Clock problems play a major role quantitative aptitude test. There is no competitive exam without the questions from this topic. We have already learned this topic in our lower classes.Even though we have been already taught this topic in our lower classes, we need to learn some more short cuts which are being used to solve the problems in the above topic.

The only thing we have to do is, we need to apply the appropriate short cut and solve the problems in a limited time. This limited time will be one minute or less than one minute in most of the competitive exams.

**Points to Remember**

The face of a clock would be a circle whose circumference is divided in to 60 equal parts called minute spaces |

Every clock would have two hands. The smaller one is called short hand or hour hand and the larger one is called long hand or minute hand. |

The face of a clock would be a circle whose circumference is divided in to 60 equal parts called minute spaces |

In every hour, both the hands coincide once. |

If the hands are in the same straight line, either they are coincident or they are opposite to each other. |

If there is 15 minute space between the hour hand and minute hand, then they are at right angle. |

If there is 30 minute space between the hour hand and minute hand, then they are in opposite directions. |

Angle traced by hour hand in 12 hours is 360° |

Angle traced by hour hand in 1 hour = 360/12 = 30° |

Angle traced by minute hand in 60 minutes is 360° |

Angle traced by minute hand in 1 minute is 360/60 = 6° |

Angle between the hour hand and minute hand = Difference between the angles traced by hour hand and minute hand. Example: At 5.25, angle traced by the hour hand is 160° and the minute hand is 120°. Hence angle between the two hands = 160 - 120 = 40°. For detailed examples, please look at the problems given below |

Too Fast :
If the correct time is 9 and a clock indicates 9.15, it is said to be 15 minutes too fast. |

Too Slow :
If the correct time is 9 and a clock indicates 8.45, it is said to be 15 minutes too slow. |

**It is nothing but the minute spaces between the two hands. **

**When we calculate minute spaces between the two hands , we have to consider the following points. **

**1. We have to measure minute spaces from hour hand to minute hand in clock wise direction. **

**2. If the two hands are coincident, and we want the minute spaces between them to be 10 minutes, then 10 minutes to be gained. **

**3. If there is 10 minutes spaces between the two hands and we want them to be coincident, then 10 minutes to be gained. **

**In the above clock, the two hands are coincident and the time is 12'o clock. The minute spaces between the two hands is zero. If the minute hand traces 60 minutes, the time will be 1'o clock. That has been shown in the clock below. **

**From the first clock to second clock, when the minute hand traces 60 minutes, the minute spaces between the hour hand and minute hand becomes 55 in the second clock from zero in the first clock. **

**From this, it is very clear, to gain 55 minutes ( head of hour hand), minute hand must trace 60 minutes. **

**So, 55 minutes are gained in 60 minutes. **

**1 minute is gained in 60/55 (=12/11) minutes. **

**No. of minutes taken > No. of minutes gained. **

Students who are preparing to improve their aptitude skills and those who are preparing for this type of competitive test must prepare this topic in order to have better score. Because, today there is no competitive exam without questions from the topic Clock problems. Whether a person is going to write placement exam to get placed or a students is going to write a competitive exam in order to get admission in university, they must be prepared to solve Clock problems. This is the reason for why people must study this topic.

As we mentioned in the above paragraph, a person who wants to get placed in a company and a students who wants to get admission in university for higher studies must write competitive exams like placement test and entrance exam. To meet the above requirement, it is very important to score more marks in the above mentioned competitive exams. To score more marks, they have to prepare the topics like clock problems. Preparing this topic would definitely improve their marks in the above exams. Preparing this topic is not difficult task. We are just going to remember the stuff that we have already learned in our lower classes

Students have to learn few basic operations in this topic Clock problems and some additional tricks. Already we are much clear with the four basic operations which we often use in math. They are addition, subtraction, multiplication and division. Even though we are much clear with these four basic operations, we have to be knowing some more stuff to do the problems which are being asked from this topic in competitive exams. The stuff which I have mentioned above is nothing but the tricks and shortcuts which need to solve the problems in a very short time.

**Here, we are going to have some Clock problems . You can check your answer online and see step by step solution.**

1. Find the angle between the hour hand and minute hand of a clock when the time is 4.20?

Angle traced by the hour hand in 12 hours = 360°

Angle traced by the hour hand in 1 hour = 360/12 = 30°

4 hrs 20 min = 4^{20}⁄_{60} = 4^{1}⁄_{3} = 13/3 hrs

Angle traced by 4 hrs 20 min = 30X13/3 = 130°

Angle traced by the minute hand in 60 minutes = 360°

Angle traced by the minute hand in 1 minute = 360/60 = 6°

Angle traced by the minute hand in 20 min = 20X6 = 120°

Angle between the two hands = Difference between the angles traced by the two hands.

