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# Elapsed Time

## Definition

Time can be defined as the dimension based on which the evolution of any system takes place. We measure time in seconds, minutes, hours, days, weeks, months and years with clocks and calendars. Let us learn about these one by one.

## Understanding Time

Time is measured in hours, minutes and seconds. How are these three related to each other?

1 minute = 60 seconds

1 hour = 60 minutes

This means that 1 hour = 60 x 60 = 3600 seconds

A second is a very short period of time, say the time that is taken to blink your eye, to snap your fingers etc. it is a smaller unit of time than a minute.

## Telling Elapsed Time to Exact Minute

To understand how to tell time to exact minute, let us consider an example.

Suppose Victor and his sister Alice were going to school. When they started from home, the following time was shown by the clock –

Victor asked her to read the time. She read it easily as 7:30. Then they reached the school and the time shown by the clock in the school is

Victor again asked her to read the time. But, she was not able to read the time to the exact minutes. Now, Victor told her it was 8: 10. Let us understand how to read time to exact minutes.

If you see carefully, there are 12 numbers and 60 divisions on the face of a clock. Therefore, we have,

1 hour = 60 minutes.

Hence, the number of divisions between two consecutive numbers is 60 ÷ 12 = 5 divisions or 5 minutes.

We will read the time on the given clock as 2 :07 if,

• The hour hand of the clock is at 2
• The minute hand of the clock is at the 2nd division after the 1st number, i..e 7 minutes ( 1 x 5 = 5) + 2 minutes = 7 minutes.

Hence, we can say that the numbers on the clock actually represent hours and the divisions between them represent the minutes.

It is important to know here that –

Quarter of an hour = 60 ÷ 4 = 15 minutes

Half of an hour = 60 ÷ 2 = 30 minutes.

12 :00 o’clock in the night is 12:00 midnight

12 :00 o’clock at the daytime is 12:00 noon

We do not write a.m. or p.m. with 12:00 o’clock time.

In what time to the hour, minute and second hands complete one round or one revolution of the clock?

• The hour hand of the clock goes round the clock in 12 hours.
• The minute hand completes one round in 1 hour.
• The second’s hand completes one round in 1 minute

## Time in a.m. and p.m.

We can observe that there are 12 numbers written on the face of a clock. We also know that a day has 24 hours. Therefore, the hour hand of the clock completes 1 round in 12 hours. So, we divide 24 hours of a day in two periods:

1. From 12:00 midnight to 12:00 noon
2. From 12:00 noon to 12:00 midnight

The time between 12:00 midnight to 12:00 noon is written as a.m. Here a.m. means ante meridiem which means before midday.

The time between 12:00 noon and 12:00 midnight is known as p.m. Here p.m. means post meridiem which means after midday.

Another thing that should be noticed is that at times we see time written in a.m. or p.m. while at times it is written as, say, 16:00 with no mention of a.m. and p.m.

## 12-hour Clock vs 24-hour Clock

There are two ways of telling the time:

The 12-hour clock runs from 1am to 12 noon and then from 1pm to 12 midnight.

The 24-hour clock uses the numbers 00:00 to 23:59 (midnight is 00:00).

Let us do a time comparison between the two styles of telling time.

## Understanding the Clock

We see different types of clocks everywhere around us. But do we define a clock? A clock is an instrument to measure time which is shown in hours, minutes and seconds. There are two types of clocks –

Analog Clock – Analog clocks are the clocks that use hands to show us the hours, minutes and seconds are called analog clocks. There are two hands in the analog clocks –

The little hand shows the hour.

The bigger hand shows the minutes.

In many clocks, there is another big constantly moving hand that shows the seconds.

Digital Clock – Digital Clocks are the clocks that use numbers to show the hours and minutes such as 5:35.

Consider the clock shown above. We can see that there is a short hand at 5 which represents the hour, while the bigger hand at 3 represents the minutes. So, the time is 5 hours and 15 minutes.

### Third Hand of a Clock

Many of the clocks have three hands instead of two. What is this third hand for? The third hand is usually long and thin, and it is called the second hand. The seconds hand takes 60 seconds to complete one round of the clock.

Therefore, 1 minute = 60 seconds.

Hence the second hand is a smaller unit of time than a minute.

Thus, second is the smallest unit and hour is the largest unit among the three units of time i.e. hour, minute and second.

When the second hand moves once around the clock the minute hand moves through 1 minute

## Visualising Elapsed Time on a Number Line

Since time is not represented using a decimal system, the difference in time between 2:05 and 1:55 is 0:10 as opposed to 0:50. In other words, conventional subtraction suggesting that 2:05 – 1:55 = 0:50 is incorrect. A good alternative to subtraction (which will still work sometimes e.g. 2:30 – 2:15) is to use an “add-on” method. This is best done using or visualizing a number line. A number line is an excellent tool when working with elapsed time.  Below is a representation of elapsed time on a number line –

Let us consider an example. What would be the elapsed time between 8:40 and 9:00? We now know that it would be 20 minutes between 8:40 and 9:00. Similarly,, what would be the elapsed time between 9:00 and 11:00. It would be 2 hours. How can we represent these on a number line? Below is the representation of these on a number line –

### Using Number Line to represent elapsed one hour

Below is the graphic representation of how a number line can be used to represent the elapsed time in an hour –

### Using Number Line to represent elapsed 2 hours or more

What if we wish to show a jump of 2 hours or 3 hours? Below is the representation of the same on the number line –

### Using points on a number line as stepping stones to understand elapsed time

We can use the points on a number line as “ stepping stones ” and add on smaller “chunks of time” to find the total elapsed time. Below is an example, showing an elapsed time of 2 hours and 30 minutes –

Another way to add-on with the number line is to take as many 1 hour steps as we can before going past the end time. In the example below, the first step is from 3:15 to 4:15. The second step is from 4:15 to 5:15. Another one hour step would take us too far along the line so we stop after 2 hour steps and then find what needs to be added on to get to the end time. Here we need a 15 minute final step to give has a total elapsed time of 2 hours 15 minutes.

