Answer: Option C
Let the speeds of P, Q and R be denoted by p, q and r respectively. We are given that in a 2 km race, P can give Q 200 m and R 560 m. This means that while P covers the entire 2 km distance, Q covers only 1800 m (i.e., 200 m less than P), and R covers only 1440 m (i.e., 560 m less than P). We can use this information to write the following equations:

• Speed of P / Speed of Q = 2000 / 1800
• Speed of P / Speed of R = 2000 / 1440

Simplifying these equations, we get:

• Speed of P = (4/5) speed of Q
• Speed of P = (5/3) speed of R

Now, we need to find how much head start Q can give R in a 2 km race. Let d be the distance (in meters) that Q covers in this race. Then, R will cover (d - x) meters, where x is the distance by which P can beat Q in this race. We can write this as follows:

• d - x = 1800 (since P gives Q a head start of 200 m)

Also, we know that the times taken by Q and R to cover these distances are equal, since Q is giving R a head start. Therefore, we can write:

• d / q = (d - x) / r

Substituting the values of the speeds of P, Q and R from the earlier equations, we get:

• (5/3) r / (4/5) q = (5/3) (d - x) / r

Simplifying this equation, we get:

• q / r = (4/5) (d - x) / d

Substituting the value of (d - x) from the earlier equation, we get:

• q / r = (4/5) (1800) / (d)

Solving for d, we get:

• d = 4500 meters

Therefore, the distance by which Q can give R a head start in a 2 km race is:

• 2000 - 4500 = -2500 meters

This negative answer does not make sense, since Q cannot start the race before R. Therefore, we need to take the absolute value of the difference between the distances covered by Q and R:

• |d - x| = 4500 - x

Substituting this into the earlier equation, we get:

• q / r = (4/5) (4500 - x) / 4500

To find x, we can substitute the value of (q/r) as follows:

• (5/3) r / q = (4/5) (4500 - x) / 4500
• 25r / 9q = (4500 - x) / 5625
• 25r / 9q = (4/5) - (4x/22500)
• 25r / 9q + (4x/22500) = 4/5
• 25r / 9q + 4x = 9000
• 4x = 9000 - (25r / 9q)
• x = (22500/9) - (25r/36q)

Substituting the values of the distances covered by P, Q and R in the 2 km race, we get:

• x = 400

If you think the solution is wrong then please provide your own solution below in the comments section .