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.