Say there are 72 units of work.

A and B together do 72/(72/5) = 5 units every day.
B and C together do 72/12 = 6 units every day.

(A worked for 8 days + B worked for 8 days) + (B worked for 4 days + C worked for 4 days) + C worked for 2 days = 72

5 * 8 + 4 * 6 + C worked for 2 days = 72

So C does 4 units of work every day. Then B does 2 units every day and A does 3 units every day.

Time taken by A = 72/3 = 24 days, by B = 72/2 = 36 days, by C = 72/4 = 18 days

Answer (A)
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The answer is A.

If I select option A --> A = 24 days, B = 36 days, C = 18 days.
Work rate is: A=1/24, B=1/36, C=1/18 task/day

Now taking this statement: After A worked for 8 days, B for 12 days C takes up and finishes it alone in 6 days.

A worked for 8 days --> A finished 8/24=1/3 of the total work
B worked for 12 days --> B finished 12/36=1/3 of the total work
C worked for 6 days --> C finished 6/18=1/3 of the total work

Other options won't fit. Correct me if I am wrong.

This seems a very decent approach!

I formed the equation

72a/5 + 72b/5 = 12b + 12c = 8a + 12b + 6c

Now we have to extract 3 equations from the above and solve for 3 variables. That gets very complicated.

Set A, B, C as the rates and treat an entire piece as 1 job. Then we have the following equations:

(1) \(A + B = \frac{5}{72}\)

(2) \(B + C = \frac{1}{12}\)

(3) \(8A + 12B + 6C = 1\)

First do (3) - 8*(1) to get: \(4B + 6C = 1 - \frac{5}{9} = \frac{4}{9}\)

Subtract 4*(2) from the above to get: \(2C = \frac{4}{9} - \frac{1}{3} = \frac{1}{9}\)

Thus \(C = \frac{1}{18}\) which means C takes 18 days alone. Then we can plug this in (2) to get \(B = \frac{1}{12} - \frac{1}{18} = \frac{1}{36}\).

Now we can confirm the answer must be A.

Ans: A
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For this question, we can benefit from first looking at the answer choice. All the option has A = 24 days. So, we have the answer for A already.

Now we know that (1/A) + (1/B) = (5/72) --> Put A = 24 and solve for B --> B = 36 days
Simillarly, (1/B) + (1/C) = (1/12) --> Put B = 36 and solve for C --> C = 18 days.

Answer is A

I totally agree with the explanations given above. Just thought of giving another approach by using the answer choices.

If you analyze the answer choices, Did you notice something interesting?

A = 24 days is common in all answer choices.

My next question is can we use it to solve the question? Yes. you can.
we can confirm that A will take 24 days to finish the work alone. Using this data will help you to save some time.

LCM approach is more convenient here. Lets assume the total work = LCM ( 72,12,24) = 72 units.

Since we are asked to calculate the time taken by A, B, and C to finish the work alone, we need to calculate the individual rate of A, B and C respectively.
By using the data from answer choices, we can easily find the rate of A.

Rate of A = 72/24 = 3 units / day . As A can complete the work in 24 days working alone.


Rate of A and B = 72/(72/5) = 5 units/day

Rate of B = 5 - 3 = 2 units/day

Rate of B and C = 72/12 = 6 units /day.

Rate of C = 6 - 2 = 4 units /day

Since we know the individual rates of B and C, it's very easy to calculate the time taken by each of them to complete the work alone.

Time taken by B alone = Total work / Rate of B = 72/2 = 36 days

Time taken by C alone = Total work / Rate of C = 72/4 = 18 days.

Option A is the answer.

Remember, you can always use the answer choices to your advantage.

Thanks,
Clifin J Francis,
GMAT QUANT SME
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@Experts
How to come up with this equation/ How it is derived?

(A worked for 8 days + B worked for 8 days) + (B worked for 4 days + C worked for 4 days) + C worked for 2 days = 72

I understood the total work is 72 units but where it is mentioned that A and B together worked for 8 days, B worked for 12 days and in total and I am confused in the question stem when C was introduced?

this took me a looooong while. but i eventually went wit hmy instincts.. my take away here.

so we have

(A+B) = (5/72) ..............................................f(1)
(C+B)=(1/12)................................................f(2)
A works for 8 days with B i.e. 8*(A+B)
B works alone for a further 4 hours i.e. 4(B)
C works alone for 6 hours 6(C)

Which gives us 8(A+B)+4(B)+6(C)=.................f(3)
Sub f(1) into f(3)

8(5/72)+4(B)+6(C)=1
simplify to 4(B)+6(C)=(4/9)............................f(4)

from f(2) B=(1/12)-C.....................................f(5)
sub f(4) in into f(5)

4[(1/12)-C]+6C=(4/9), which simplifies to C=1/18, i.e. rate for C meaning C alone takes 18 days

sub C=1/18 in to f(2) for B=1/36, i rate for B alone meaning B takes 36 days

sub B=1/36 into f(1) for a=1/24. i.e. rate for A , meaning A takes 24 days

Answer A.

..............................fin.............