What is the two-digit number? (I) The difference between the two digits is 9.…

Concept: A statement in a Data Sufficiency question is sufficient by itself only if it pins down exactly one value once the question's own built-in constraints are applied. For a two-digit number, the built-in constraint is that the tens digit cannot be 0. Each statement must be checked on its own, before ever combining them, to see whether it alone narrows the digits to a single case.

1. Let the tens digit be x and the units digit be y, so x is from 1 to 9 (a two-digit number's leading digit is never 0) and y is from 0 to 9.

2. Statement I alone: the difference between the digits is 9, so |x - y| = 9. Since x and y are both single digits, y - x = 9 would need x = 0, which is not allowed, so the only working case is x - y = 9.

3. Solving x - y = 9 with x from 1 to 9 gives exactly one pair: x = 9, y = 0. No other digit pair fits, so Statement I alone fixes a single number.

4. Statement II alone: the sum of the digits equals the difference between them, so x + y = |x - y|. If x < y, this needs x = 0, which is invalid, so x >= y and the equation becomes x + y = x - y, which simplifies to y = 0.

5. With y = 0, x can be any digit from 1 to 9, so nine different values of x each give a valid two-digit number - nine numbers in all, not one. Statement II alone cannot narrow the digits to a single number.

Cross-check: the number obtained from Statement I (x = 9, y = 0) is one of the nine numbers that also satisfy Statement II's equation, which is why combining both statements still lands on the same case - but that overlap does not mean Statement II is needed: Statement I already narrows the digits to one pair entirely on its own.

So Statement I alone is sufficient to answer the question, while Statement II alone is not sufficient.