Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. The mean temperature of Monday to Wednesday was $37^\circ\text{C}$ and of Tuesday to Thursday was $34^\circ\text{C}$. What was the temperature on Thursday? I. The temperature on Thursday was $\frac{4}{5}$th that of Monday. II. The mean temperature of Monday and Thursday was $40.5^\circ\text{C}$. III. The difference between the temperature on Monday and that on Thursday was $9^\circ\text{C}$.

### Concept & Logic The core logic relies on understanding how overlapping averages can be algebraically manipulated to find the difference between non-overlapping elements. The sum of a set is calculated as: $$Sum = Average \times Number of Elements$$ By finding the difference between the two overlapping temperature sets, we can isolate a relationship between Monday and Thursday. We then evaluate which additional statements provide enough independent information to solve for Thursday. ### Step-by-Step Solution **Given:** * Mean temperature of Monday (M), Tuesday (T), and Wednesday (W) = $37^\circ\text{C}$. * Mean temperature of Tuesday (T), Wednesday (W), and Thursday (Th) = $34^\circ\text{C}$. **Calculation / Deduction:** 1. Formulate equations for the sums: * $M + T + W = 37 \times 3 = 111$ (Equation A) * $T + W + Th = 34 \times 3 = 102$ (Equation B) 2. Subtract Equation B from Equation A to eliminate T and W: * $(M + T + W) - (T + W + Th) = 111 - 102$ * $M - Th = 9$ (We now know Monday is $9^\circ\text{C}$ hotter than Thursday). 3. Evaluate the statements to see which can solve for $Th$ when combined with $M - Th = 9$: * **Statement I:** $Th = \frac{4}{5}M$. This is an independent linear equation. Substituting this into our derived equation ($M - \frac{4}{5}M = 9 \Rightarrow \frac{1}{5}M = 9 \Rightarrow M = 45$, so $Th = 36$). **Sufficient.** * **Statement II:** $\frac{M + Th}{2} = 40.5 \Rightarrow M + Th = 81$. This is also an independent linear equation. Combining with $M - Th = 9$ allows us to solve for $Th$. **Sufficient.** * **Statement III:** $M - Th = 9$ (or $Th - M = 9$). This merely restates the deduction we already made from the main question. It provides no new information to solve for the exact values. **Not sufficient.** Since Statement I alone is sufficient, and Statement II alone is sufficient, we need *either* I or II. ### Exam Strategy & Shortcut In Data Sufficiency, **never actually solve the equations** unless absolutely necessary. Once you deduce $M - Th = 9$, you simply need one more unique, independent equation involving $M$ and $Th$ to find their values (a system of two variables needs two distinct linear equations). Statement I gives a ratio (independent). Statement II gives a sum (independent). Statement III gives a difference (which is identical to our existing equation, thus dependent/redundant). You can instantly conclude "Either I or II" without calculating the final temperature of $36^\circ\text{C}$. ### Common Pitfall Students often waste precious time calculating the exact temperature on Thursday ($36^\circ\text{C}$) for both Statement I and Statement II to "prove" they work. Remember that in Data Sufficiency, identifying that a unique solution *exists* is enough; calculating it is a time trap. Another pitfall is thinking Statement III is useful because it contains the number 9, failing to realize it is mathematically redundant. ### Final Answer Therefore, the correct answer is **Either I or II**.