Data Sufficiency — Clock Angles Is it exactly 9 o’clock now? I. After 30 minutes from now, the hour and minute hands will be exactly 90° apart. II. Exactly 15 minutes ago, the hour and minute hands coincided.

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Solve this multiple-choice question and choose the correct option.

Accepted Answer

Introduction / Context:We must decide whether the current time is exactly 9:00 using two angle-based facts about a clock. Given Data / Assumptions: I: In 30 minutes the hands will be 90° apart. II: 15 minutes ago the hands coincided. Concept / Approach:Coincidences occur every 65 5/11 minutes; 90° separations occur multiple times per hour. Many present times satisfy either condition without uniquely fixing “now”. To prove sufficiency we would need a unique present time equal to 9:00. Step-by-Step Solution: 1) From I alone: infinitely many “now” times yield a right angle 30 minutes later (e.g., around each hour), so not unique to 9:00.2) From II alone: “15 minutes after a coincidence” also occurs many times and does not map uniquely to 9:00.3) Combining I and II does not eliminate multiplicity; solving the simultaneous conditions yields candidate times that are not uniquely 9:00. Verification / Alternative check:Numerical angle equations show multiple feasible “now” times satisfy both constraints. Why Other Options Are Wrong: I alone or II alone: clearly insufficient. Both together sufficient: not true; still multiple possibilities. Either alone sufficient: false. Common Pitfalls:Assuming coincidences occur exactly at hh:00; or assuming a single 90° event per hour. Final Answer:Not sufficient even together.

Data Sufficiency — Clock Angles Is it exactly 9 o’clock now? I. After 30 minutes from now, the hour and minute hands will be exactly 90° apart. II. Exactly 15 minutes ago, the hour and minute hands coincided.

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