problems, breaking down the speeds and using proportional relationships can often simplify the process of finding time, speed, or distance.