Time,Speed, Distance-Conceptual ; Difficulty: Hard

[GMATGuruNY]

Steve & Bill leave points A and B respectively at the same time and travel towards each other on the same road. They meet at point C, between A and B and proceed towards their respective destinations. After meeting Bill, Steve takes 16 minutes more to reach his destination. After meeting Steve, Bill takes 9 minutes more to reach his destination.
How long did Steve take to travel from A to B, if they did not spend any time at point C?

A) 25 mins
B) 23 mins
C) 28 mins
D) 144 mins
E) Cannot be determined with the given data.

Let t = the time for Steve and Bill to meet.
Let the distance = 1 mile.

Combined rate for Steve and Bill:
Since Steve and Bill together take t minutes to cover the 1 mile between them, the combined rate for Steve and Bill = 1/t.

Steve's rate:
Since Steve takes 16 more minutes after Steve and Bill meet, Steve's time to travel the entire 1 mile = t+16.
Thus, Steve's rate = 1/(t+16).

Bill's rate:
Since Bill takes 9 more minutes after Steve and Bill meet, Bill's time to travel the entire 1 mile = t+9.
Thus, Bill's rate = 1/(t+9).

Since the sum of Steve's rate and Bill's rate must be equal to their combined rate, we get:
1/(t+16) + 1/(t+9) = 1/t

[(t+9) + (t+16)] / [(t+9)(t+16)] = 1/t

(2t + 25)/(t² + 25t + 144) = 1/t

2t² + 25t = t² + 25t + 144

t² = 144

t = 12.

Thus, Steve's time to travel the entire 1 mile = t+16 = 12+16 = 28 minutes.

The correct answer is C.

[regor60]

(1) Steve's Rate: Ds/T = Db/16 > Ds and Db are Steve's and Bill's distance traveled upon meeting at C after Time T and equality reflects rate is the same after passing C

(2) Bill's Rate: Db/T = Ds/9

From (1) Ds/Db = T/16

From (2) Ds/Db = 9/T

Equating the two reflects T^2 = 144 > T = 12> Time to meet at C

Steve traveled for 16 more minutes, so total time is 12 + 16 = 28

[Matt@VeritasPrep]

Let's say the total distance = m miles and that it takes t minutes for the men to meet at point C.

We know that Steve travels the m miles in (16 + t) minutes, so Steve's rate = m/(16 + t).

We know that Bill travels the m miles in (9 + t) minutes, so Bill's rate = m/(9 + t).

We know that when the two men meet, Bill has traveled m/(9 + t) * t miles. We know that it will take Steve 16 minutes to travel that distance after the meeting, so

16 * Steve's Rate = m/(9+t) * t

16 * m/(16 + t) = m/(9+t) * t

16/(16 + t) = t/(9 + t)

16*(9 + t) = t*(16 + t)

0 = t² - 144

0 = (t + 12) * (t - 12)

We can only have a positive solution, so t = 12. Steve's total time = 16 + t, or 16 + 12, so C.