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Work/Time

by kartikshah » Fri Jul 27, 2012 5:57 pm
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

I assumed x as the time taken by Dave and Diana to work together. Then Dave will be x+20 and Diana will be x+5
Dave/Diana = (x+20)/(x+5)
= (x+5)+15/(x+5)

But How do I solve further?!
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by tisrar02 » Fri Jul 27, 2012 7:04 pm
Here's how I would solve this problem:

Set benchmarks

If Dave alone would work 20 hours more than them both together and Diana would work 5 hours more than them together, pick a number to represent both of them working together!

Both= 10
Dave= 20+10=30 hours alone
Diana= 10+5= 15 hours alone

Ratio of Dave/Diana= 30/15 or 2 to 1

Answer B
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by tutorphd » Fri Jul 27, 2012 7:14 pm
The above solution gives the correct result by accident, if you pick another time for them working together, you will get a different answer. The problem fixes the time they work together, so you cannot pick it (see at the end).

Using your solution notation:
x = time of them working together
x+20 = time of Dave; A = rate of Dave
x+5 = time of Diana; B = rate of Diana

Dave completes the same work as them working together: A(x+20) = (A+B)x -> 20A=Bx
Diana completes the same work as them working together: B(x+5) = (A+B)x -> 5B = Ax

Divide the two simplified equations to cancel x: 4A/B = B/A -> (B/A)^2 = 4 -> B/A = 2

That means Diana's rate is twice Dave's rate so Dave time is twice bigger than Diana's: 2:1.

The time they work together to complete the task is fixed by the problem to x = 5*B/A = 10 hours. That is why, if you 'pick' x to be 10 you will get the correct answer, otherwise you won't.
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by kartikshah » Fri Jul 27, 2012 7:22 pm
@tutorphd: Yes, I did plug in values and got different answers each time which is what made me stop. But after translating the problem into an equation, I couldn't think further.

Thanks for the solution! I understand the problem better now.
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by tisrar02 » Fri Jul 27, 2012 7:25 pm
Thank you for noting a flaw in my strategy. My initial reaction was to pick 10 I guess because I timed myself but I will be more careful next time
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by GMATGuruNY » Fri Jul 27, 2012 8:28 pm
kartikshah wrote: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
Since Dave takes 20 more hours and Diana takes 5 more hours, the difference between their times is 15 hours.
All of the values in the problem are multiple of 5.
It is almost certain that Dave's time alone and Diana's time alone are also multiples of 5, implying the following options for Dave : Diana:
20:5 = 4:1.
25:10 = 5:2
30:15 = 2:1.
35:20 = 7:4.
40:25 = 8:5.
Only the ratios in red are among the answer choices (A and B).
The first ratio is not viable: if Diana alone = 5, then the time for Dave and Diana together = 0.
Eliminate A.
The correct answer almost certainly is B.

Answer choice B: Dave = 30 hours, Diana = 15 hours, for a ratio of 2:1.
Let the job = 30 units.
Dave's rate = 30/30 = 1 unit per hour.
Diana's rate = 30/15 = 2 units per hour.
Time for Dave and Diana together = 30/(1+2) = 10 hours.
Success!
Dave's time alone is 20 more hours than the time for Dave and Diana together: 30-10 = 20.
Diana time alone is 5 more hours: 15-10 = 5.

The correct answer is B.
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