Data Sufficiency

[GMAT math practice question]

When the members of group A are divided into groups of 13 people, m subgroups are formed. When the members of group B are divided into groups of 11 people, n subgroups are formed, and 8 people are left over. What is the number of members of group B?

1) m = n
2) The numbers of members of groups A and B are the same.

Statement 1: It is not sufficient, as for m=n, there can be infinite values of m that we will get.
Statement 2: It is not sufficient, as there are many values of m and n for which, 13m= 11n+8
Statement 1 and Statement 2 together: It is Sufficient, as 13m=11m+8, will have only 1 solution

=>

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)
The conditions give the equations m = n and 13m = 11n + 8.
Plugging the first equation into the second yields 2n = 8 or n = 4.
Thus, group B has 11*4 + 8 = 52 members.
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)
The equation m = n is not sufficient for determining the value of n.
Thus, condition 1) alone is not sufficient.

Condition 2)
13m = 11n + 8 does not give enough information to find the value of n.
Thus, condition 2) is not sufficient.

Therefore, the answer is C.

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.

Let the members be x and y
therefore, x = 13m
y = 11n + 8

Question: Find y [number of members in group B]

Statement 1: m = n
y = 11m + 8
There is no specific information about variable m.
Statement 1 is in sufficient.

Statement 2: Number of members of group A and B are the same.
x = y
13m = 11n +8
Value of n is not sufficient combing statement 1 and 2 together.
13m = 11n + 8 and m = n
Therefore, 13n = 11n + 8
13n - 11 = 8
$$\frac{22}{2\ }=\frac{8}{2}$$
$$n\ =\frac{8}{2}$$
n = 4

Statement 1 and 2 are not sufficient alone but they are sufficient together.
Answer is Option C.