So, the angle between the two hands = 130 - 120 = 10°

At 4.20, the angle between the hour hand and minute hand is 10°

Angle traced by the hour hand in 1 hour = 360/12 = 30°

4 hrs 20 min = 4

Angle traced by 4 hrs 20 min = 30X13/3 = 130°

Angle traced by the minute hand in 60 minutes = 360°

Angle traced by the minute hand in 1 minute = 360/60 = 6°

Angle traced by the minute hand in 20 min = 20X6 = 120°

Angle between the two hands = Difference between the angles traced by the two hands.

So, the angle between the two hands = 130 - 120 = 10°

At 4.20, the angle between the hour hand and minute hand is 10°

2. At what time, in minutes, between 3 o'clock and 4 o'clock, both the needles will coincide each other?

At 3 o'clock, the min.spaces between minute hand and hour hand is 15 minutes.

To be coincident, the minute hand must must gain 15 min.spaces.

1 minute is gained in 12/11 minutes.(see the notes)

15 min.are gained in 15X12/11 min. = 180/11 = 16^{4}⁄_{11}min.

hence, the hands are coincident at 3 hrs 16^{4}⁄_{11}minutes

To be coincident, the minute hand must must gain 15 min.spaces.

1 minute is gained in 12/11 minutes.(see the notes)

15 min.are gained in 15X12/11 min. = 180/11 = 16

hence, the hands are coincident at 3 hrs 16

3.How many times do the hands of a clock coincide in a day?

The hands of a clock coincide in the following timings.

12'o clock --->1

1 to 2 ---->2

2 to 3 ---->3

3 to 4 ---->4

4 to 5 ---->5

5 to 6 ---->6

6 to 7 ---->7

7 to 8 ---->8

8 to 9 ---->9

9 to 10 ---->10

10 to 11 ---->11

From the above calculation, it is very clear that the hands coincide 11 times in 12 hours.

In 24 hours, they coincide 22 times.

Hence, the hands of a clock coincide 22 times in a day.

12'o clock --->1

1 to 2 ---->2

2 to 3 ---->3

3 to 4 ---->4

4 to 5 ---->5

5 to 6 ---->6

6 to 7 ---->7

7 to 8 ---->8

8 to 9 ---->9

9 to 10 ---->10

10 to 11 ---->11

From the above calculation, it is very clear that the hands coincide 11 times in 12 hours.

In 24 hours, they coincide 22 times.

Hence, the hands of a clock coincide 22 times in a day.

4. How many times in a day, are the hands of a clock in straight line but opposite in direction?

The hands of a clock point in opposite direction in the following timings.

6'o clock --->1

7 to 8 ---->2

8 to 9 ---->3

9 to 10 ---->4

10 to 11 ---->5

11 to 12 ---->6

12 to 1 ---->7

1 to 2 ---->8

2 to 3 ---->9

3 to 4 ---->10

4 to 5 ---->11

From the above calculation, it is very clear that the hands point in opposite directions 11 times in 12 hours.

In 24 hours, they point in opposite directions in 22 times.

Hence, they point in opposite directions 22 times in a day.

6'o clock --->1

7 to 8 ---->2

8 to 9 ---->3

9 to 10 ---->4

10 to 11 ---->5

11 to 12 ---->6

12 to 1 ---->7

1 to 2 ---->8

2 to 3 ---->9

3 to 4 ---->10

4 to 5 ---->11

From the above calculation, it is very clear that the hands point in opposite directions 11 times in 12 hours.

In 24 hours, they point in opposite directions in 22 times.

Hence, they point in opposite directions 22 times in a day.

5.At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?

At 4 o'clock, min. spaces between hour hand and minute hand is 20.

To be in opposite directions, there must be 30 min.spaces between the two hands .

If 20 minutes are gained, the hands would be coincident. But we need 30 min. spaces between the hands for being opposite in directions.

So, 30 more minutes to be gained when the are coincident.

In total, 50 minutes (20+30 = 50) to be gained for the hands to be opposite in directions.

1 minute is gained in 12/11 minutes.

50 minutes are gained in 50X12/11 minutes = 54^{6}⁄_{11}

Hence, between 4 and 5 o'clock the hands of a clock point in opposite directions at 4 hrs 54^{6}⁄_{11}minutes

To be in opposite directions, there must be 30 min.spaces between the two hands .

If 20 minutes are gained, the hands would be coincident. But we need 30 min. spaces between the hands for being opposite in directions.

So, 30 more minutes to be gained when the are coincident.

In total, 50 minutes (20+30 = 50) to be gained for the hands to be opposite in directions.

1 minute is gained in 12/11 minutes.

50 minutes are gained in 50X12/11 minutes = 54

Hence, between 4 and 5 o'clock the hands of a clock point in opposite directions at 4 hrs 54

6. How many times are the hands of a clock at right angle in a day?

Number of times of right angle between the two hands in the following timings.

