Right angle between 7 and 8 o’clock: At what time between 7 and 8 o’clock will the hands of a clock be at right angles?

Introduction / Context:
For clock problems, the angle between hands at time H:M is given by |30H − 5.5M| degrees. Right angles occur when this magnitude equals 90°. Between any two consecutive hours, there are typically two right-angle instants.

Given Data / Assumptions:

• Hour between 7 and 8 ⇒ H = 7.
• Right angle condition: |30H − 5.5M| = 90.

Concept / Approach:
Set up |210 − 5.5M| = 90 and solve the two linear cases to find the pair of minutes M in (0, 60). Convert fractional minutes to a mixed number of minutes and 11ths (since 5.5 = 11/2 commonly yields denominators of 11).

Step-by-Step Solution:

Verification / Alternative check:
Substitute each M to get |210 − 5.5M| = 90 exactly.

Why Other Options Are Wrong:
19 5/11 or 21 5/11 are not the correct solutions; 18 min is approximate and off. Since none match either 21 9/11 or 54 6/11, “None of these” is correct.

Common Pitfalls:
Dropping the negative case or rounding fractions prematurely; there are two answers within the hour.

Final Answer:
None of these