diebeatsthegmat wrote:If Dev works alone he will take 20 more hours to complete a task than if he worked with Tina to complete the task. If Tina works alone, she will take 5 more hours to complete the complete the task, then if she worked with Dev to complete the task? What is the ratio of the time taken by Dev to that taken by Tina if each of them worked alone to complete the task?
Difference between time for Tina alone and time for Dev alone = 20-5 = 15 hours.
We can plug in times for Tina alone and for Dev alone until we find a combination that works.
Given that all the numbers in the problem are multiples of 5, the correct times for Tina alone and for Dev alone will probably also be multiples of 5.
Let Tina alone = 10 hours, Dev alone = 25 hours.
Let job = 50 units.
Rate for Tina alone = w/t = 50/10 = 5 units per hour
Rate for Dev alone = w/t = 50/25 = 2 units per hour.
Combined rate for Tina and Dev = 5+2 = 7 units per hour.
The job (50 units) is not a multiple of the combined rate (7).
Doesn't work.
Let Tina alone = 15 hours, Dev alone = 30 hours.
Let job = 30 units.
Rate for Tina alone = w/t = 30/15 = 2 units per hour
Rate for Dev alone = w/t = 30/30 = 1 unit per hour.
Combined rate for Tina and Dev = 2+1 = 3 units per hour.
Time for Tina and Dev together = w/r = 30/3 = 10 hours.
Success! Both conditions are satisfied:
Time for Tina alone - Time for Tina and Dev together = 15-10 = 5 hours.
Time for Dev alone - Time for Tina and Dev together = 30-10 = 20 hours.
Answer:
Time for Dev alone : time for Tina alone = 30:15 = 2:1.
Here's an algebraic approach:
Let t = time for Dev and Tina working together.
Rate for Dev and Tina together = 1/t
Rate for Dev alone = 1/(t+20)
Rate for Tina alone = 1/(t+5)
Rate for Dev alone + Rate for Tina alone = Rate for Dev and Tina together:
1/(t+20) + 1/(t+5) = 1/t
(t+5)+(t+20) / (t+20)(t+5) = 1/t
(2t + 25) / (t² + 25t + 100) = 1/t
Cross-multiplying, we get:
2t² + 25t = t² + 25t + 100
t² = 100
t = 10.
Time for Dev alone = t+20 = 10+20 = 30.
Time for Tina alone = t+5 = 10+5 = 15.
Ratio = 30:15 = 2:1.
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