Problem Solving

Q. A & B run a kilometer and A wins by 25 sec. A & C run a kilometer & A wins by 275 m. when B & C run the same distance B wins by 30 sec. The time taken by A to run a Km is-
A) 2 min 25 sec
B) 2 min 50 sec
C) 3 min 20 sec
D) 3 min 30 sec
E) None

Joy Shaha wrote:Q. A & B run a kilometer and A wins by 25 sec. A & C run a kilometer & A wins by 275 m. when B & C run the same distance B wins by 30 sec. The time taken by A to run a Km is-
A) 2 min 25 sec
B) 2 min 50 sec
C) 3 min 20 sec
D) 3 min 30 sec
E) None

Let A = A's rate, B = B's rate and C = C's rate.

Time for A to run 1000 meters = d/r = 1000/A.
Time for B to run 1000 meters = d/r = 1000/B.
Time for C to run 1000 meters = d/r = 1000/C.

The time taken by A to run a Km is:
In other words, what is the value of 1000/A?

A & B run a kilometer and A wins by 25 sec.
Since B's time is 25 seconds longer than A's time, we get:
1000/B - 1000/A = 25.

B & C run the same distance B wins by 30 sec.
Since C's time is 30 seconds longer than B's time, we get:
1000/C - 1000/B = 30.

Adding together the two equations above, we get:
(1000/B - 1000/A) + (1000/C - 1000/B) = 25 + 30
1000/C - 1000/A = 55.

A & C run a kilometer & A wins by 275 m.
Implication:
A's time to run 1000 meters is equal to C's time to run 725 meters, with the result that A finishes 275 meters ahead of C.
Thus:
1000/A = 725/C.

Substituting 1000/A = 725/C into 1000/C - 1000/A = 55, we get:
1000/C - 725/C = 55
275/C = 55
275 = 55C
C = 275/55 = 5.

Substituting C=5 into 1000/A = 725/C, we get:
1000/A = 725/5 = 145 seconds = 2 minutes, 25 seconds.

The correct answer is A.