A takes 5 more days than B to do a certain job and 9 days more than C;

Bunuel, I think the answer options are numbers and hence need to be in ascending or descending order, as per the PS questions on the GMAT.

Let the total work = 1 unit.

Let time taken by B to complete the work be b days; then, time taken by A will be b + 5 days.

Let time taken by C to complete the work be c days; then, time taken by A will be c + 9 days.

Therefore, b + 5 = c + 9 OR c = b – 4.

A and B can together do the job in the same time as C. Therefore, combined rate of A and B is equal to rate of C.

\(\frac{1 }{ a}\) + \(\frac{1 }{ b}\) = \(\frac{1 }{ c}\)

Substituting the values of a and c, we have,

\(\frac{1 }{ (b + 5) }\)+\( \frac{1 }{ b}\) =\( \frac{1 }{ (b – 4)} \)

Taking the LCM on the LHS and simplifying, we have,

\(\frac{b + b + 5 }{ b (b + 5)}\) = \(\frac{1 }{ (b – 4)}\)

Cross multiplying and simplifying, we have,

2\(b^2\) – 3b – 20 = \(b^2\) + 5b

Simplifying further, we have,

\(b^2\) – 8b – 20 = 0.

Factorising, (b-10) (b+2) = 0.
Therefore, b = 10 since b = -2 is not a valid proposition for the time taken by B.

If b = 10, a = b + 5 = 15.
A would take 15 days to complete the job.

The correct answer option is E.