1. **State the problem:** We want to find the exact time between 10:00 and 11:00 such that six minutes from now, the minute hand is exactly opposite the position where the hour hand was three minutes ago. 2. **Define variables:** Let the current time be $10:x$ minutes. 3. **Positions of hands:** - The hour hand moves $0.5$ degrees per minute (since it moves 30 degrees per hour). - The minute hand moves $6$ degrees per minute. 4. **Calculate hour hand position 3 minutes ago:** The hour hand position at time $10:x$ is: $$H = 30 \times 10 + 0.5 \times x = 300 + 0.5x$$ degrees. Three minutes ago, the hour hand was at: $$H_{-3} = 300 + 0.5(x - 3) = 300 + 0.5x - 1.5 = 298.5 + 0.5x$$ degrees. 5. **Calculate minute hand position 6 minutes from now:** The minute hand position at time $10:x$ is: $$M = 6x$$ degrees. Six minutes from now, the minute hand will be at: $$M_{+6} = 6(x + 6) = 6x + 36$$ degrees. 6. **Condition for opposite positions:** The minute hand 6 minutes from now is exactly opposite the hour hand 3 minutes ago, so their positions differ by 180 degrees: $$|M_{+6} - H_{-3}| = 180$$ Substitute values: $$|6x + 36 - (298.5 + 0.5x)| = 180$$ Simplify inside absolute value: $$|6x + 36 - 298.5 - 0.5x| = |5.5x - 262.5| = 180$$ 7. **Solve the absolute value equation:** Two cases: Case 1: $$5.5x - 262.5 = 180$$ $$5.5x = 442.5$$ $$x = \frac{442.5}{5.5} = 80.4545...$$ (not possible since $x$ must be between 0 and 60) Case 2: $$5.5x - 262.5 = -180$$ $$5.5x = 82.5$$ $$x = \frac{82.5}{5.5} = 15$$ 8. **Check if $x=15$ is valid:** Yes, $x=15$ minutes is between 0 and 60. 9. **Final answer:** The exact time now is $\boxed{10:15}$.