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Manager  Joined: 12 Oct 2009
Posts: 99
Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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27 00:00

Difficulty:   95% (hard)

Question Stats: 59% (02:51) correct 41% (03:08) wrong based on 580 sessions

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Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10
Math Expert V
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Posts: 53709
Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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10
14
asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

Responding to a pm.

First 2 days all three of them worked together, thus they did 2*(1/10 + 1/12 + 1/15) = 1/2 of the work.

Last 3 days only Carla worked, thus she did 3/15 = 1/5 of the work.

1 - 1/2 - 1/5 = 3/10 of the work was done by Ben and Carla: (time)*(combined rate)=(job done) --> t*(1/12 + 1/15) = 3/10 --> t = 2 days. So, we have that Ben and Carla worked together for 2 days.

Total days = 2 + 3 + 2 = 7.

Hope it's clear.
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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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4
1
Let the work be completed in 't'.

Then again "Rate * Time = Work"

Rate(A)) = 1/10
Rate(B)=1/12
Rate(C)=1/15

Since A worked for 2 Days Work done by A= 2/10
Since B worked for 3 Days before work was completed work done B= (t-3)/12
Since C worked for full number of days = t/15

Adding them gives total work which is 1 unit.

2/10 + (t-3)/12 + t/15 = 1

Hence t=7
##### General Discussion
VP  Joined: 05 Mar 2008
Posts: 1398
Re: Word problem-work rate  [#permalink]

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3
2
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

B. 7

1/10 + 1/12 + 15 = 15/60 of the work done each day.

60 - 15/60-15/60 = 30/60 (abel then leaves)

Ben and carla working together finish 9/60 each day. Carla alone finished 4/60 each day so I figured the amount carla finishes alone will be a multiple of 4

30/60 - 9/60 = 21/60 (3 days total)
21/60-9/60 = 12/60 (4 days total; and coincidentally a multiple of 4) so assume carla works alone after this
12/60-4/60 = 8/60 (5 days)
8/60 - 4/60 = 4/60 (6 days)
4/60-4/60 = 0 (7 days)
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Re: Word problem-work rate  [#permalink]

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1
(1/A + 1/B + 1/C)*2 + (1/B + 1/C) ( N-2-3 ) + (1/C)*3 = 1

Solve for N, we get N = 7
Economist GMAT Tutor Instructor Joined: 01 Oct 2013
Posts: 69
Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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3
1
asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

Not to sound like a broken record from some of my earlier posts, but, worst case, you could always plug in answer choices for this problem.

Start with C and you get 2/10 of a job from Abel (which, notice, will always be the case), (8-3)/12 from Ben, and 8/15 from Carla.

LCM and add these up, and you get 12/60+25/60+32/60. Too much!

Do the same with B. Abel stays at 2/10, Ben is now 4/12, and Carla is 7/15. So, 12/60+20/60+28/60 = 60/60.

This approach could take longer in some circumstances, but it's always a default strategy where you have answer choices like these and no idea how to proceed.
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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

Abel in the 2 days that he worked completed 1/5 of the job = 4/5 remains
Then if Ben had to leave 3 days before the completion, this means that Carla had to work alone for these 3 days in which she completed 1/5 of the job.

Now together, Ben and Carla completed the job in (1/12 + 1/15)(t) = 3/5

3/20 (t) = 3/5 ---> t = 4

Therefore, these 4 days worked plus the 3 days that Carla had to work by herself add to 7 days

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Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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Bunuel wrote:
asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

Responding to a pm.

First 2 days all three of them worked together, thus they did 2*(1/10 + 1/12 + 1/15) = 1/2 of the work.

Last 3 days only Carla worked, thus she did 3/15 = 1/5 of the work.

1 - 1/2 - 1/5 = 3/10 of the work was done by Ben and Carla: (time)*(combined rate)=(job done) --> t*(1/12 + 1/15) = 3/10 --> t = 2 days. So, we have that Ben and Carla worked together for 2 days.

Total days = 2 + 3 + 2 = 7.

Hope it's clear.

