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.

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:

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.