Max@Math Revolution wrote:[GMAT math practice question] 6.25

If m and n are positive integers, m+n=?

1) m-n=-1
2) n^2<5

We need to determine the sum of m and n given that they are both positive integers.

Statement One Alone:

m-n=-1

That means m = n - 1. However, without knowing the value of either one of them, we can't determine their sum.

Statement Two Alone:

n^2<5

That means n is either 1 or 2. However, without knowing the value of m, we can't determine the sum of m and n.

Statements One and Two Together:

From statement two, n can be only 1 or 2 and from statement one, m is 1 less than n. If n = 1, then m = 0, but we are given that m is positive. So n can't be 1. If n = 2, then m = 1. So, the sum of m and n is 1 + 2 = 3.

Answer: C

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (m and n) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since 0 < n^2<5, we must have n = 1 or n = 2.

Case 1: n = 1
Since m - n = m - 1 = -1, we have m = 0, which is not positive.
So, n is not equal to 1.

Case 2: n = 2
Since m - n = m - 2 = -1, we have m = 1.
Thus m + n = 1 + 2 = 3.
Because we have a unique solution, both conditions together are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If m = 1 and n = 2, then m + n = 3.
If m = 2 and n = 3, then m + n = 5.
Since we don't have a unique solution, condition 1) is not sufficient.

Condition 2)
Since we don't have any information about m, condition 2) is not sufficient.


Therefore, C is the answer.

Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.