Working Problem (need help)

[mariofelixpasku]

Working alone Thomas can do a job in 8 hours longer than if both Thomas and Kevin worked together. If Kevin worked alone, he would take 4.5 hours more to complete the job than when he works with Thomas. Which of the following could be the amount of hours if they work together?
a) 4
b) 6
c) 8
d) 12
e) 15

Answer: B

[GMATGuruNY]

Working alone Thomas can do a job in 8 hours longer than if both Thomas and Kevin worked together. If Kevin worked alone, he would take 4.5 hours more to complete the job than when he works with Thomas. Which of the following could be the amount of hours if they work together?
a) 4
b) 6
c) 8
d) 12
e) 15

We can plug in the answers, which represent the time needed when Thomas and Kevin work together.

Answer choice C: 8 hours
Since Thomas working alone needs 8 more hours, Thomas's time alone = 8+8 = 16 hours.
In this case:
(Thomas and Kevin's time together) : (Thomas's time alone) = 8:16 = 1:2.
Since time and rate are RECIPROCALS:
(Thomas and Kevin's combined rate) : (Thomas's rate alone) = 2:1.
Thus, Kevin's rate alone = (combined rate) - (Thomas's rate) = 2-1 = 1 unit per hour.

Here, Thomas and Kevin are working at the SAME RATE.
Not possible.
Since Thomas's time alone must be LONGER than Kevin's time alone, Thomas must work MORE SLOWLY.
Thus, the ratio of the additional time needed when Thomas works alone (8 hours) to the time needed when the two work together (8 hours in answer choice C) must INCREASE.
Thus, the time needed when the two work together must be LESS than 8 hours.
Eliminate C, D, and E.

Answer choice B: 6 hours
Since Thomas working alone needs 8 more hours, Thomas's time alone = 6+8 = 14 hours.
In this case:
(Thomas and Kevin's time together) : (Thomas's time alone) = 6:14 = 3:7.
Since time and rate are RECIPROCALS:
(Thomas and Kevin's combined rate) : (Thomas's rate alone) = 7:3.
Thus, Kevin's rate alone = (combined rate) - (Thomas's rate) = 7-3 = 4 units per hour.

Since their combined rate = 7 units per hour, the amount of work completed by Thomas and Kevin in 6 hours = r*t = 7*6 = 42 units.
Time for Kevin alone to produce 42 units = w/r = 42/4 = 10.5 hours -- 4.5 hours longer than the time needed when Thomas and Kevin work together.
Success!

The correct answer is B.

Algebraically:
Let t = the time needed when Thomas and Kevin work together.

Rate for Thomas alone = 1/(t+8).
Rate for Kevin alone = 1/(t+4.5).
Rate for Thomas and Kevin together = 1/t.

Since the sum of Thomas's rate and Kevin's rate is equal to their combined rate, we get:.
1/(t+8) + 1/(t+4.5) = 1/t
(t+4.5) + (t+8) / (t+8)(t+4.5) = 1/t
(2t + 12.5) / (t² + 12.5t + 36) = 1/t.

Cross-multiplying, we get:
2t² + 12.5t = t² + 12.5t + 36
t² = 36
t = 6.