Different variables mean different numbers always?

[hcgoldberg]

In both problem solving and data sufficiency questions, should we assume that different variables (e.g., x, y, A, B) are different numbers always? Or can they be the same sometime? The CAT problem that I just got wrong says that they can be the same same, but I really am surprised by this claim.


(Question below, completely reproduced given that this is from MGMAT on their own forum. Apologies if this is against the rules, please message me if you must delete it so that I can restate the request without the question below...)



Question
ABC
+ BCB
CDD

In the addition shown above, A, B, C, and D represent the nonzero digits of three 3-digit numbers. What is the largest possible value of the product of A and B ?

8

10

12

14

18

Explanation
First, note that A, B, C, and D have to be digits between 1 and 9 (and the problem does not prevent some letters from having the same value).

The two rightmost columns both contain C + B = D. From that information, you can deduce that B + C < 9. (If B + C were 10 or more, then the rightmost column would “carry over” into the next column, making the tens digit into D + 1 rather than D.)

Next, A + B = C. A larger value of C, then, will reduce B (because B + C < 9) and therefore increase A (because A + B = C).

Since the questions asks to maximize the product for A and B, consider only the cases in which B + C = 9. Further, since A + B = C, you also know that B must be less than C. (Not sure why? Test a couple of real numbers to figure it out.) That leaves only a few cases, so write them out:

If B = 1 and C = 8, then A = 7. In this case, the product of A and B is 7.
If B = 2 and C = 7, then A = 5. In this case, the product of A and B is 10.
If B = 3 and C = 6, then A = 3. In this case, the product of A and B is 9.
If B = 4 and C = 5, then A = 1. In this case, the product of A and B is 4.

That’s all of the possible cases. The largest possible product is 10.

The correct answer is B.

[RonPurewal]

the default is that "two unknowns" may or may not be the same.

this is an absolutely universal default -- i.e., this is the assumed meaning of "two unknowns" or "twe variables" absolutely everywhere in the world, in every language in whith modern mathematics is formalized.

if the problem intends two different quantities, then it wil EXPLICITLY SAY the word "different" or "distinct".

__

note, further, that the outcome to this problem doesn't even depend on this issue anyway. (in the "winning" case, the letters all stand for different digits.)

[RonPurewal]

furthermore -- if this problem is from MPrep, then it belongs in either the "MPrep CAT math" folder or the "MPrep non-CAT math" folder, depending on whether it comes from our practice tests (the former) or from any of our other materials (the latter).

since this thread is posted in the wrong folder, it is now locked. if you have any further questions about this problem, please post them in the appropriate folder.

thank you.