I've noticed that one of the common DS traps is to give you an equation that seems like it needs two equations to solve but it really only needs one. For example:
Word translation = 31X+25Y=?
1)ratio is of X to Y is blah blah
2)sum is 280
Answer here is B. It comes out to 5 each. My question is on whether there is a shortcut to know if there is only one answer. testing all of the cases for a sum of 280 takes a lot of time. Can you be assured there is only one solution to the problem (if there is one at all, or course) if the coefficients dont share any factors?
First of all, 31*x+25*y=280 has infinitely many
solutions, not just one.
It will have only one solution x=y=5 if x and y are restricted to positive integers
only. In this case 31*x+25*y=280 is a special kind of equation (Diophantine equation) and you can encounter them in problems where x and y must be positive (nonnegative) integers only, for example when they represent # of people/items. Now, trial and error along with some common sense is pretty much the only way you should approach such kind of problems on the GMAT. You won't get some very tough numbers to manipulate with or there will be some shortcut available, based on multiples concept or on the answer choices. So generally you would have to try just couple of values to get the answer.
For example for this problem since 25*y and 280 are both multiples of 5, then 31y, or simply y, is also a multiple of 5. So you need to check ONLY one value: y=5 (since if y=10 then 31y=310>280) and see whether it yields integer solution for x (well if its given that x and y are positive integers and 31*x+25*y=280 then since only possible value of y is 5 then it must yield integer solution for x, in order the given statements to be true).
Hope it helps.
Thanks for clarifying that both are positive. I understand how to do te trial and error method and was looking for time savers. Once again you delivered. I never thought to use the factor of x + factor of x concept on this problem type...brilliant. Kudos time again.