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Problems in angles

Problems on incorrect clocks Problems in angles

Method :1

Before we actually start solving problems on angles, we need to know couple of basic facts clear:

Speed of the hour hand = 0.5 degrees per minute (dpm)

Speed of the minute hand = 6 dpm

At n o clock, the angle of the hour hand from the vertical is 30n

The questions based upon these could be of the following types

Example : 1

What is the angle between the hands of the clock at 7:20

At 7 o clock, the hour hand is at 210 degrees from the vertical.

In 20 minutes,

Hour hand = 210 + 20*(0.5) = 210 + 10 = 220 {The hour hand moves at 0.5 dpm}

Minute hand = 20*(6) = 120 {The minute hand moves at 6 dpm}

Difference or angle between the hands = 220 120 = 100 degrees

Method : 2

Example :2

Find the reflex angle between the hands of a clock at 05.30?

The above problem are solved by the bellow formula

Angle between X and Y =|(X*30)-((Y*11)/2)|

Angle between hands at 5:30

Step 1: X=5 , Y=30

Step 2: 5*30=150

Step 3: (30*11)/2 = 165

Step 4: 165-150=15

Thus, angle between hands at 5:30 is 15 degrees.

Method : 3

Example : 3

At what time 3&4o clock in the hands of clock together. Approximately we know at 03:15 hands of the clock together

So 15*60/55=16.36 min

Problems on incorrect clocks

Such sort of problems arise when a clock runs faster or slower than expected pace. When solving these problems it is best to keep track of the correct clock.

Example : 4

A watch gains 5 seconds in 3 minutes and was set right at 8 AM. What time will it show at 10 PM on the same day?

The watch gains 5 seconds in 3 minutes = 100 seconds in 1 hour.

From 8 AM to 10 PM on the same day, time passed is 14 hours.

In 14 hours, the watch would have gained 1400 seconds or 23 minutes 20 seconds.

So, when the correct time is 10 PM, the watch would show 10 : 23 : 20 PM

Important Notes

Two right angles per hour(Right angle = 90, Straight angle=180)

Forty four right angles per day

Between every two hours the hands of the clock coincide with each other for one time except between 11, 12 and 12, 1.In a day they coincide for 22 times.

Between every two hours they are perpendicular to each other two times except between 2, 3 and 3, 4 and 8, 9 and 9, 10.In a day they will be perpendicular for 44 times.

Between every two hours they will be opposite to each other one time except between 5, 6 and 6, 7.In a day they will be opposite for 22 times.

Type 1: Finding the time when the angle between the two hands is given.

Type 2: Finding the angle between the 2 hands at a given time.

Type 3: Questions on clocks gaining/losing time.

We will cover the first two types in this article.

Basic Concept of Clocks:

A clock is a complete circle having 360 degrees. It is divided into 12 equal parts i.e. each part is 360/12 = 30.

As the minute hand takes a complete round in one hour, it covers 360 in 60 minutes.

In 1 minute it covers 360/60 = 6/ minute.

Also, as the hour hand covers just one part out of the given 12 parts in one hour. This implies it covers 30 in 60 minutes i.e. per minute. This implies that the relative speed of the minute hand is 6 - = 5 degrees.

We will use the concept of relative speed and relative distance while solving problems on clocks.

Some facts about clocks:

Every hour, both the hands coincide once. In 12 hours, they will coincide 11 times. It happens due to only one such incident between 12 and 1'o clock.

The hands are in the same straight line when they are coincident or opposite to each other.

When the two hands are at a right angle, they are 15-minute spaces apart. In one hour, they will form two right angles and in 12 hours there are only 22 right angles. It happens due to right angles formed by the minute and hour hand at 3o clock and 9'o clock.

When the hands are in opposite directions, they are 30-minute spaces apart. If both the hour hand and minute hand move at their normal speeds, then both the hands meet after 65 minutes.

Now, let's apply the above concept to some questions.

Important Formulas - Clock

1. Minute Spaces

The face or dial of clock is a circle whose circumference is divided into 60 equal parts, named minute spaces.

2. Hour hand and minute hand

A clock has two hands. The smaller hand is called the hour hand or short hand and the larger one is called minute hand or long hand.

3. In 60 minutes, minute hand gains 55 minute spaces over the hour hand.

(In 60 minutes, hour hand will move 5 minute spaces while the minute hand will move 60 minute spaces. In effect the space gain of minute hand with respect to hour hand will be 60 - 5 = 55 minutes.)

4. Both the hands of a clock coincide once in every hour.

5. The hands of a clock are in the same straight line when they are coincident or opposite to each other.

6. When the two hands of a clock are at right angles, they are 15 minute spaces apart.

7. When the hands of a clock are in opposite directions, they are 30 minute spaces apart.

8. Angle traced by hour hand in 12 hrs = 360

9. Angle traced by minute hand in 60 min. = 360.

10. If a watch or a clock indicates 9.15, when the correct time is 9, it is said to be 15 minutes too fast.

11. If a watch or a clock indicates 8.45, when the correct time is 9, it is said to be 15 minutes too slow.

12. The hands of a clock will be in straight line but opposite in direction, 22 times in a day. 13. The hands of a clock coincide 22 times in a day.

14. The hands of a clock are straight 44 times in a day.

15. The hands of a clock are at right angles 44 times in a day.

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