Speed/Distance

[Joy Shaha]

Q. P can give Q a start of 20 seconds in a kilometer race. P can give R a start of 200 meters in the same kilometer race. And Q can give R a start of 20 seconds in the same kilometer race. How long does P take to run the kilometer?


(1) 160 seconds (2) 140 seconds
(3) 200 seconds (4) 240 seconds
(5) 260 seconds

[GMATGuruNY]

I've rearranged the answer choices in ascending order, as they would appear on the GMAT.
I believe that the following expresses the intent of the problem:

In a 1-kilometer race, P gives Q a head start of 20 seconds and R a head start of 200 meters. Q gives R a head start of 20 seconds. All three runners finish the race at exactly the same time. How long does P take to run 1 kilometer?

(1) 140 seconds (2) 160 seconds
(3) 200 seconds (4) 240 seconds
(5) 260 seconds

Since R requires a head start of 200 meters to finish at the same time as P, R runs only 800 meters in the time it takes P to run the entire 1 kilometer.
We can PLUG IN THE ANSWERS, which represent P's time to run 1 kilometer.
When the correct answer choice is plugged in, R will run 800 meters -- 4/5 of a kilometer -- in the time that P takes to run 1 kilometer.

D: 240 seconds
Since P gives Q a 20-second head start, Q runs for 20 seconds longer than P:
Q = 240+20 = 260 seconds.
Since Q gives R a 20-second head start, R runs for 20 seconds longer than Q:
R = 260+20 = 280 seconds.
Thus, R's rate = d/t = 1/280 kilometer per second.
In P's time of 240 seconds, the distance traveled by R at a rate of 1/280 kilometer per second = rt = (1/280)(240) = 6/7 of a kilometer.
The distance is too great.
Eliminate D.

B: 160 seconds
Since P gives Q a 20-second head start, Q runs for 20 seconds longer than P:
Q = 160+20 = 180 seconds.
Since Q gives R a 20-second head start, R runs for 20 seconds longer than Q:
R = 180+20 = 200 seconds.
Thus, R's rate = d/t = 1/200 kilometer per second.
In P's time of 160 seconds, the distance traveled by R at a rate of 1/200 kilometer per second = rt = (1/200)(160) = 4/5 of a kilometer.
Success!

The correct answer is B.