Problem Solving

coolhabhi wrote:C is 20% more efficient that A. A and B can do a piece of work in 16 days. B and C can do it in 15 days. in how many days can A alone do the work?

a)36
b)42
c)45
d)48
e)54

AnsD

I prefer Mitch's approach, but here's another way to do it...

For work questions, there are two useful rules:

Rule #1: If a person can complete an entire job in k hours, then in one hour, the person can complete 1/k of the job
Example: If it takes Sue 5 hours to complete a job, then in one hour, she can complete 1/5 of the job. In other words, her work rate is 1/5 of the job per hour

Rule #2: If a person completes a/b of the job in one hour, then it will take b/a hours to complete the entire job
Example: If Sam can complete 1/8 of the job in one hour, then it will take him 8/1 hours to complete the job.
Likewise, if Joe can complete 2/3 of the job in one hour, then it will take him 3/2 hours to complete the job.

Let's use these rules to solve the question. . . .

A and B can do a piece of work in 16 days.
So according to Rule #1, IN ONE DAY, A and B can complete 1/16 of the job
In other words, the COMBINED rate for A and B is 1/16 of the job EACH DAY
If we let A = the A's DAILY rate and let B = the B's DAILY rate, we can write: A + B = 1/16

B and C can do it in 15 days
So according to Rule #1, IN ONE DAY, B and C can complete 1/15 of the job
In other words, the COMBINED rate for B and C is 1/15 of the job EACH DAY
If we let C = the C's DAILY rate and let B = the B's DAILY rate, we can write: C + B = 1/15

So, we have:
C + B = 1/15
A + B = 1/16

Subtract the bottom equation from the top equation to get: C - A = 1/15 - 1/16 = 1/240

Finally, we have: C is 20% more efficient that A
So, we can write: C = 1.2A

Now take C - A = 1/240, and replace C with 1.2A
We get: 1.2A - A = 1/240
Simplify: 0.2A = 1/240
Or: (1/5)A = 1/240
Multiply both sides by 5 to get: A = 1/48
So, in ONE DAY, A can complete 1/48 of the job

Applying Rule #2 (above), we see that A can complete the entire job in 48 days.

Answer: D

Cheers,
Brent

Hi coolhabhi,

The first sentence in this question is oddly-worded, but this question can be solved by TESTing THE ANSWERS. Your ability to recognize the patterns involved and do some thick-looking (but ultimately straight-forward) calculations will also help you to work through this question in an efficient manner.

To start, this is a "Work Formula" question and often the 'numbers' involved in these types of questions are NOT 'nice' integers. Here though, EVERY number is an integer - and that's interesting. It's likely that the values for A, B and C are also all integers - and that pattern can help us to quickly eliminate answers.

Work = (X)(Y)/(X + Y) where X and Y are the individual times it takes to complete a job.

With the information in the prompt, we can create the following equations:

(A)(B)/(A+B) = 16
(B)(C)/(B+C) = 15

Those equations can be re-written as..
(A)(B) = 16A + 16B
(B)(C) = 15B + 15C

Again, the first sentence in the prompt is awkwardly worded, but we'll come back to that in a moment. The question asks for the value of A.

Let's TEST Answer B: 42

IF... A = 42...
42(B) = 16(42) + 16(B)
26B = 16(42)
13B = 8(42)

At this point, I'm going to stop. 13 does NOT divide evenly into 8(42), so B would be a non-integer and that does not seem likely to be correct in this prompt. Eliminate Answer B.

Let's TEST Answer D: 48

IF... A = 48...
48(B) = 16(48) + 16(B)
32B = 16(48)
2B = (48)
B = 24

Now we have an integer value for B, which is what we were looking for. Taking THAT value and plugging it into the other equation, we have...

24(C) = 15(24) + 15(C)
9C = 15(24)
3C = 5(24)
C = 5(8)
C = 40

Now we also have an integer value for C. The three values in this situation are...

A = 48
B = 24
C = 40

The value for A is 20% greater than the value for C, which is likely what the author of this question meant to imply when he/she wrote the first sentence. Thus, this is almost certainly the correct answer.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich