Clock — AFCAT Numerical Ability Formulas and Traps

How Fast the Hands Turn

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}

The speeds of the two hands

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.

Common traps

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.

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.

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.

The Angle Between the Hands at a Given Time

Learn this subtopic in the notes

The angle on the hour

On the hour

\theta = 30h \;\text{ or }\; 360 - 30h

The angle at any time

Angle between the hands

\theta = |30h - 5.5m|

Above 180°: the smaller angle and the reflex angle

Smaller and reflex angle

\theta_{\text{small}} = \min(\theta,\ 360 - \theta), \qquad \theta_{\text{reflex}} = 360 - \theta_{\text{small}}

Naming, comparing and changing angles

Change in the angle

\Delta\theta = |\theta_2 - \theta_1|

Common traps

Count the short way

At 11:00 the hour hand is 11 gaps from 12 one way but only 1 the other. The angle between the hands is

30°

, not

330°

.

The hour hand has moved

At 5:40 the hour hand is

20°

past the 5, at

170°

. Taking it at

150°

gives

90°

instead of the true

70°

.

Half past is not on the number

At 7:30 the hour hand is halfway between 7 and 8, at

225°

. The angle is

|210 - 165| = 45°

, not

30°

.

Above 180 is the long way round

At 12:44 the formula gives

242°

. The hands are

118°

apart the short way;

242°

is the reflex angle, the answer only when the question asks for it.

Reflex is 360 minus the smaller angle

If the smaller angle is

70°

, the reflex angle is

290°

. Adding 180 gives

250°

, which is neither angle.

Work the angles, do not judge the times

6:15 and 6:45 look alike, but the angles are

97.5°

and

67.5°

: one obtuse, one acute.

Near 90 still needs the exact value

An angle of

87.5°

is acute even though it is close to a right angle. Name the type from the worked value, never from a rough picture.

When the Hands Meet, Point Opposite or Form an Angle

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When the hands coincide or point opposite

Coinciding and opposite hands

m = \dfrac{60h}{11} \;\text{(together)}, \qquad m = \dfrac{2}{11}\left(30h \pm 180\right) \;\text{(opposite)}

When the hands are at a given angle

Time for a given angle

m = \dfrac{2}{11}\left(30h \pm \theta\right)

How many times a position comes round

Position	In 12 hours	In 24 hours	How often
Coinciding	11	22	every $65\tfrac{5}{11}$ min Not 12 in 12 hours: the meeting between 11 and 12 is 12:00 itself.
Opposite ( $180°$ )	11	22	every $65\tfrac{5}{11}$ min
In a straight line (together or opposite)	22	44	every $32\tfrac{8}{11}$ min
At right angles	22	44	every $32\tfrac{8}{11}$ min Not 48 in a day: 3:00 and 9:00 are each shared by two neighbouring hours.
Any other angle between $0°$ and $180°$	22	44	twice in each lap

Common traps

Eleven meetings, not twelve

The hands meet once in each hour except between 11 and 12, where the meeting is 12:00 itself. That is why a meeting comes every

65\tfrac{5}{11}

minutes.

A straight line includes together

If a question asks only for 'in a straight line', the moment the hands coincide counts too. Between 1 and 2 that is about 1:05 as well as about 1:38.

Drop a value outside the hour

Between 10 and 11,

\tfrac{2}{11}(300 + 90) = 70\tfrac{10}{11}

is past the hour, so only

\tfrac{2}{11}(300 - 90) = 38\tfrac{2}{11}

counts.

Minute spaces are not degrees

Hands 5 minute spaces apart are

30°

apart. Putting

\theta = 5

into the formula answers a different question.

Count a stretch by listing

From 2:00 to 5:00 is 3 hours, but

3 \times \tfrac{11}{12} = 2.75

is not a count. List the meetings, about 2:11, 3:16 and 4:22, and there are 3.

44 right angles a day, not 48

Two right angles an hour would make 48. Around 3 and 9 the right angles of neighbouring hours fall on the same moment, 3:00 or 9:00, so 12 hours hold 22 and a day 44.

