*What is the angle between the two hands of the clock at 5.40?*

Sounds familiar? In almost every competitive exam, you will find either a direct question or a question based on application of clocks. All these questions, in reality, are application of circular motion concepts. In this article, we will look at the basic concepts of clocks and how to solve these questions quickly!

{As all the questions are limited to hour hand and minute hand, we will restrict the scope of this article. We will not be taking into consideration second hand here. Once you learn the concepts, they can be applied to second hand as well.}

A clock is nothing but a circle which is divided in twelve parts (from point of view of hour hand) and sixty parts (from point of view of minute hand). Hour hand, the smaller one of the two, is also called short hand; Minute hand, the longer one of the two, is also called long hand.

In one hour or 60 minutes, minute hand covers one entire round of 360^{0}

Hence, **the speed of the minute hand = 360/60 = 6 ^{0} per minute**

In one hour or 60 minutes, hour hand covers one sector out of the 12 equal sectors

Hence, **the speed of the hour hand = 30/60 = 1/2 ^{0} per minute**

Therefore, we can say that the relative speed of hour hand and minute hand (traveling in the same direction) is the difference of their respective speeds = 5.5^{0} per minute

This is quite useful when you are solving questions of the type: When will the two hands meet?

Let’s solve a few questions:

**What is the angle between the two hands of the clock at 5.45?**

There are two ways of solving this problem.

Method 1: At 5.45, the minute hand is at 9 and the hour hand is between 5 and 6. So we just need to find the distance that hour hand has covered from its initial position of 5 o clock. Hour hand covers ½ degree per minute and hence, in 45 minutes, it will cover 22.5 degrees. So from 6 o clock position, this difference is 30 – 22.5 = 7.5 degrees. And the angle between 6 and 9 is 90 degrees. Hence, the angle between the two hands of the clock at 5.45 = 90 + 7.5 = 97.5 degrees

Method 2: Let the initial position be 5 o clock where the hour hand is at 5 and minute hand is at 12. The angle between them is 30*5 = 150 degrees. From here,

Minute hand travels for 45 minutes covering 45*6 = 270 degrees

Hour hand travels for 45 minutes covering 45*1/2 = 22.5 degrees

270 – 150 = 120 – 22.5 = 97.5 degrees

**After 4 o clock, at what time, will the two hands of the clock meet?**

For all such questions, visualize what the initial stage is. At 4 o clock, minute hand is at 12 and hour hand is at 4. The distance separating them is 30*4 = 120 degrees. The relative speed of the two hands is 5.5 degrees.

Distance/speed = 120/5.5 = 240/11 minutes (21.81 minutes)

[Cross checking. In 240/11 minutes, hour hand will cover 120/11 distance from the initial position of 4 (which from 12 o clock position is 120 + 120/11 = 1440/11). In 240/11 minutes, minute hand will cover 6*240/11 = 1440/11. QED]

**At what time between 7 and 8 o clock, will the angle between hour hand and minute hand be 105 degrees?**

For such questions, one must understand if there is a single possibility of this or two possibilities. What I mean by two possibilities is: 1. where minute hand is before hour hand, clockwise. 2. Where minute hand is after hour hand, clockwise.

How do you find that? What is the upper time limit? 8 o clock. At 8 o clock, the angle between hour hand and minute hand is 8*30 = 240 or 4*30 = 120 degrees. This 120 degrees, is the limit after minute hand crosses the hour hand. (Had the question been between 8 and 9, the limit will be 90 degrees. Get it?)

As the question is 105 degrees which is under our limit of 120, there are two possibilities.

Case 1: Minute hand before hour hand.

In this case, from 7 o clock which was the initial position and the angle between the two hands was 30*7 = 210 degrees, minute hand needs to reduce this distance to 105 at relative speed of 5.5 degrees per minute

210 – 105 = 105/5.5 = 210/11 = 19.09 minutes [19.09 minutes after 7, the angle between two hands will be 105 degrees]

Case 2: Minute hand after hour hand.

In this case, minute hand needs to cover the initial distance separating the two hands and travel additional 105 degrees at relative speed of 5.5 degrees per minute.

210 + 105 = 315/5.5 = 630/11 = 57.27 minutes [57.27 minutes after 7, the angle between two hands will be 105 degrees]

You should be able to solve any clock based question with this knowledge. The most important thing is visualizing what is being asked and then using the right set of values to arrive at the answer. If you have any queries, feel free to drop a line in the comments section. Have a clock based question which you think is difficult? Send it to us and we’ll answer it for you!

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