another work problem

ace_gre wrote:Deep and Ana decided to work together & complete the construction of a road.However, as Deep went out of town, Ana started work alone and finished 1/4 th of the road and took 'x' days more than what they would have taken if they worked together. After this, Ana fell ill and Deep working alone, completed the construction of 1/3 of the remaining road and took 25/16 x days more than what they would have taken if they had worked together.Finally both of them worked together and completed the remaining road the whole process taking a total of 121 days.
In how many days can deep alone complete the construction of entire road?
1. 180 days
2. 194
3. 160

Sorry no OA and taken from another forum. Please help solve. Thanks!

This problem is completely absurd; there is no way to solve it without some kind of computational assistance. If I saw it on a real GMAT, I'd try to find a shortcut: presumably x needs to be an integer, and so does (25/16)x, so my first assumption would be that x is divisible by 16. Since x needs to be quite small here (since the entire process only takes 121 days), I'd guess that x=16 and proceed from there. On a real GMAT question, that would almost certainly lead to the correct answer, but here that doesn't work.

You can do the problem algebraically as follows: we know that if they worked together the entire time, they would have saved x + (25/16)x days, or (41/16)x days. So

A+D do 1 job in 121 - (41/16)x days
A does 1/4 job in (121 - (41/16)x)/4 + x days --> A does 1 job in 121 - (41/16)x + 4x = 121 + (23/16)x days
D does 1/3 job in (121 - (41/16)x)/3 + (25/16)x days --> D does 1 job in 121 - (41/16)x + (75/16)x = 121 + (34/16)x days

Summarizing:

A+D do 1 job in 121 - (41/16)x days
A does 1 job in 121 + (23/16)x days
D does 1 job in 121 + (34/16)x days

You can then plug these values into the rates formula, but this produces an absolutely ridiculous quadratic equation, and the only way to get an answer is with a calculator. In the end, you find that x is approximately 17.555, that Deep takes roughly 158.3 days alone, that Ana takes roughly 146.2 days alone, and that together they would take roughly 76 days. You'd never see a question like this on the real GMAT.