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# Data Sufficiency: The Question Stem - What is the Issue?

$$Q$$ is the set of all integers between $$A$$ and $$B$$, inclusive. The average (arithmetic mean) of $$Q$$ is $$m$$. If $$A \lt B$$ and $$B=190$$, what is the value of $$A$$? >(1) $$Q$$ contains 40 terms that are greater than $$m$$. >(2) $$m=150$$
Incorrect. [[snippet]] Stat. (2): If $$m$$ is an integer, it follows that $$m$$ is also a member of set $$Q$$. Since the average is also the median, then the number of terms smaller than $$m$$ is equal to the number of terms greater than $$m$$, until $$B=190$$. It is therefore possible to count down the same number of consecutive integers from 150 and to find a single value for $$A$$. Stat.(2) → S → B. Alternative explanation for Stat. (2) Since the average of a set of consecutive integers is the average of the first and last members of the set, >$$\frac{A + B}{2} = m$$. Since $$B=190$$ and $$m=150$$, this leads to a single equation with $$A$$. >$$\frac{A + 190}{2} = 150$$ From here you can determine a single value for $$A$$.
Stat. (2): If $$m$$ is an integer, it follows that $$m$$ is also a member of set $$Q$$. Since the average is also the median, then the number of terms smaller than $$m$$ is equal to the number of terms greater than $$m$$, until $$B=190$$. It is therefore possible to count down the same number of consecutive integers from 150 and to find a single value for $$A$$. Stat.(2) → S → B. Alternative explanation for Stat. (2): Since the average of a set of consecutive integers is the average of the first and last members of the set, >$$\frac{A + B}{2} = m$$. Since $$B=190$$ and $$m=150$$, this leads to a single equation with $$A$$. >$$\frac{A + 190}{2} = 150$$ From here you can determine a single value for $$A$$.
Incorrect. [[snippet]] Stat. (1): Since $$m$$ is the middle of the set, it follows that there are 40 members smaller than $$m$$. This statement wants to lead you to the conclusion that there are 81 members in set $$Q$$: 40 members greater than $$m$$, 40 members smaller than $$m$$, and $$m$$ itself. So $$A$$ can be calculated by counting down 81 consecutive integers from $$B=190$$ ($$A$$ in this case would be 110, but you don't really have to know that). But is that the only possible value for $$A$$? The problem lies with the assumption that $$m$$ itself is a member of set $$Q$$, but that fact is not stated anywhere. In fact, there is even no guarantee that $$m$$ is an integer. If set $$Q$$ has an even number of members (i.e., only 80 terms), then $$m$$, as the median of the set, is calculated by averaging the pair of middle terms; $$m$$ would still have 40 members greater and 40 members smaller than it, but will not itself be a member of set $$Q$$. However, since in this case set $$Q$$ has only 80 members, this will determine a different value for $$A$$ (i.e., $$A=111$$). Therefore, no single value can be determined for $$A$$, so Stat.(1) → IS → BCE.
Correct. [[snippet]] Stat. (1): Since $$m$$ is the middle of the set, it follows that there are 40 members smaller than $$m$$. This statement wants to lead you to the conclusion that there are 81 members in set $$Q$$: 40 members greater than $$m$$, 40 members smaller than $$m$$, and $$m$$ itself. So $$A$$ can be calculated by counting down 81 consecutive integers from $$B=190$$ ($$A$$ in this case would be 110, but you don't really have to know that). But is that the only possible value for $$A$$? The problem lies with the assumption that $$m$$ itself is a member of set $$Q$$, but that fact is not stated anywhere. In fact, there is even no guarantee that $$m$$ is an integer. If set $$Q$$ has an even number of members (i.e., only 80 terms), then $$m$$, as the median of the set, is calculated by averaging the pair of middle terms; $$m$$ would still have 40 members greater and 40 members smaller than it, but will not itself be a member of set $$Q$$. However, since in this case set $$Q$$ has only 80 members, this will determine a different value for $$A$$ (i.e., $$A=111$$). Therefore, no single value can be determined for $$A$$, so Stat.(1) → IS → BCE.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.