Mirror Images of a Clock

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The image time is 11:60 minus the real time

Mirror time

\text{image} = 11{:}60 - \text{real time}

Finding the real time from the image

Real time from the image

\text{real} = 11{:}60 - \text{image}

Times after 11, time passing and other twists

Reading 0 hours

11{:}60 - 11{:}m = 0{:}(60 - m) = 12{:}(60 - m)

Common traps

Subtract from 11:60, not 12:00

Subtracting 2:40 from 12:00 digit by digit tempts you to 10:20. With 11:60 there is nothing to borrow:

11{:}60 - 2{:}40 = 9{:}20

.

60 minutes is the next hour

11{:}60 - 5{:}00 = 6{:}60

, which a clock shows as 7:00. Writing 6:60 as the answer matches no option.

The image is not 6 hours off

An image of 1:20 is not 7:20 in real time. Only 12:00 minus the image works:

11{:}60 - 1{:}20 = 10{:}40

.

Move the real clock, not the image

Two hours ago the image showed 3:10, so the real time then was 8:50 and now it is 10:50. Adding the two hours to the image first gives 5:10, whose reflection is 6:50: wrong, because the image runs backwards.

There is no 0 on the dial

11{:}60 - 11{:}50

is 0:10, which a 12-hour dial shows as 12:10.

Clocks That Gain or Lose Time

Learn this subtopic in the notes

What a gaining or losing clock shows

Time shown

\text{shown} = \text{real} \pm \text{rate} \times \text{hours}

When a fast or slow clock shows the right time

Hours until right

t = \dfrac{\text{error}}{\text{rate}}

Turning shown time into real time

Shown and real time

\text{real} = \text{shown} \times \dfrac{60}{60 \pm g}

When a faulty clock is right again

Right again after

t = \dfrac{720}{\text{drift per hour}} \text{ hours}

Common traps

Carry into the hour

A clock 75 minutes ahead at 4:00 PM shows 5:15 PM, not 4:75.

Count the real hours across noon

From 9:00 AM to 2:00 PM is 5 hours. Taking

9 - 2 = 7

hours counts the wrong way and overstates the drift.

A third of an hour is 20 minutes

6\tfrac{1}{3}

hours is 6 hours 20 minutes, not 6 hours 33 minutes. Multiply the fraction by 60.

A fast clock that gains never catches up

A clock 10 minutes fast that gains 2 minutes an hour only gets further ahead. It looks right again only after gaining a whole 12 hours, the last concept on this page.

The rate is per real hour

A watch gaining 12 minutes an hour shows 6 hours gone. Taking off

6 \times 12 = 72

minutes gives 4 hours 48 minutes, but those were the watch's hours. The ratio gives

360 \times \tfrac{60}{72} = 300

minutes, exactly 5 hours.

An hour of gain is not enough

A clock 60 minutes ahead shows the wrong hour. Only a gain of 12 hours, 720 minutes, brings both hands back to the right place.

Both gain: subtract the rates

Clocks gaining 5 and 3 minutes an hour pull apart by only 2 minutes an hour, so they match again after 360 hours, not

720 \div 8 = 90

.

Clock formulas

How Fast the Hands Turn

How far a hand turns in a given time

The minute hand gains 5.5° a minute

The speeds of the two hands

Common traps

The Angle Between the Hands at a Given Time

The angle on the hour

The angle at any time

Above 180°: the smaller angle and the reflex angle

Naming, comparing and changing angles

Common traps

When the Hands Meet, Point Opposite or Form an Angle

When the hands coincide or point opposite

When the hands are at a given angle

How many times a position comes round

Common traps

Mirror Images of a Clock

The image time is 11:60 minus the real time

Finding the real time from the image

Times after 11, time passing and other twists

Common traps

Clocks That Gain or Lose Time

What a gaining or losing clock shows

When a fast or slow clock shows the right time

Turning shown time into real time

When a faulty clock is right again

Common traps

More AFCAT Numerical Ability formula sheets