GETTING STARTED
☛ A clock has two hands – Minute hand and Hour Hand. The face of a clock is divided into 60 parts- known as ‘minute spaces’. Every minute space corresponds to 6°.
☛ Minute hand (longer hand) starts from 12 and covers one full circle i.e. 360 ° in 1 hour
☛ The Hour hand starts at 12 and comes back to 12 (complete one full circle) in 12 hours
☛ In one hour (60 minutes), minute hand covers 360° whereas the hour hand covers only 30°
☛ In one minute, minute hand covers 6° whereas hour hand covers only 0.5°(Dividing 360° and 30° by 60 to get values for 1 minute)
Note: The above fact is important and will come in handy while solving questions
☛ In span of 60 minutes or 1 hour, minute hand covers 330° more than that of hour hand. Therefore, minute hand covers 5.5° more than that of hour hand
One of the most frequently asked question in clocks is:
Steps Involved:
1. Find angle traced by each – minute hand and hour hand
2. Subtract smaller angle from larger angle
- Finding the angle formed between two hands of the clock
Eg. Finding angle between hands of the clock at 2:55 PM Step 1: Angle traced by hour hand = Angle covered from 12:00 to 2:00 + Angle covered in 55 minutes (from 2:00 to 2:55) = [2 x 30°+55 x 0.5°] ( Hour hand covers 0.5° in 1 minute) = 87.5° Angle traced by minute hand = Angle covered in 55 minutes from 2:00 to 2:55 = [55 x 6°] = 330° Step 2: Difference in angle = 330° – 87.5° = 242.5°or 117.5° |
NOTE:
In clock problems, an angle can be expressed in two ways : x° or (360-x)°. The smaller of the two is considered as the angle and the larger one is known as the ‘reflex angle’. The angle between the two hands is therefore, 117.5°
Let us consider another example:
Eg. Finding angle between hands of the clock at 5:30 PM Step 1: Angle traced by hour hand = Angle covered from 12:00 to 5:00 + Angle covered in 30 minutes (from 5:00 to 5:30) = [5 x 30° + 30 x 0.5°] (Hour hand covers 0.5° in 1 minute) = 165° Angle traced by minute hand = Angle covered in 30 minutes from 5:00 to 5:30 = [30 x 6°] = 180° Step 2: Difference in angle = 180° – 165° = 15°or 345°. Thus, 15° is the angle between two hands and the reflex angle is 345° |
Another type of question can be finding the time at which a particular angle is formed:
- Finding the time when angle will be x°
CASE-I: Required angle less than angle at start of hour
Let us find the time when the angle between hands is 30° between 4 PM and 5 PM.
Consider both the hands to runners in the clock. At 4 PM, the hour hand is leading by 120° whereas minute hand is at 0°. This is a circular track.
The angle between the hands will be 30° when:
CASE-II: Required angle more than angle at start of hour
Let us find the time when the angle between hands is 160° between 4 PM and 5 PM.
In previous example, required angle was less than the angle formed at the start of the hour. Here, 160° is more than 120°, angle at start of hour
The angle between the hands will be 160° when:
Therefore, if the x° is less than the angle formed by the hands at the start of the hour, use Case-I
If x° is more than the angle formed by the hands at the start of the hour, use Case -II
- Hands of clock forming 0°, 180° and 90°
Number of times hour hand coincides with hour hand in a day = 22 times
Number of times hour hand is exactly opposite to hour hand in a day = 22 times
Number of times hour hand and minute hand are at 90° in a day = 44 times
The Detailed Explanation for above fact is: DETAILED EXPLANATION_Clocks