If Dave works alone he will take 20 more hours to complete a

Question

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

A. 4 : 1
B. 2 : 1
C. 10 : 1
D. 3 : 1
E. 1 : 2

KarishmaB · Accepted Answer

Note that Wd is the "time taken by Dave alone".
Also Wt is "time taken by both together"
We are given that Dave takes 20 more hours when he works alone. So Wd = Wt + 20

You use 1/Wd and 1/Wn when working with rates. 
The rate of work of Dave = 1/Wd
The rate of work of Diana = 1/Wn
Combined rate of work = 1/Wd + 1/Wn = 1/Wt
The combined rate of work will be more than the rate of work of each person alone. 
This is not correct: (1/Wd)- (1/Wt)= 1/20
1/Wt - 1/Wd = 1/Wn ----> We don't know what 1/Wn is. When Dave finishes the rest of the work, he takes 20 hrs. We can't say that Diana takes 20 hrs to finish the work alone.

Another Method :
Instead, say time taken together is T hrs. Dave working alone takes T+20 hrs and Diana working alone takes T+5 hrs.
So 1/(T+20) + 1/(T+5) = 1/T
(Rates are additive)
Here you get T = 10 so time taken by Dave/time taken by Diana = 30/15 = 2/1

Yet another Method: 
Or think logically
They take T hrs together to complete the work. Dave takes 20 extra hrs because Diana is not working with him. If she were working with him, in T hrs, she would have finished the work that Dave does in 20 hrs.
Time taken by Dave:Time taken by Diana = 20/T
Also, in 5 hrs, Diana completes the work that Dave was doing in T hrs.
So Time taken by Dave:Time taken by Diana = T/5

20/T = T/5
T = 10

time taken by Dave/time taken by Diana = 20/10 = 2/1

shahideh · Answer

we know that:

\frac{1}{T (diana)}+\frac{1}{T (dave)}=\frac{1}{T (together)}

\frac{1}{T+5}+\frac{1}{T+20}=\frac{1}{T}

==>  T=10

So,  \frac{T+20}{T+5}=\frac{30}{15}=2

Picking numbers works better and faster though.

meaningful · Answer

B. 2: 1

Wd = Time to complete when Dave works alone 
Wn = Time to complete when Donna works alone 
Wt = Time to complete when work together

Wd-Wt = 20 ---(1)
Wn-Wt = 5 ----(2)
1/Wd + 1/Wn = 1/Wt ---(3)

solve these 3 equations , you'll get (Wd)^2 = 20 (Wn)^2 = 5
so Wd/Wn = 2:1

mdfrahim · Answer

Lets say Dave takes 50 hrs.
then Dave + Diane takes 30 hrs.
Since Diane take 5 hrs less so she takes 25 hrs.

Ratio of Dave/Diane = 50/25 = 2/1.

IanStewart · Answer

I agree with your setup, but you should not at the end find that (Wd)^2 = 20 or that (Wn)^2 = 5 -- clearly in that case Diana doesn't take 15 hours less time than Dave. You should find that together they take 10 hours, that Diana alone takes 15, and that Dave alone takes 30.

meaningful is just using the standard combined rates formula here: if Dave finishes a job in x hours, and Diana finishes a job in y hours, and together they finish the job in t hours, then

1/x + 1/y = 1/t

Here, x = t + 20, and y = t + 5. Solve the following:

1/(t+20) + 1/(t+5) = 1/t

and you'll find that t = 10.

______

There's another way one could look at the problem. If together they take t hours, then Diana takes t+5, and Dave takes t+20. If we had two workers like Diana, they would take (t+5)/2 hours. We don't - Dave is slower than Diana - so t must be greater than (t+5)/2. Similarly, two workers like Dave would take (t + 20)/2 hours, but we don't have two workers like Dave - Diana is faster - so t must be less than (t + 20)/2. So,

(t+5)/2 < t < (t + 20)/2
5 < t < 20

Notice then that Diana's time (t+5) is between 10 and 25, Dave's time (t+20) is between 25 and 40, and the ratio of their times must be between 25/10 and 40/25. Only one answer is in that range.

