**Formulas for Clock**

**A) Angle between hands of a clock**

1. When minute hand is behind the hour hand, the angle between the two hands at M minutes past H o’clock.

= 30 [H -(M/5)] + M/2 degree

= 30H – (11M/2)

2. In the case where the minute hand is ahead of the hour hand, the angle between the two hands at M minutes past H ‘o clock will be calculated as

= 30 (M/5-H ) -M/2 degree

= 11/2M – 30H

**B) To calculate x minute space gain by the minute hand over the hour hand,**

** = x60/55**

** = x12/11**

**C) Both the two hands of the clock will be at the right angles between H and (H+1) o’clock at ****(5H ± 15) minutes H<’o clock.**

**D) If the minute hand of a clock overtakes the hour hand at the interval of M minutes when the time is correct, the clock gains or loses a day by**

**= [(720/11)-M] [(60×24)/M] minutes.**

**E) Between H and H+1 o’clock, the two hands of the clock are M minutes apart at ****(5H ± M)12/11 minutes past H o’ clock.**

**Clocks Formulas & Properties**

- The dial of the clock is circular in shape and is divided into 60 equal minute spaces.
- 60-minute spaces traces an angle of 360
^{0}. Therefore, one minute space traverses an angle of 6^{0}. - The hands of a clock are perpendicular when they are 15-minute space apart.
- Hands of a clock are in straight line and are opposite to one another when they are 30 minutes space apart.
- Hands of a clock are perpendicular to each other 22 times in 12 hours and 44 times in 24 hours.
- Hands of the clock are opposite to each other for 11 times in 12 hours and 22 times in a day.
- The minute hand gains 55 minutes over hour hand every hour.

**Type 1: How To Solve Quickly Clocks**

**Question 1. Calculate the angle between the two hands of a clock when the time shown by the clock is 7:30 a.m.**

**Options:**

**A. 100 ^{0}**

**B. 155 ^{0}**

**C.200 ^{0}**

**D. None of the above**

**Solution **From the formula

Angle = 11/2M – 30H, where M =30, and H =7

= 11/2*30 – 30 x 7

= |165 – 210|, taking the mode of the values, we get

= 155^{0}

**Correct Answer: Option B**

**Question 2.** **Calculate the degree at which the hour hand moves when the minute’s hand moves 750 times.**

**Options:**

**A. 13 ^{0}**

**B. 12*(3/6)°**

**C. 15(3/6)°**

**D. Cannot be determined**

**Solution: **Degrees covered by hour hand in one minute = 1/60

The minute hand will cover 750/60 degrees in 650 minutes.

This can be written as = 12 (3/6)° degrees.

**Correct Answer: Option B**

**Question 3. What is the angle an hour hand has to trace when it covers the time from 8 A.M in the morning to 2 o’clock in the afternoon?**

**Options**

**A. 120**

**B.180**

**C.250**

**D. None of the above**

**Solution: **Time between 8 A.M to 2 P.M = 6 hours.

Therefore, the angle traced by the hour hand = 360/12 x 6

= 180^{0}

**Correct Answer: Option B**

**Type 2: Solve Quickly Clocks Ques.**

**Ques 1. Determine the time at which the minute and the hour hand will be in the same straight line facing opposite directions between 2 and 3 o’clock.**

**Options:**

**A. 2:43(7/11) P.M.**

**B. 2:45 P.M.**

**C.2:50(1/13) P.M.**

**D. None of the above**

**Solution: **Degrees in a straight line = 180^{0}

When we apply the formula, 11/2M – 30H

= (180+60)/2 = 11M

Or M =43(7/11)

**Correct Answer: Option A**

**Question 2. A clock is set to correct timing at 8 a.m. In the next 24 hours, it gains 10 minutes. What will be the true time when the clock displays time to be 1 P.M?**

**Options**

**A. 12: 48 P.M**

**B. 2:50 P.M.**

**C. 1: 30 A.M.**

**D. Cannot Be Determined**

**Solution: **Number of hours from 8 A.M on a day to 1 P.M the next day = 29

24 hours 10 minutes of this clock = 24 hours of a correct clock

145/6 hours of this incorrect clock = 24 hours of the correct clock

29 hours of incorrect clock = 24 x 6/145 x 29 hours of the correct clock

= 28 hours and 48 minutes of the correct clock

Therefore, the correct time is 28 hours and 48 minutes after 8 A.M

Which means, the true time will be 12:48 P.M.

**Correct Answer: Option A**

**Question 3. What is the time between 4 and 5 o’clock, when the hands of the watch points in the opposite direction?**

**Options**

**A. 4:54 o’clock**

**B. 5 o’clock**

**C. 10 o’clock**

**D. Cannot be Determined**

**Solution: **Hands of the clock are 20 min. spaces apart when the time is 4 o’clock

They must be 30 min. Spaces apart to be in opposite directions.

Minute hand needs to gain 50 min. spaces

A clock gains 55 min. spaces in 60 minutes

50 min spaces are gained in

= 60/55 x 50 minutes

= 54(6/11)

The needed time = 54(6/11) past 4, or

= 4: 54

**Correct Answer: Option A**

### Best Tips & Tricks on Clocks problems to solve easily

- Below are some easy tips and tricks for you on problems based on clocks which efficiently help in the competitive exam . It is very important topic for recruitment drive.
- Speed of the minute hand= 6° per minute.
- Speed of the hour hand= 0.5° per minute.
- Relative Speed between hour and minute hand
**.**Relative speed of minute hand is 6 – 0.5 dpm = 5.5 dpm. Where dpm = dial per minute.

### Type 1: Tricks and Tips and Shortcuts on Clocks.

**Question 1. What is the angle between the minute hand and the hour hand when the time is 5:30.**

**Options:**

**A.15 °**

**B. 10°**

**C. 20°**

**D. None of the above**

**Solution: **Tip: Simple formula to calculate the angle between the minute and the hour hand = (X*30)-((Y*11)/2)

Multiplying hours according to formula = 5 x 30 = 150

Applying the formula, we get (Yx11) / 2

= 30 x 11/ 2

= 165

When we subtract the two values, we get,

= 165 – 150

= 15°

**Correct Option: A**

### Type 2: Shortcuts, Tricks to Solve Clocks

**Question 1 Calculate the time between 7 and 8 o’clock when the hands of a clock are in the same straight line but are not together?**

**Options:**

**A. 7:05(5/11 ) o’clock**

**B. 7:50 (1/13) o’clock**

**C. 7:20 o’clock**

**D. Cannot be determined**

**Solution: **Tip: The short formula to calculate the time when the angle is given is,

Angle = 30 (hours) – 11/2 (minutes)

Using the above formula, we get

180 = 30 (hours) – 11/2 (minutes)

180 = 30 (7) – 11/2 (minutes)

11/2 minutes = 210-180

Minutes = 30*2/11

= 5( 1/5)

Time = 7:05(5/11) o’clock

**Correct Option: A**