Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:

Since the average of *x*, *y*, and *z *is the sum of the three variables, divided by 3, we can rephrase the question as “what is the value of *x *+ *y *+ *z*?” We also note the restrictions on the possible values of *x*, *y*, and *z *– the variables must be integers in ascending order from *x *to *z *(not necessarily consecutive). Moreover, they must be different integers, since the inequality *x *< *y *< *z *indicates no equality among any of the variables. We note these conditions, but at this stage there is no simple way to apply them in a further rephrasing of the question.

Statement (2): INSUFFICIENT. We start with statement (2), which is the easier statement (it only contains 2 of the 3 variables). We can quickly come up with sets of values that satisfy this statement and the given conditions but that have different sums (or averages). For instance, the set *x *= 0, *y *= 1, and *z *= 2 meets all conditions (*x *+ *z *= 2 < 3, all variables are integers and in ascending order), with *x *+ *y *+ *z *= 3. Another set (*x *= -1, *y *= 0, and *z *= 1) also meets all conditions but sums to 0. Thus, there is no single value determined by this statement.

Statement (1): INSUFFICIENT. The equation states that *x *+ *y *(which must be an integer) multiplied by *z *(another integer) equals 5. Since 5 is a prime number, there are only 2 pairs of integers that multiply together to 5: 1 and 5, and -1 and -5. (Don’t forget about the negative possibilities) Keeping the conditions that *x *< *y *< *z*, we can construct the only sets that work:

*x *+ *y *= 1 and *z *= 5 (There’s no way to assign z = 1 and x + y = 5 while preserving x < y < z.)

*x* = 0, *y *= 1, *z *= 5 sum = 6

*x* = -1, *y *= 2, *z *= 5 sum = 6

*x* = -2, *y *= 3, *z *= 5 sum = 6

*x* = -3, *y *= 4, *z *= 5 sum = 6

*x* + *y *= -5 and *z *= -1

*x* = -3, *y *= -2, *z *= -1 sum = -6

Since there are 2 possible sums, this statement is insufficient.

Statements (1) and (2) together: INSUFFICIENT. Using the sets determined with Statement (1), we check the value of __x__ + z for each case, keeping only the cases in which *x* + *z* is less than 3. Two cases remain.

Case 1: *x *= -3, *y *= 4, *z *= 5 *x* + *z *= 2 < 3 *x* + *y* + *z* = 6

Case 2: *x *= -3, *y *= -2, *z *= -1 *x* + *z* = -4 < 3 *x* + *y* + *z* = -6

Since the two cases yield different sums, we cannot determine a single value for that sum.

The correct answer is E: The two statements together are insufficient.