Bunuel's answer is the quickest way to think about the structure of the solution. Just want to add a small tip: Because right off the bat, you notice you're adding 3 fractions. Instead of doing the usual cross multiply trick to add fractions, I would find the LCM immediately by sketching out a quick venn diagram:

10 = 2*5
12 = 2*2*3
15 = 3*5

2 is common, 3 is common, and 5 is common - leftover is just one 2. Therefore, the LCM is 2*3*5*2 = 60. Rewrite all of the fractions with a denominator of 60 and this problem can be solved in under 2 minutes.

All working together = 6/60 + 5/60 + 4/60 = 15/60 for 2 days = 30/60.
B and C working together = 5/60 + 4/60 = 9/60 for (x) days.
C working alone = 4/60 for 3 days = 12/60.

30 + 9x+12 = 60; 42 + 9x = 60; 9x = 18; x =2.

Therefore, total number of days = x+2+3 = 7 days.

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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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5
1
asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

Another option is to convert it to units of work if you don't want to work with fractions.

Say, the work involves 60 units (LCM of 10, 12 and 15).
So Abel does 60/10 = 6 units a day, Ben does 60/12 = 5 units a day and Carla does 60/15 = 4 units a day.
Together, they do 6+5+4 = 15 units a day.

In 2 days, they complete 15*2 = 30 units and are left with 30 units. Then only Ben and Carla are working and doing 5+4 = 9 units a day.
The last 3 days Carla works alone and does 4*3 = 12 units of the 30 units so Ben and Carla together do 30 - 12 = 18 units. Hence Ben and Carla work together in the middle at the rate of 9 units per day for 18/9 = 2 days.

The work lasts for 2 + 2 + 3 = 7 days.

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Director  Joined: 07 Aug 2011
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Concentration: International Business, Technology
GMAT 1: 630 Q49 V27 Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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1
asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

speed of A=1.2B =1.5C .

combined speed of B and C = $$\frac{1}{B} +\frac{1}{C} = \frac{B+C}{BC} = \frac{3}{20}$$
$$1-(\frac{1}{5} +\frac{1}{6}+ \frac{2}{15} + \frac{3}{15} )$$ = work done together by B and C= $$\frac{9}{30}.$$

so time take by B and C together (in A's absence) = 2 days

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2
Amount of work Carla completed + Amount of work Abel completed + Amount of work Ben completed = Total amount of work completed
t/15 + 2/10 + (t-3)/12 = 1
t = 7

The hardest part of this problem (at least for me) was interpreting the amount of time Ben spent working.
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Posts: 1158
Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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1
let d=total days
2(15/60)+(d-5)(9/60)+3(4/60)=1
d=7
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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

Let the complete work be 60 units..
Per day..
A does = 60/10 = 6 units
B = 5 units
C = 4 units

The timeline of the work is

ABC for the first 2 days
Work done = 2(6+5+4) = 30 units

BC for some days..let it be x
Work done = x(5+4) = 9x

C alone for last three days
Work done = 3(4) = 12 units

So..

30 + 9x + 12 = 60..

x = 2 days..
and total days = 2 + 2 + 3 = 7 days(B)

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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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easy way to solve the problem:-

Assume T :- time taken for all the three to complete the work

GENERAL CONCEPT:-
abel's rate will be T/10 (Abel's rate T/10 - total time taken by all the three (T)/abel's time to complete(10))

Ben's rate will be :T/12

Carla's rate will be : T/15

Given in problem:-

But because Abel leaves in 2 days after the start of work ,his rate will be : 2/10

Ben leaves 3 days before the completion of work so his rate will be :(T-3)/12

carla stays till the end of completion so her rate will be : T/15

all of them complete 1 task

so equation is 2/10 + (T-3)/12 + T/15 = 1

Hence T = 7
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Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

1. (time worked by Abel)/(time taken by Abel working alone)+ (time worked by Ben)/(time taken by Ben working alone) + (time worked by Carla)/(time taken by Carla working alone) = 1

2. 2/10 + (x-3)/12 + x/15 =1

So x=7.
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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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Let the work be completed in 't'.