12 to 1 ----> 2 times

1 to 2 ----> 2 times

2 to 3 ----> 2 times (right angle at 3'o clock included)

3 to 4 ----> 1 time (right angle at 3'o clock not included)

4 to 5 ----> 2 times

5 to 6 ----> 2 times

6 to 7 ----> 2 times

7 to 8 ----> 2 times

8 to 9 ----> 2 times (right angle at 9'o clock included)

9 to 10 ----> 1 time (right angle at 9'o clock not included)

10 to 11 ----> 2 times

11 to 12 ----> 2 times

From the above calculation, it is clear that the hands are at right angle 22 times in 12 hours.

In 24 hours, the hands are at right angle in 44 times.

Hence, the hands of a clock at right angle 44 times in a day

12 to 1 ----> 2 times

1 to 2 ----> 2 times

2 to 3 ----> 2 times (right angle at 3'o clock included)

3 to 4 ----> 1 time (right angle at 3'o clock not included)

4 to 5 ----> 2 times

5 to 6 ----> 2 times

6 to 7 ----> 2 times

7 to 8 ----> 2 times

8 to 9 ----> 2 times (right angle at 9'o clock included)

9 to 10 ----> 1 time (right angle at 9'o clock not included)

10 to 11 ----> 2 times

11 to 12 ----> 2 times

From the above calculation, it is clear that the hands are at right angle 22 times in 12 hours.

In 24 hours, the hands are at right angle in 44 times.

Hence, the hands of a clock at right angle 44 times in a day

7. The reflex angle between the hands of a clock at 10.25 is:

Hint: To find the reflex angle between the hands, find the actual

angle between the hands and subtract it from 360°.

10 hrs 25 min = 10^{25}⁄_{60} = 10^{5}⁄_{12} = 125/12 hrs

Angle traced by the hour hand in 1 hour = 360/12 = 30°

Angle traced by 10 hrs 25 min = (125/12)X30 =312^{1}⁄_{2}°

Angle traced by the minute hand in 1 minutes = 360/60 = 6°

Angle traced by the minute hand in 25 min = 25X6 = 150°

Angle between the two hands = Difference between the angles traced by the two hands.

So, the angle between the hands = 312^{1}⁄_{2}° - 150° = 162^{1}⁄_{2}°

Reflex Angle = 360°-162^{1}⁄_{2}°° = 197^{1}⁄_{2}°°

angle between the hands and subtract it from 360°.

10 hrs 25 min = 10

Angle traced by the hour hand in 1 hour = 360/12 = 30°

Angle traced by 10 hrs 25 min = (125/12)X30 =312

Angle traced by the minute hand in 1 minutes = 360/60 = 6°

Angle traced by the minute hand in 25 min = 25X6 = 150°

Angle between the two hands = Difference between the angles traced by the two hands.

So, the angle between the hands = 312

Reflex Angle = 360°-162

8. At what time between 4 and 5'o clock will the hands of a clock be at right angle (approximately)?

At 4'o clock, the minute hand will be 20 min. spaces behind the hour hand.

If we want them to be at right angle, they have to be 15 min. spaces apart.

So they would be at right angle in the following two cases.

Case1:

When the minute hand is 15 min. spaces behind the hour hand,minute hand must gain (20-15) = 5 minutes.

1 minute is gained in 12/11 minutes.

5 minutes are gained in 5X12/11 min = 5^{5}⁄_{11} minutes.

They are at right angles at 4 hrs 5^{5}⁄_{11}

Case1:

When the minute hand is 15 min. spaces ahead of the hour hand,minute hand must gain (20+15) = 35 minutes.

1 minute is gained in 12/11 minutes.

35 minutes are gained in 35X12/11 min = 38^{2}⁄_{11} minutes.

They are at right angles at 4 hrs 38^{2}⁄_{11}

Hence, the hands would be at right angles at 4 hrs 5^{5}⁄_{11} and 4 hrs 38^{2}⁄_{11}

If we want them to be at right angle, they have to be 15 min. spaces apart.

So they would be at right angle in the following two cases.

Case1:

When the minute hand is 15 min. spaces behind the hour hand,minute hand must gain (20-15) = 5 minutes.

1 minute is gained in 12/11 minutes.

5 minutes are gained in 5X12/11 min = 5

They are at right angles at 4 hrs 5

Case1:

When the minute hand is 15 min. spaces ahead of the hour hand,minute hand must gain (20+15) = 35 minutes.

1 minute is gained in 12/11 minutes.

35 minutes are gained in 35X12/11 min = 38

They are at right angles at 4 hrs 38

Hence, the hands would be at right angles at 4 hrs 5

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