Times when hands are at right angles between 4 PM and 5 PM — At what time(s) between 4 and 5 are the hands 90° apart?

Introduction / Context:
Within a given hour, the hands are at right angles twice. Solving the right-angle condition gives two solutions. The general formulas for minutes past H:00 are t = (60H ± 15) / 11. We must compute the exact times for H = 4 and compare with the options listed.

Given Data / Assumptions:

• Right-angle condition: |6t − (30H + 0.5t)| = 90.
• Derived formulas: t1 = (60H + 15)/11, t2 = (60H − 15)/11 (both must lie between 0 and 60).
• Hour window: 4:00–5:00.

Concept / Approach:
Substitute H = 4 and compute both exact fractional minute values. Convert to mixed fractions to see if any provided option matches exactly.

Step-by-Step Solution:
t1 = (60 × 4 + 15)/11 = 255/11 = 23 2/11 minutes ⇒ 4 : 23 2/11.t2 = (60 × 4 − 15)/11 = 225/11 = 20 5/11 minutes ⇒ 4 : 20 5/11.

Verification / Alternative check:
Plugging either time into angles confirms a 90° separation (either ahead or behind). Both fall between 4:00 and 5:00 as required.

Why Other Options Are Wrong:
4 : 75/11 (≈ 6 9/11), 4 : 45/11 (≈ 4 1/11), 4 : 95/11 (≈ 8 7/11) do not match 20 5/11 or 23 2/11 minutes. Hence “None of these” is correct.

Common Pitfalls:
Using only one of the ± cases or rounding fractional minutes to a nearby but incorrect fraction.

Final Answer:
None of these (correct times = 4 : 20 5/11 and 4 : 23 2/11)