Time

Introduction

There are 60 x 60 = 3 600 seconds in an hour

So for example,

The number of seconds in 4 hours \[= 4 \times 60 \times 60\] \[= 14\ 400\ \mbox{seconds} \]

The number of hours corresponding to 25 200 seconds \[ = \frac{25\ 200}{3\ 600} \]
= 7 hours

Quick Quiz

How many seconds are there in
  1. \[ 1) \ \ 5\ \mbox{hours} \]
  2. \[ 2) \ \ 2\ \mbox{hours } 30\ \mbox{min} \]

How many minutes are there in
  1. \[3) \ \ 12\ \mbox{hours} \]

Convert to hours
  1. \[ 4) \ \ 14400 \ \mbox{seconds} \]
  2. \[ 5) \ \ 780\ \mbox{minutes} \]
  3. \[ 6) \ \ 1440\ \mbox{minutes} \]

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24 Hour Clock






No doubt you will already have seen the 24-hour clock in use on railway timetables, for example (Other countries make much more use of it than in Britain).

The '12 hour' clock requires that times are followed by either 'a.m.' or 'p.m' in order to avoid ambiguity.

Times on the 24 hour clock are always represented by four numbers - but nothing else is required to be added.

  • Some people just write four numbers, e.g. 0452, 0930, others insert a colon in the middle, e.g. 04:52, 09:30.
  • Times like 0900, 1300 are commonly stated as 9 hundred hours or 13 hundred hours
  • Midnight on the 24 hour clock is either 2400 or 0000, but times in the first hour after midnight will always be of the form 0049, 0001, etc. Having said that, note that 0000 is often the only notation used for midnight on things like Internet booking sites where it is essential to remove any ambiguity

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Decimal Form

Note that

7 hrs 30 mins = 7.5 hrs

It is definitely NOT 7.3 hours.

Converting time to a decimal form is necessary if the time is to be used in arithmetic calculations.

Example

If 720 items are produced in a factory in 7 hours 30 mins, how many are produced per hour on average

\[ \frac{720}{7.5} \] = 96

You definitely do not state the denominator as 7.3 hours, this would give you the wrong answer. The correct denominator is 7.5 hours.


To express minutes as a decimal (of an hour) you might find it easier to first convert to a fraction, and then convert this fraction to a decimal, e.g. to express 12 mins as a decimal (of an hour)

  • First state as a fraction of an hour

    \[ \frac{12}{60} \]

  • Convert to a decimal by treating this fraction as a division

    0.2

With experience, you can introduce shortcuts, e.g. you could recognize the fact that 6 mins is 0.1 hours, so

24 minutes = 0.1 x 4 = 0.4 hours

You might like to view this conversion chart

Quick Quiz
Express in decimal form
  1. 5 hours 45 mins
  2. 2 hours 30 mins
  3. 2 hours 40 mins
  4. 5 hours 48 mins

Express as hours and minutes
  1. 5.45 hours
  2. 3.2 hours
  3. 9.05 hours


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Time Intervals


In everyday life, time is not usually represented in a decimal form, so we need to use a slightly different technique to find the interval between two specific times.


Example

To find the time between 0320 and 0945, the technique would be :

  • count the whole hours between 0400 and 0900

  • add the extra minutes on either side of 0400 and 0900

    • On the '0400' side we need to add 40 minutes (the difference between 0320 and 0400)

    • On the '0900' side, we add 45 minutes (the difference between 0900 and 0945)

So the calculation is :

5 hours + 40 minutes + 45 minutes

= 5 hours + 1 hour + 25 minutes

= 6 hours + 25 minutes

Alternatively you could just say :- from 0320 to 0920 is 6 hours and from 0920 to 0945 is another 25 minutes, therefore the total time interval is 6 hours and 25 minutes

Quick Quiz
Find the time interval between
  1. 0934 and 1245
  2. 2145 and 2234
  3. 0454 and 2145

  1. 9.23 a.m. and 11.56 a.m.
  2. 5.45 a.m. and 9.25 p.m.
  3. 11.45 a.m. and 6.12 p.m.

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Average Speed

Average Speed = Distance Travelled ÷ Time Taken

From which we can derive

Time Taken = Distance Travelled ÷ Average Speed

Distance Travelled = Average Speed × Time Taken

I have used simple algebra to derive the last two equations, but you could derive these formulae separately, without algebra, by just thinking about the physical situation. If you are not too happy with algebra, just spend a bit of time convincing yourself that the last two formulae are correct, and think how you would be able to produce them to solve a given question ( by that, I mean if you can understand the logic, you should be able to reproduce them without necessarily committing the formulae to memory).


Example : If a car covers 720 km in 7 hours 30 mins, what is its average velocity ?

\[ \mbox{Average Velocity} = \frac{\mbox{Distance}}{\mbox{Time}} \]

\[ = \frac{750}{7 h\ 30 m} \]

\[ = \frac{750}{7.5} = 100\ km / h\]

Example : If a train travels for 3 hours at an average speed of 35 km/h, how far has it travelled ?

Distance Travelled = Average Speed × Time Taken

= 35 × 3

= 105 kms

Example : If a journey involves a distance of 3000 kilometres hours at an average speed of 35 km/h, how long will the journey take ?

Time Taken = Distance Travelled ÷ Average Speed

= 3000 ÷ 35

= 85.71 hours (to 2 d.p.)

Quick Quiz
  1. 1. If a car travels 120 kilometres in 3 hours, what is its average speed ?
  2. 2. If a train travels 450 kilometres in 1hour 40 mins, what is its average speed ?
  3. 3. If an aircraft travels 2 000 kilometres at an average speed of 400 km/hr, how long will the journey take ?
  4. 4. To travel a distance of 300 kilometres at an average speed of 80 km/hr will require what length of time ?
  5. 5. If a train has been travelling at a constant speed of 225 km/hr for 3 hrs 30 mins, how far has it travelled ?
  6. 6. If you travel up the motorway at a constant speed of 115 km/hr, how far will you have travelled in 5 hours ?

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Average Speed - Part 2


A more complicated example

A car travels 60 km at 30 km/hr, and then a further 120 km at an average speed of 40 km/hr. Work out its average speed for the entire journey.

Given that the formula is

Average Speed for Entire Journey = Total Distance Travelled ÷ Total Time Taken

We need to know the

Total Distance Travelled ( which is 60 + 120 = 180 km)


What we don't know yet is the total time taken, which is more complicated.

So consider each stage separately.

  1. To travel 60 kms at 30 km / hr requires
    \[ \frac{60}{30} = 2\ \mbox{hours} \]
  2. To travel 120 kms at 40 km / hr requires
    \[ \frac{120}{40} = 3\ \mbox{hours} \]

So the Total Time Taken = 2 + 3 = 5 hours.

So finally the average speed is given by

Average Speed for Entire Journey = Total Distance Travelled ÷ Total Time Taken


\[ = \frac{180}{5} = 36\ km / h \]

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Past Test Questions


The average speed for a journey of 300 miles was 50 miles per hour (mph). The average speed for the first 150 miles was 60 mph. What was the average speed for the second 150 miles of the journey ?

  • A    between 55 and 60 mph
  • B    between 50 and 55 mph
  • C    between 45 and 50 mph
  • D    between 40 and 45 mph

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