Each of the questions given below consists of a statement and / or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is / are sufficient to answer the given question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. Find the average score for all the juniors and seniors combined. I. The average of the scores was 85 for juniors and 89 for seniors. II. The groups are of equal size.

### Concept & Logic To find the combined average of two distinct groups, you need the individual averages of both groups AND the ratio of their respective sizes. The combined average is a weighted average. $$ \text{Combined Average} = \frac{n_1A_1 + n_2A_2}{n_1 + n_2} $$ ### Step-by-Step Solution * **From Statement I:** We are given the individual averages. Let $j$ be the number of juniors and $s$ be the number of seniors. We know the average for juniors ($A_1$) is 85 and for seniors ($A_2$) is 89. We cannot find the combined average without knowing the exact values or ratio of $j$ to $s$. Thus, Statement I alone is NOT sufficient. * **From Statement II:** We are told the groups are of equal size ($j = s$). However, we do not know their average scores. Thus, Statement II alone is NOT sufficient. * **Combining Statements I and II:** Using both statements, we know $A_1 = 85$, $A_2 = 89$, and $j = s$. $$ \text{Combined Average} = \frac{85j + 89j}{j + j} = \frac{174j}{2j} = 87 $$ Since we can find a unique answer by combining both statements, both are necessary. ### Exam Strategy & Shortcut Data Sufficiency questions on weighted averages often follow a predictable pattern: you need both the constituent averages and the weightings (ratio of quantities). If Statement I gives the averages and Statement II gives the ratio (equal size means 1:1), instantly conclude that both statements are needed. You don't actually need to calculate the average of 87 to mark the answer. ### Common Pitfall A major mistake is assuming that the combined average is always just the simple average of the two group averages ($\frac{85 + 89}{2} = 87$) without needing Statement II. This is ONLY true if the groups are of equal size, which is why Statement II is strictly necessary. ### Final Answer **Therefore, the correct answer is Data in both Statements I and II together are necessary.**