Clock: AFCAT Numerical Ability

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)

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

Formulas (4)

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

Watch out for (7)

• Count the short way
  → The angle on the hour
• The hour hand has moved
  → The angle at any time
• Half past is not on the number
  → The angle at any time
• Above 180 is the long way round
  → Above 180°: the smaller angle and the reflex angle
• Reflex is 360 minus the smaller angle
  → Above 180°: the smaller angle and the reflex angle
• Work the angles, do not judge the times
  → Naming, comparing and changing angles
• Near 90 still needs the exact value
  → Naming, comparing and changing angles

Formulas (2)

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

Reference tables (1)

Watch out for (6)

• Eleven meetings, not twelve
  → When the hands coincide or point opposite
• A straight line includes together
  → When the hands coincide or point opposite
• Drop a value outside the hour
  → When the hands are at a given angle
• Minute spaces are not degrees
  → When the hands are at a given angle
• Count a stretch by listing
  → How many times a position comes round
• 44 right angles a day, not 48
  → How many times a position comes round

Formulas (3)

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

Watch out for (5)

• Subtract from 11:60, not 12:00
  → The image time is 11:60 minus the real time
• 60 minutes is the next hour
  → The image time is 11:60 minus the real time
• The image is not 6 hours off
  → Finding the real time from the image
• Move the real clock, not the image
  → Times after 11, time passing and other twists
• There is no 0 on the dial
  → Times after 11, time passing and other twists

Formulas (4)

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

Watch out for (7)

• Carry into the hour
  → What a gaining or losing clock shows
• Count the real hours across noon
  → What a gaining or losing clock shows
• A third of an hour is 20 minutes
  → When a fast or slow clock shows the right time
• A fast clock that gains never catches up
  → When a fast or slow clock shows the right time
• The rate is per real hour
  → Turning shown time into real time
• An hour of gain is not enough
  → When a faulty clock is right again
• Both gain: subtract the rates
  → When a faulty clock is right again