NUMERICAL — TIER 3

Six small topics · half a day total, Day 8 afternoon

112 questions across all six combined — less than Number System alone. This file is deliberately short. Read it once, learn the formulas, do not spend a full day here.

These are still worth the 3–4 hours because each one is formula-driven: almost no thinking required once you know the line. They are cheap marks, just not many of them.

§1 · CLOCKS & CALENDARS — 27 questions

The largest of the Tier 3 topics. Four ideas, all mechanical.

Angle between the hands — one formula

Angle = |30H − 5.5M|      (H = hour, M = minutes)
If the result exceeds 180, subtract from 360.

At 4:20 → |30(4) − 5.5(20)| = |120 − 110| = 10°.

Where it comes from: the hour hand moves 30° per hour and 0.5° per minute; the minute hand moves 6° per minute. The 5.5 is 6 − 0.5.

How often the hands do things — the counts are the question

Coincide (overlap)  → 11 times in 12 hours,  22 in a day   (NOT 12/24)
Opposite (180°)     → 11 times in 12 hours,  22 in a day
At right angles     → 22 times in 12 hours,  44 in a day

The hands coincide 11 times, not 12, because between 11:00 and 1:00 they only overlap once (at 12:00). This is the trap.

The mechanism — makes the counts stick and solves faulty-clock questions

Minute hand: 6° per minute        Hour hand: 0.5° per minute
→ the minute hand GAINS 55 minutes on the hour hand every 60 minutes
→ so the hands coincide every 65 5/11 minutes (≈ 65.45 min)
→ 720 / 65.45 ≈ 11  →  hence 11 coincidences in 12 hours

This also answers faulty-clock questions directly: a clock whose hands meet every 65 minutes exactly is running fast (they should meet every 65.45).