Then again "Rate * Time = Work"

Rate(A)) = 1/10
Rate(B)=1/12
Rate(C)=1/15

Since A worked for 2 Days Work done by A= 2/10
Since B worked for 3 Days before work was completed work done B= (t-3)/12
Since C worked for full number of days = t/15

Adding them gives total work which is 1 unit.

2/10 + (t-3)/12 + t/15 = 1

Hence t=7
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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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When Abel Ben and Carla started together, they finished half of the job in 2 days. So the part left is 1/2. This should be done by Ben and Carla only; and they should take 10/3 days to complete the remaining half portion, but Ben left 3 days prior to completion, that means Ben and Carla worked for only 1/3 days. In this 1/3 days Ben and Carla together finished 1/20 part, so the remaining 19/20 part of the work has to be done by the Carla alone. Carla should take 14 and 1/4 days. Hence the total duration of the work = 2+1/3+14 and 1/4 days.

Every answer so far shown here assumed that Carla should finish the job in 3 days after Ben left, but when Abel left, Ben and Carla started together, the time taken to complete the remaining job will be changed, so "T" is not fixed here. Every time "T" changes when some one leaves the job, so the answer 7 days is quite confusing.

Can any body explain with proper logic about the concept of "T" ?
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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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piyali1979g wrote:
When Abel Ben and Carla started together, they finished half of the job in 2 days. So the part left is 1/2. This should be done by Ben and Carla only; and they should take 10/3 days to complete the remaining half portion, but Ben left 3 days prior to completion, that means Ben and Carla worked for only 1/3 days. In this 1/3 days Ben and Carla together finished 1/20 part, so the remaining 19/20 part of the work has to be done by the Carla alone. Carla should take 14 and 1/4 days. Hence the total duration of the work = 2+1/3+14 and 1/4 days.

Every answer so far shown here assumed that Carla should finish the job in 3 days after Ben left, but when Abel left, Ben and Carla started together, the time taken to complete the remaining job will be changed, so "T" is not fixed here. Every time "T" changes when some one leaves the job, so the answer 7 days is quite confusing.

Can any body explain with proper logic about the concept of "T" ?

Hi piyali1979g ,

You must know, Work=Rate*Time and Total work =1

Working sequence:-

Work sequence Who worked-------Time(in no of days)=How many days?---------------Rate-------------------------Work done
1---------------- A+B+C--------------- First 2 days-----------------------------------------1/10+1/12+1/15=1/4---------2*1/4=1/2
2------------------B+C---------------------x(say)----------------------------------------------1/12+1/15---------------------x*(1/12+1/15)
3-------------------C----------------------Last 3 days--------------------------------------------1/15----------------------------3*1/15=1/5

Total work=1
So, $$\frac{1}{2}+x*(\frac{1}{12}+\frac{1}{15})+\frac{1}{5}=1$$
Or, $$x*\frac{9}{60}=\frac{1}{2}-\frac{1}{5}=\frac{3}{10}$$
Or, $$x=\frac{3}{10}*\frac{60}{9}= 2$$

So, total no of days taken by ABC to complete the work=$$2+x+3=2+2+3=7$$
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Re: Abel can complete a work in 10 days, Ben in 12 days and Carl  [#permalink]

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asterixmatrix wrote:
Abel can complete a work in 10 days, Ben in 12 days and Carla in 15 days. All of them began the work together, but Abel had to leave after 2 days and Ben 3 days before the completion of the work. How long did the work last?

A. 6
B. 7
C. 8
D. 9
E. 10

Let the total work be 60 units (LCM of 10,12 & 60)

Thus, the efficiency of A = 6 units/day ; efficiency of B = 5 units/day & efficiency of C = 4 units/day

Let the total time taken for completion of the work be "t"

$$2*6 + 5 ( t - 3) + 4*t = 60$$

Or, $$12 + 5t -15 + 4t = 60$$

Or, $$9t = 63$$

So, $$t = 7$$, Answer must be (B) 7 days.
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# Abel can complete a work in 10 days, Ben in 12 days and Carl

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