kartikc11 wrote:Machines X and Y work at their respective constant rates. How many more hours does it take Machine Y, working alone, to fill a production order of a certain size than it takes Machine X, working alone?

1) Machine X and Y, working together, fill a production order of this size in two thirds the time that Machine X, working alone, does.
2) Machine Y, working alone, fills a production order of this size in twice the time it takes Machine X, working alone, does.

Statement 1:
Time and rate are RECIPROCALS.
Since X and Y together take 2/3 as long as X alone, X and Y together work 3/2 as fast as X alone.
Thus, if X's rate alone = 2 units per hour, then X and Y's combined rate = (3/2)(2) = 3 units per hour, implying that Y's rate alone = 3-2 = 1 unit per hour.

Case 1: Job = 6 widgets
At a rate of 2 units per hour, the time for X alone = w/r = 6/2 = 3 hours.
At a rate of 1 unit per hour, the time for Y alone = w/r = 6/1 = 6 hours.
In this case, Y's time - X's time = 6-3 = 3 hours.

Case 2: Job = 600 widgets
At a rate of 2 units per hour, the time for X alone = w/r = 600/2 = 300 hours.
At a rate of 1 unit per hour, the time for Y alone = w/r = 600/1 = 600 hours.
In this case, Y's time - X's time = 600-300 = 300 hours.

Since the time difference can be different values, INSUFFICIENT.

Cases 1 and 2 also satisfy statement 2:
In each case, Y's time is twice X's time.
Thus, even when both statements are satisfied, the time difference can be different values, implying that the two statements combined are INSUFFICIENT.

The correct answer is E.

thank you! That makes perfect sense