## Spanning 12 o’clock

Not surprisingly there can be difficulty overcoming the ..9, 10, 11, 12 , 1, 2.. the sequence of hours. The time number line helps as we can use 12 as a “stepping stone” and break the elapsed time into manageable “chunks of time.” Below we have an example, showing elapsed time between 8:30 and 2:15 –

It is important to remember here that there are no rules to these methods. We can identify and use whatever “chunks of time” are easiest for us. For instance, we can represent the elapsed time between 7:45 and 10:21 as follows –

## Solved Examples

Example 1 Mother says that the dinner will be ready in 45 minutes. It is 6:30 p.m. now. What time will dinner be ready?

Solution We have been given that the mother says that the dinner will be ready in 45 minutes and it is 6:30 p.m. now. We need to find out what time the dinner will be ready?

The first thing that we would do when solving for elapsed time is to figure out our start time and end time. Here, our start time is 6:30 pm and the end time is 45 minutes after the start time.

Next, we are going to count to the nearest hour after the start time, i.e. 6:30 pm.  Looking at the clock, we know that the next hour after 6:30 p.m. would be 7:00 p.m.  So counting from 6:30 p.m. to 7:00 p.m. we see that there are 30 minutes from 6:30 p.m. and 7:00 p.m.

So, we have now counted 30 minutes out of the given 45 minutes interval. Now, we shall count the remaining fifteen minutes after 7:00 p.m. again, looking at the clock we can see that 15 minutes after 7:00 p.m. would be 7:15 p,m. So, what would be the time 45 minutes after 6:30 p.m? It would be 7:15 p.m. Hence, the dinner would be ready by 7:30 p.m.

Example 2 Change the 24-hour clock time to 12-hour clock time

a) 11 : 15 hours

b) 16 : 40 hours

Solution We have been given time in 24-hour clock and we need to change it into 12-hour clock. Let us do them one by one.

a) 11 : 15 hours

We know that 11 : 00 in 12 hour format means 11 : 00 am. Therefore 11 : 15 hours in 12 hour format will mean 11 : 15 a.m.

b) We have been given 16 : 40 hours.

Now, 16 = 12 + 4

Therefore, 16 : 00 hours  = 4 : 00 p.m.

Hence,  16 : 40 hours = 4 : 40 p.m.

Example 3 Alicia’s family is traveling from Atlanta, Georgia to New York by plane. Their flight leaves at 9:15 a.m. and should take 2 hours and 15 minutes. What time will their plane arrive in New York?

Solution We have been given that Alicia’s family is traveling from Atlanta, Georgia to New York by plane. Their flight leaves at 9:15 a.m. and should take 2 hours and 15 minutes. We need to find the time at which their plane arrives at New York.

The first thing that we would do when solving for elapsed time is to figure out our start time and end time. Here, our start time is 9:15 a.m. and the end time is 2 hours and 15 minutes after the start time.

Next, we are going to count to the nearest hour after the start time, i.e. 9:15 a.m.  Looking at the clock, we know that the next hour after 9:15 a.m.. would be 10:15 a.m.  So counting from 9:15 a.m. to 10:15 a.m.. we see that there is one hour from 9:15 a.m. and 10:15 a.m. Since we are required to find the time 2 hours and 15 minutes after 9:15 a.m., therefore we shall now jump another 1 hour from 10:15 a.m. Looking at the clock, we know that the next hour after 10:15 a.m.. would be 11:15 a.m.  So counting from 10:15 a.m. to 11:15 a.m.. We see that there is one hour from 10:15 a.m. and 11:15 a.m.

So, we have now counted 2 hours from 9:15 a.m. and reached at 11:15 a.m. we are left to count 15 minutes now. Looking at the clock again, we can see that 15 minutes after 11:15 a.m. would be 11:30 a.m. So, what would be the time 15 minutes after 11:15 a.m? It would be 11:30 a.m. Hence, the plane would arrive in New York at 11:30 a.m. which is 2 hours and 15 minutes after it left Atlanta.

## Key Facts and Summary

1. Time can be defined as the dimension based on which the evolution of any system takes place.
2. A number line is an excellent tool when working with elapsed time.
3. 1 minute = 60 seconds and 1 hour = 60 minutes
4. A second is a very short period of time, say the time taken to blink your eye, to snap your fingers etc.
5. A clock is an instrument to measure time which is shown in hours, minutes and seconds.
6. Quarter of an hour = 60 ÷ 4 = 15 minutes
7. Half of an hour = 60 ÷ 2 = 30 minutes.
8. 12 :00 o’clock in the night is 12:00 midnight
9. 12 :00 o’clock in the daytime is 12:00 noon
10. We do not write a.m. or p.m. with 12:00 o’clock time.
11. The numbers on the clock actually represent hours and the divisions between them represent the minutes.
12. Second is the smallest unit and hour is the largest unit among the three units of time i.e. hour, minute and second. When the second hand moves once around the clock the minute hand moves through 1 minute
13. The time between 12:00 midnight to 12:00 noon is written as a.m. Here a.m. means ante meridiem which means before midday.
14. The time between 12:00 noon and 12:00 midnight is known as p.m. Here p.m. means post meridiem which means after midday.
15. The hour hand of the clock goes round the clock in 12 hours.
16. The minute hand completes one round in 1 hour.
17. The second’s hand completes one round in 1 minute
18. We can use the points on a number line as “ stepping stones ” and add on smaller “chunks of time” to find the total elapsed time.