Nivigmat · Answer

i dont understand why we are taking like this 
Wd-Wt = 20 ---(1)
Wn-Wt = 5 ----(2)

Coz I wrote this step like (1/Wd)- (1/Wt)= 1/20
similarly 1/Wn - 1/Wt = 1/5

i dont understand when to use 1/x + 1/y and when to take it as x+y

Please explain

TroyfontaineMacon · Answer

I am so FRUSTRATED :evil:

I keep getting \frac{1}{t+20} + \frac{1}{t+5}= \frac{1}{4} because \frac{1}{20}+ \frac{1}{5}=\frac{1}{4} none of these explanations are clicking furthermore, Dave has the slowest rate...and if it took them both 10 hours to complete the task, it took Dave alone 30 hours....Diana 15 hours that's why its 2:1 none of this crap you guys are saying...its logic based

KarishmaB · Answer

There is one thing wrong here:  \frac{1}{t+20} + \frac{1}{t+5}=  \frac{1}{4}

You are adding the rates of Dave and Diana which is perfectly fine but how do they add up to 1/4? They add up to 1/t because we are assuming that it takes them t hrs to complete the work together. You can write this: 
 \frac{1}{t+20} + \frac{1}{t+5}= \frac{1}{t} 
and then you will get t = 10

(This is 'Another Method' discussed in my post above)

ak1802 · Answer

Can't grasp what anyone is explaining here so here's my explanation:

To complete a job Dave must work: 1/(t+20)

To complete the same job, Diana must work: 1/(t+5)

Therefore 1/(t+20) + 1/(t+5) = 1/t Jobs

Solve the equation and you get that time = 10

1/(10+20) + 1/(10+5) = 1/10 job

Therefore it takes Dave time of 30 and Diana time of 15.

This works nicely and follows the standard Work/Rate formulas.

Kudos if this helped, quote & yell at me if this doesn't make sense

:lol:

bradburyj · Answer

For 1/(t+20)+1(t+5) = 1/t .... how does t = 10? I don't know how to calculate it.

ak1802 · Answer

Hey, I hope this helps:

+   = 1/t

(t+20+t+5) /   = 1/t

(t+5)*(t+20) = t(t+20) + t(t+5)

t^2 + 25t + 100 = t^2 + 20t + t^2 + 5t

100 = t^2

10 = t , plug 10 = t back into the original set-up.

shreyas · Answer

short cut for such problems:

time taken to complete the task working together = sqrt (20*5) = 10.
Dave takes 20 more hours = 10 + 20 = 30; Diane takes 5 more hours = 5 + 10 = 15.

Ratio = 30/15 = 2:1

KingMike782 · Answer

Could you please explain this step in detail?

Bunuel · Answer

\frac{t+20+t+5}{(t+5)*(t+20)} = \frac{1}{t} ;

Cross-multiply:  ((t+20)+(t+5))t=(t+5)*(t+20) ;

Expand the left-hand side:  t*(t+20)+t*(t+5)=(t+5)*(t+20) .

GMATinsight · Answer

Please find the solution as attached.

SVaidyaraman · Answer

1. Let x be the time required when both work together
2. Dave's time is x+20 and Diana's time is x+5
3. 1/(x+20) + 1/(x+5) = 1/x 
x=10. So the ratio is 30:15 or 2:1

crampipan · Answer

Does anyone know any similar shortcut had Diana taken 5 hours less to complete the task?

adewale223 · Answer

Let x be time required for both to complete task.
Working alone Dave will complete task in 20+x

And Diana in x+ 5 hours.

Therefore

1/20+x + 1/x+5 = 1/x

x2 +5x +20x + x2 = 20x + x2 + 100 + 5x

x2 = 100
x = 10.

So Dave: Diana

30:15
2:1.

Posted from my mobile device

samarpan.g28 · Answer

Let Dave and Diana together take x hrs i.e. their 1 hr work is 1/x.
If Dave works alone then he will take x+20 hrs i.e. Dave's 1 hr work is 1/(x+20).
Therefore, Diana's 1 hr work will be 1/x - 1/(x+20) = 20/x(x+20).
Also given, Diana alone will take x+5 hrs i.e. her 1 hr work will be 1/(x+5).

20/x(x+20) = 1/(x+5) or, x=10. So, Dave will take 10+20=30hrs to finish the complete work and Diana will take 10+5=15hrs to finish the complete work.
30/15=2/1. Option (B) is correct.

If Dave works alone he will take 20 more hours to complete a

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