Work rate

If Dave works alone he will take 20 more hours to complete a task than if he worked with Diana to complete the task. If Diana works alone, she will take 5 more hours to complete the complete the task, then if she worked with Dave to complete the task? What is the ratio of the time taken by Dave to that taken by Diana if each of them worked alone to complete the task?


A. 4 : 1


B. 2 : 1


C. 10 : 1


D. 3 : 1


E. 1 : 2

TIME and RATE are RECIPROCALS.
This reciprocal relationship suggests another way to PLUG IN THE ANSWERS.
When the correct answer choice is plugged in, Dave's time alone must be 20 hours longer than the time for Dave and Diana working together.

Answer choice D: 3:1
Since (Dave's time)/(Diana's time) = 3/1, (Dave's rate)/(Diana's rate) = 1/3.
Implication:
If Dave produces 1 unit per hour, then Diana produces 3 units per hour, for a total of 4 units per hour when they work together.

Let the job = the LCM of 1, 3 and 4 = 12 units.
Since Dave's rate = 1 unit per hour, the time for Dave alone = 12/1 = 12 hours.
Since Diana's rate = 3 units per hour, the time for Diana alone = 12/3 = 4 hours.
Since their combined rate = 4 units per hour, the time for Dave and Diana working together = 12/4 = 3 hours.
Here, Diana's time alone (4 hours) is only 1 HOUR LONGER than the time for Dave and Diana working together (3 hours).

Since Diana's time alone must be 5 HOURS LONGER than the time for Dave and Diana working together, all of the times above must be multiplied by a FACTOR OF 5, implying the following values:
Time for Dave alone = 12*5 = 60 hours.
Time for Diana alone = 4*5 = 20 hours.
Time for Dave and Diana working together = 3*5 = 15 hours.
Doesn't work:
Here, Dave's time alone (60 hours) is not 20 hours longer than the time for Dave and Diana working together (15 hours).
Eliminate D.

Answer choice B: 2:1
Since (Dave's time)/(Diana's time) = 2/1, (Dave's rate)/(Diana's rate) = 1/2.
Implication:
If Dave produces 1 unit per hour, then Diana produces 2 units per hour, for a total of 3 units per hour when they work together.

Let the job = the LCM of 1, 2 and 3 = 6 units.
Since Dave's rate = 1 unit per hour, the time for Dave alone = 6/1 = 6 hours.
Since Diana's rate = 2 units per hour, the time for Diana alone = 6/2 = 3 hours.
Since their combined rate = 3 units per hour, the time for Dave and Diana working together = 6/3 = 2 hours.
Here, Diana's time alone (3 hours) is only 1 HOUR LONGER than the time for Dave and Diana working together (2 hours).

Since Diana's time alone must be 5 HOURS LONGER than the time for Dave and Diana working together, all of the times above must be multiplied by a FACTOR OF 5, implying the following values:
Time for Dave alone = 6*5 = 30 hours.
Time for Diana alone = 3*5 = 15 hours.
Time for Dave and Diana working together = 2*5 = 10 hours.
Success!
Here, Dave's time alone (30 hours) is 20 hours longer than the time for Dave and Diana working together (10 hours).

The correct answer is B.