Directions : Each of these questions is followed by three statements. You have to study the question and all the three statements given and decide whether any information provided in the statement(s) is/are redundant and can be dispensed with while answering the questions. A $25\text{ m}$ long wire is cut into 3 pieces. How long is the longest piece? I. Two pieces are each $1\text{ m}$ shorter that the longest piece. II. Two pieces of the wire are of the same length. III. The longest piece of the wire is $12.5\%$ more than the smallest piece.

### Concept & Logic This is a Data Sufficiency problem where we need to identify the **redundant** statements. A statement is redundant if the question can be uniquely answered without it. We need to find the length of the longest piece of a $25\text{ m}$ wire cut into 3 pieces. Let the pieces be $x, y, z$. ### Step-by-Step Solution **Evaluating Statement I alone:** * The statement says two pieces are each $1\text{ m}$ shorter than the longest piece. * Let the longest piece be $x$. The other two pieces are $(x - 1)$ and $(x - 1)$. * Total length $= x + (x - 1) + (x - 1) = 25$ * $3x - 2 = 25 \implies 3x = 27 \implies x = 9\text{ m}$. * Statement I alone gives a unique, definite answer. **Evaluating Statements II and III together:** * Since Statement I is sufficient, we must check if II and III are also capable of solving it independently (which would make I redundant instead). * Statement II: Two pieces are of the same length. * Statement III: Longest piece $= 112.5\%$ of smallest piece $\implies$ Longest $= \frac{9}{8} \times$ smallest. * **Case A:** If the two smallest pieces are equal, let them be $8k$ and $8k$. Longest is $9k$. Sum $= 25k = 25 \implies k=1$. Pieces are $8, 8, 9$. Longest is $9\text{ m}$. * **Case B:** If the two longest pieces are equal, let them be $9k$ and $9k$. Smallest is $8k$. Sum $= 26k = 25 \implies k = \frac{25}{26}$. Longest is $9 \times \frac{25}{26} = \frac{225}{26}\text{ m}$. * Statements II and III together do not yield a unique answer without making arbitrary assumptions about which pieces are equal. **Conclusion:** * Only Statement I is sufficient to answer the question uniquely. * Therefore, Statements II and III are completely redundant and can be dispensed with. ### Exam Strategy & Shortcut When a statement defines the entire system of variables in terms of a single unknown (like Statement I does), it is almost always sufficient. The moment you see $x + 2(x-1) = 25$, you know $x$ is perfectly solvable. You can instantly mark II and III as redundant without fully calculating the complex $12.5\%$ fraction cases. ### Common Pitfall Students often assume "longest piece" in Statement III implies there is strictly only *one* longest piece, ruling out Case B. However, mathematical phrasing allows for two pieces to tie for the maximum length unless explicitly stated otherwise. Assuming Case A is the only valid scenario leads to incorrectly thinking II and III are also uniquely sufficient. ### Final Answer **Therefore, the correct answer is Only II and III together.**