Clocks: Concepts, Formulas, and Clear Examples

Clock MCQs treat the hour and minute hands as two moving lines on a circle. You compute relative speed, coincidence times, right angles, and mirror/reflection images using angular formulas—not by drawing every tick.

Placement papers repeat the same toolkit: angle at H hours M minutes, time when hands overlap, how many times hands are perpendicular in 12 hours. Master relative speed of the minute hand over the hour hand and the classic angle formula with $|30H - 5.5M|$ (adjust for obtuse/reflex when asked).

Hand speeds

Minute hand

$$6^\circ \text{ per minute}$$

Hour hand

$$0.5^\circ \text{ per minute} \quad (30^\circ \text{ per hour})$$

Relative (minute vs hour)

$$5.5^\circ \text{ per minute}$$

• In one hour, minute hand gains 360° on hour hand → gain 55 min-spaces on a 60-space dial.

Angle between hands

Angle at H : M (12h clock)

$$\theta = \left| 30H - 5.5M \right| \text{ (take smaller angle if } \theta > 180^\circ \text{ unless reflex asked)}$$

Hour hand position

$$30H + 0.5M$$

Minute hand position

$$6M$$

Example — Angle at 3:15

$\theta = |30 \times 3 - 5.5 \times 15| = |90 - 82.5| = 7.5^\circ$.

Not zero—common trap thinking ‘quarter past’ looks straight.

Coincidence (overlap) times

Hands coincide when relative gain is a multiple of 360°. Starting from 12:00, overlaps occur roughly every $\frac{720}{11}$ minutes ≈ 65.45 min.

First overlap after H o'clock (approx method)

$$\text{At } H \text{ o'clock, first coincidence: } \frac{60H}{11} \text{ minutes past } H$$

Valid for H = 1..10 from 12; adjust for 11 and 12 with same relative-speed idea.

Right angle times

Hands perpendicular when $|30H - 5.5M| = 90^\circ$ or $270^\circ$. Solve for M in each hour window; typically two right angles per hour except special cases.

Wrong clock / gain or loss

• If a clock gains x minutes per hour, true time maps by proportion.
• Wrong clock showing correct time twice daily: use relative error formula from your material.

Mirror and reflection

• Mirror on 12–6 axis swaps left–right on dial; convert to angle arithmetic or ‘minutes subtract from 11:60’ style shortcuts for vertical mirror.

Step-by-step method for exam questions

1. Convert time to H and M; confirm 12h vs 24h not needed for angle.
2. Compute hour position $30H + 0.5M$ and minute $6M$.
3. Apply $|30H - 5.5M|$; if question wants smaller angle, use $\min(\theta, 360-\theta)$.
4. For overlap/perpendicular, set up linear equation in M for that hour.
5. Check boundary: M between 0 and 60.
6. Pick option; verify with one nearby minute if close.

Formula cheat sheet

Frequently asked questions

Why is 3:15 not 0°?

Hour hand moves past 3 while minute hand moves to 3; $|30H-5.5M|$ captures both motions.

Reflex angle requested?

If $\theta < 180$ but they want reflex, use $360 - \theta$.

How many overlaps in 12 hours?

11 (not 12)—no overlap at 12:00 counted twice in a naive count).

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.