# Clocks — Angle Between Hands, Overlaps and Wrong Clocks

Source: https://skills101.in/learn/reasoning/clocks — Skills101 free tutorial (Logical Reasoning). Updated 2026-11-09.

A clock face has 360°, 12 hours and 60 minutes. Every clock question uses these speeds:

- **Minute hand:** 360° in 60 min → **6° per minute**
- **Hour hand:** 30° in 60 min → **½° per minute**
- The minute hand gains **5.5° per minute** on the hour hand.

## 1. Angle between the hands

![A clock at 3:40. The hour hand is at 110° and the minute hand at 240°, so the angle is 130°](https://skills101.in/learn-img/reasoning/clock-angle.svg)

> **Angle = | 30H − 5.5M |**   (H = hour, M = minutes)
> If the answer is more than 180°, subtract it from 360° to get the smaller angle.

### Example 1
*What is the angle between the hands at 3:40?*

|30 × 3 − 5.5 × 40| = |90 − 220| = **130°**.

### Example 2
*At 7:20?*

|210 − 110| = **100°**.

### Example 3
*At 9:00?* |270 − 0| = 270 → 360 − 270 = **90°**.

> **Why the hour hand matters:** at 3:40 the hour hand is not on 3 — it has moved ⅔ of the way to 4. Students who forget this get 120° instead of 130°.

## 2. When do the hands overlap?

The hands are together when 30H = 5.5M → **M = 60H/11**.

### Example 4
*At what time between 4 and 5 are the hands together?*

M = 240/11 = **21 9/11 minutes past 4**.

Useful facts in **12 hours**:
- Hands overlap **11 times** (not 12 — between 11 and 1 they overlap only once, at 12).
- Hands are at a right angle (90°) **22 times**.
- Hands are opposite (180°) **11 times**.

In 24 hours: 22 overlaps, 44 right angles, 22 opposite.

## 3. Hands opposite each other

### Example 5
*Between 2 and 3, when are the hands opposite?*

|30 × 2 − 5.5M| = 180 → 5.5M = 240 → M = 480/11 = **43 7/11 minutes past 2**.

## 4. Mirror images

The time seen in a mirror = **11:60 − actual time** (for times between 1:00 and 11:00).

### Example 6
*A clock shows 3:25. What does its mirror image show?* 11:60 − 3:25 = **8:35**.

## 5. Fast and slow clocks

### Example 7
*A clock gains 5 minutes every hour. It is set right at 8 am. What will it show at 2 pm (actual time)?*

6 hours × 5 min = 30 min gained → it shows **2:30 pm**.

## Common mistakes

- Treating the hour hand as fixed on the hour number.
- Saying the hands overlap 12 times in 12 hours (it is 11).
- Forgetting to take 360° minus the angle when it is over 180°.

## Quick recap

- Minute hand 6°/min, hour hand ½°/min, gap closes at 5.5°/min.
- Angle = |30H − 5.5M|.
- Overlap at M = 60H/11. 11 overlaps in 12 hours.
- Mirror time = 11:60 − time.


Practice quiz and free certificate: https://skills101.in/learn/reasoning/clocks