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A, B and C working alone can complete a work in 24

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A, B and C working alone can complete a work in 24

by vinni.k » Sat Sep 07, 2013 7:48 am
A, B and C working alone can complete a work in 24, 32, and 48 days respectively. "A" started the work and "B" joined him after 4 days and "C" joined them after 4 more days. If they were paid $10,800 for the work done, how much did "A" get ?

(A) 2800
(B) 3600
(C) 5400
(D) 6400
(E) 7600

OA is D

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by Anindya Madhudor » Sat Sep 07, 2013 8:32 am
In 4 days, A completes 4/24 part of the whole work. Remaining work is 20/24 after 4 days.
When B joins after 4 days, A and B together do (1/24 + 1/32)*4 = 7/24. So remaining work after 8 days is (20/24 -7/24) =13/24.
When C joins after another 4 days, A, B and C can do (1/24 + 1/32+ 1/48) = 3/32 in 1 day. So, they need (32/3 * 13/24) = 52/9 days to complete the task.
This means, A works for a total of (4+4+52/9) days and in this he completes (4+4+52/9)*(1/24)=31/54 part of the whole task. So, A gets 10,800 * (31/54) = 6200.
I think choice D should be 6200.
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by Brent@GMATPrepNow » Sat Sep 07, 2013 10:18 am
vinni.k wrote:A, B and C working alone can complete a work in 24, 32, and 48 days respectively. "A" started the work and "B" joined him after 4 days and "C" joined them after 4 more days. If they were paid $10,800 for the work done, how much did "A" get ?

(A) 2800
(B) 3600
(C) 5400
(D) 6400
(E) 7600
Another approach is to assign the entire job a certain number of units.
Since the least common multiple of 24, 32 and 48 is 96, let's say that the entire job consists of 96 work units.

So, if A can complete the entire job (i.e., 96 work units) in 24 days, then A's rate is 4 units/day
Likewise, if B can complete the entire job (i.e., 96 work units) in 32 days, then B's rate is 3 units/day
And if C can complete the entire job in 48 days, then C's rate is 2 units/day

A works alone for 4 days. So, during this time, A completes 16 units of work.
This leaves us with 80 work units remaining (96 - 16 = 80)

A & B work together for 4 days. Their combined rate is 7 units/day (4+3=7). So, during this time, A+B complete 28 units of work.
This leaves us with 52 work units remaining (80 - 28 = 52)

A, B & C work together to complete job. Their combined rate is 9 units/day (4+3+2=9).
Since there are 52 work units remaining in the job, the number of days to complete the job = 52/9

IMPORTANT: 52/9 is not a nice number. However, since the answer choices are somewhat spread apart, I'm going to be somewhat aggressive with my estimation/rounding.

So, let's say that 52/9 = 6 (close enough).
This means that it takes A,B and C about 6 days to complete the rest of the job.

So, in total, A worked for approximately 14 days (4+4+6=14)
Since A works at a rate of 4 units/day, A completed a total of 56 work units (since 14x4=56).

The ENTIRE job was 96 work units. So, A did 56/96 of the work.
This means that A should get 56/96 of the $10,800 payment.
(56/96)(10,800) = (7/12)(10,800)
= $6300
Of course, none of the answer choices match $6300, but that's because we did some estimating/rounding earlier.
The closest answer choice is $6400, so the correct answer must be D

Cheers,
Brent
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by vinni.k » Sat Sep 07, 2013 11:02 am
Thanks Brent. Your solution is crystal clear, and i also believe that the word "approx" must be mentioned here.

Vinni
Brent@GMATPrepNow wrote:
Another approach is to assign the entire job a certain number of units.
Since the least common multiple of 24, 32 and 48 is 96, let's say that the entire job consists of 96 work units.

So, if A can complete the entire job (i.e., 96 work units) in 24 days, then A's rate is 4 units/day
Likewise, if B can complete the entire job (i.e., 96 work units) in 32 days, then B's rate is 3 units/day
And if C can complete the entire job in 48 days, then C's rate is 2 units/day

A works alone for 4 days. So, during this time, A completes 16 units of work.
This leaves us with 80 work units remaining (96 - 16 = 80)

A & B work together for 4 days. Their combined rate is 7 units/day (4+3=7). So, during this time, A+B complete 28 units of work.
This leaves us with 52 work units remaining (80 - 28 = 52)

A, B & C work together to complete job. Their combined rate is 9 units/day (4+3+2=9).
Since there are 52 work units remaining in the job, the number of days to complete the job = 52/9

IMPORTANT: 52/9 is not a nice number. However, since the answer choices are somewhat spread apart, I'm going to be somewhat aggressive with my estimation/rounding.

So, let's say that 52/9 = 6 (close enough).
This means that it takes A,B and C about 6 days to complete the rest of the job.

So, in total, A worked for approximately 14 days (4+4+6=14)
Since A works at a rate of 4 units/day, A completed a total of 56 work units (since 14x4=56).

The ENTIRE job was 96 work units. So, A did 56/96 of the work.
This means that A should get 56/96 of the $10,800 payment.
(56/96)(10,800) = (7/12)(10,800)
= $6300
Of course, none of the answer choices match $6300, but that's because we did some estimating/rounding earlier.
The closest answer choice is $6400, so the correct answer must be D

Cheers,
Brent
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by GMATGuruNY » Sun Sep 08, 2013 7:15 am
The solution below is similar to Brent's, but with one suggestion:
To make the math easier, we can divide all of the times by 4:
A, B and C working alone can complete a work in 6, 8, and 12 days respectively. "A" started the work and "B" joined him after 1 day and "C" joined them after 1 more day. If they were paid $10,800 for the work done, how much did "A" get ?

(A) 2800
(B) 3600
(C) 5400
(D) 6200
(E) 7600
Let the job = the LCM of 6, 8 and 12 = 24 units.
Pay per unit = 10800/24 = $450.

A's rate = w/t = 24/6 = 4 units per day.
B's rate = w/t = 24/8 = 3 units per day.
C's rate = w/t = 24/12 = 2 units per day.

Work produced by A on the first day = 4 units.
Work produced by A on the second day = 4 units.
Work produced by B on the second day = 3 units.

Remaining work = 24-4-4-3 = 13 units.
When A, B and C work together, their combined rate = 4+3+2 = 9 units, of which 4 will be produced by A.
Thus, A will produces 4/9 of the remaining 13 units:
(4/9) * 13 = 52/9 ≈ 6.

Total number of units produced by A ≈ 4+4+6 = 14.
Thus:
A's pay ≈ 14*450 ≈ 6300.

The correct answer is D.

Please note that A's exact pay = $6200 (not $6400 as indicated in the problem as posted above):
(4 + 4 + 52/9) * 450 = 6200.
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