AFCAT Numerical Ability · Clock

How Fast the Hands Turn

The minute hand turns 6° a minute and the hour hand half a degree, so the minute hand gains 5.5° on the hour hand every minute.

All Clock notes

Why this matters

This page holds the three speeds every other clock question is built on, and most of its questions are one multiplication. The marks go when the hour hand is taken to sit still on its number instead of creeping half a degree each minute, or when a gap is divided by 6 instead of 5.5.

Concept 1 of 3: The speeds of the two hands

A hand's speed is one full turn, 360°, divided by the time that turn takes. The minute hand needs an hour for it and the hour hand needs 12 hours, so the minute hand is 12 times as fast.

Definition

The dial has 12 hour marks, so two neighbouring numbers are $30°$ apart.
It has 60 minute marks, so two neighbouring minute marks are $6°$ apart.
The hour hand does not jump from number to number. Between 4 and 5 it moves steadily, a little each minute.

Hand	One full turn takes	Degrees per hour	Degrees per minute
Minute hand	60 minutes	$360°$	$6°$
Hour hand	12 hours	$30°$	$0.5°$ Half a degree a minute: in 20 minutes the hour hand moves 10°, a third of the way to the next number.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Clock · Angle Covered by the Hands

What is the speed of the hour hand of a clock in degrees per minute?

6° a minute is the minute hand

The hour hand covers only

30°

in a whole hour, so it moves

30 \div 60 = 0.5°

a minute. Giving it

6°

a minute makes it as fast as the minute hand.

Per hour or per minute

The minute hand turns

360°

an hour and

6°

a minute. Read the unit in the question before picking, because the options often carry both.

Drill 4 more on the speeds of the two hands

Concept 2 of 3: How far a hand turns in a given time

Degrees turned are speed times time. For the hour hand measured from 12 o'clock, count the full hours at

30°

each and add half a degree for every minute past the hour.

Definition

Minute hand: degrees turned $= 6 \times$ minutes.
Hour hand from 12 o'clock at $h{:}m$ : $30h + 0.5m$ degrees.
Between two clock times, count the time first, going forward round the dial (11 o'clock to 2 o'clock is 3 hours), then multiply.

Degrees turned

\theta_{\text{hour}} = 30h + 0.5m, \qquad \theta_{\text{minute}} = 6m

Worked example

How far has the hour hand turned from 12 o'clock at 5:20? How far does the minute hand turn between 5:20 and 5:55?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Clock · Angle Covered by the Hands

9:15

, how many degrees has the hour hand moved from

12

o'clock position?

The hour hand is past its number

At 2:40 the hour hand is not on the 2. It has moved

40 \times 0.5 = 20°

beyond it, so it is

80°

from 12, not

60°

Count the hours forward

From 10 o'clock to 2 o'clock is 4 hours (10 to 12, then 12 to 2), so the hour hand turns

120°

. Taking

10 - 2 = 8

hours counts the wrong way round.

Drill 6 more on how far a hand turns in a given time

Concept 3 of 3: The minute hand gains 5.5° a minute

Both hands turn the same way. Each minute the minute hand turns

6°

and the hour hand

0.5°

, so the gap between them changes by

5.5°

a minute, like a faster runner gaining on a slower one.

Definition

Relative speed $= 6 - 0.5 = 5.5°$ per minute, which is $330°$ per hour.
Time to open or close a gap of $\theta$ degrees: $\theta \div 5.5 = \tfrac{2\theta}{11}$ minutes.
A full lap of $360°$ takes $\tfrac{720}{11} = 65\tfrac{5}{11}$ minutes, a little more than an hour.
Dividing by 5.5 is multiplying by $\tfrac{2}{11}$ . The elevenths in clock answers come from here.

Time to change the gap

t = \dfrac{\theta}{5.5} = \dfrac{2\theta}{11} \text{ minutes}

Worked example

At 12:00 the hands are together. After how many minutes are they

66°

apart for the first time, and what time is it then?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Clock · Angle Covered by the Hands

Starting at

12:00

when hands coincide, after how many minutes will they be perpendicular for the first time?

Use 5.5, not 6

The gap does not grow at

6°

a minute, because the hour hand moves too. From 12:00 a gap of

44°

takes

44 \div 5.5 = 8

minutes, not

44 \div 6

A lap takes more than 60 minutes

In 60 minutes the minute hand gains only

330°

, short of a full lap. The hands meet every

65\tfrac{5}{11}

minutes, not every hour.

Drill 4 more on the minute hand gains 5.5° a minute

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

How far a hand turns in a given time
Degrees turned
$\theta_{\text{hour}} = 30h + 0.5m, \qquad \theta_{\text{minute}} = 6m$
The minute hand gains 5.5° a minute
Time to change the gap
$t = \dfrac{\theta}{5.5} = \dfrac{2\theta}{11} \text{ minutes}$

Reference tables (1)

The speeds of the two hands2 rows

Hand	One full turn takes	Degrees per hour	Degrees per minute
Minute hand	60 minutes	$360°$	$6°$
Hour hand	12 hours	$30°$	$0.5°$ Half a degree a minute: in 20 minutes the hour hand moves 10°, a third of the way to the next number.

Watch out for (6)

6° a minute is the minute hand
→ The speeds of the two hands
Per hour or per minute
→ The speeds of the two hands
The hour hand is past its number
→ How far a hand turns in a given time
Count the hours forward
→ How far a hand turns in a given time
Use 5.5, not 6
→ The minute hand gains 5.5° a minute
A lap takes more than 60 minutes
→ The minute hand gains 5.5° a minute

Drill the questions on this topic

17 questions from the bank, with cart and Word export.

Drill all 17 questions