In clock problems you usually calculate time and angles.

Clock Hands

- Minute Hand (Base)
- Hour Hand
- Seconds Hand

Minute Hand | Second Hand | Minute Hand | Hour Hand |
---|---|---|---|

1 min | 60 min | 1 min | 1/12 min |

2 min | 120 min | 2 min | 1/6 min |

30 min | 1800 min | 30 min | 5/2 min |

... | ... | ... | ... |

n min | 60n min | n min | n/12 min |

This simply means that, if the minute hand moves n minutes the second hand moves 60n minutes and the hour hand moves n/12 minutes.

Minutes | Angle (degree) from 12 |
---|---|

15 | 90 |

30 | 180 |

60 | 360 |

Use this as a conversion unit, so if you like to calculate how many degrees is 12 minutes. 12 x 90 / 15 = 72 degrees

Basic Principle of Clock Calculation:
Draw and label the figure
Create two equations(sum along the first circle : forward = backwards)

What is the angle between the hour hand and the minute hand at 3:30?

Solution:

sum forward (blue line) = \( 15 + 30/12 + \theta \)

sum backwards (black line) = 30

\(15 +30/12 +\theta \) = 30

\( \theta\) = 25 minutes = 75 degrees

Using the hour hand, **rewrite it as described by minute hand**. From 12 to 3 that's 15 minutes, from 3 to where it is now is 30/12 minutes(*refer to the table above*) and the rest up to 6 is unknown.

The minute hand just moves 30 minutes.

Another Solution:

Convert all minutes to degrees

15 minutes = 90 degrees

30/12 minutes = 15 degrees

30 minutes = 180 degrees

sum forward (blue line) = \( 90 + 15 + \theta \)

sum backwards (black line) = 180

\( 90 + 15 + \theta \) = 180

\( \theta\) = 75 degrees