# Calculating with Time

See also: NumbersEven otherwise highly numerate people have been known to throw up their hands in horror at the thought of adding together times.

A system like time, which is not decimal, can be counter-intuitive and requires concentration.

However, with a bit of application, you too can learn how to calculate with time, and gain confidence in manipulating hours, minutes and seconds.

### Basic Units of Time

**The basic units of time, which allow you to do calculations involving time, are:**

Unit | Notes | |

Second | The second is the International System of Units (SI) measurement for time. The symbol 's' is used to denote a second. Seconds are also commonly abbreviated to sec. | |

Minute | 60 Seconds | A minute nearly always has sixty seconds. However, very occasionally (approx. once every 18 months) a minute can have 61 seconds. These 'leap seconds' are used to keep our clocks aligned to the earth's rotation around the sun. |

Hour | 60 Minutes | It is common to talk about 'half an hour' (30 minutes) and 'quarter of an hour' (15 minutes). |

Day | 24 hours | |

Week | 7 days | It is also common for people to talk about a work week, usually 5 days (Monday - Friday) and weekends (Saturday and Sunday). |

Month | 28,29, 30 or 31 days. | Different months have different numbers of days. All months have either 30 or 31 days except for February. February has 28 days in a common year and an extra day in a leap year; the 29^{th} day of February is called the leap day. |

Year | 12 Months | A year always has 12 months. A year has approximately 52 weeks. A common year has 365 days and a leap year (occurring mostly every 4 years) has 366 days. |

Decade | 10 Years | |

Century | 100 Years | |

Millennium | 1,000 Years | |

**Years are commonly split into quarters, especially in business and education settings, with each quarter being three months or approximately 90 days. Quarters in a specific business or sector may not necessarily be the same as those indicated below:**

**The number of days in a month varies. All months have the same number of days each year except for February which has 28 days in a common year and 29 in a leap year.**

Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |||||||

Month | Days | Month | Days | Month | Days | Month | Days | |||

January | 31 | April | 30 | July | 31 | October | 31 | |||

February | 28/29 | May | 31 | August | 31 | November | 30 | |||

March | 31 | June | 30 | September | 30 | December | 31 |

**As the number of days in a month varies, this also means that the number of weeks in a month varies too**.

People often use approximations when calculating the number of days in months. For example, it is common to use the assumption that a month contains four weeks, even though only February in a common year actually does so.

This makes it hard to convert from months to weeks and vice versa, unless you know which months are being used, because there is no immediate ‘conversion factor’.

Top Tip!

If you are converting between months and weeks, the best way to do so is via years, because they always contain 12 months (even in a leap year).

- To convert months to weeks, divide by 12 and multiply by 52.
- To convert weeks to months, divide by 52 and multiply by 12.

The answers you get will be good approximations.

## Writing Time

There are many different ways to write time. The simplest (digital) form is as written hh.mm or hh:mm, for example 10.21 or 10:21.

Hours may be either divided into 24 (known as the *24-hour clock*) or two lots of 12 (often referred to as the *12-hour clock*).

**When using the 24-hour clock**, the 24 hour cycle starts with the time 00:00 (midnight) and runs all the way through to 23:59, before rolling over to 00:00 again. We never write ‘24:00’. You will often see the time in 24-hour clock written without a separator (‘.’ or ‘:’) between the hours and minutes. For example 1400, which may be referred to as *fourteen hundred hours*. This is especially common in military and seafaring situations.

**When using the 12-hour clock** the 24 hour period is divided into two 12-hour periods. Both midnight and midday are 12:00. Time before midday (noon) is designated * am* (

*ante meridiem*) and time after noon is referred to as

*(*

__pm__*post meridiem*). For example, 4:10pm is the same time as 16:10. Note that we never write ‘16:10pm’ The use of

*am*and

*pm*is only for the 12-hour clock.

You can also describe time in words, such as 'ten minutes past/after twelve', or 'ten past twelve'. Up to half way through an hour, we describe ‘minutes past’ the hour; beyond ‘half past’ we talk about minutes to/before the next hour. A time ‘on the hour’, for example 2:00pm, can be referred to as ‘two pm’, ‘fourteen hundred’, or ‘two o’clock’.

Calculations of How Much Time has Passed

You may need to do calculations of how much time has passed, for example, to work out the end time of an examination, how long you will have to wait for a train, or perhaps to calculate race results.

If time is described in words, you will need to convert it into a digital form in order to do any calculations.

It is also simpler to use the 24 hour clock to avoid any confusion, unless all your figures are either am or pm.

### Example time formats:

## Calculating the Passage of Time

In the normal way of things, to calculate how much one thing is greater than another you would simply subtract one from another. However, subtracting time is complicated because it’s not decimal. Instead of the columns being hundreds, tens and units, they are hours, minutes and seconds.

Top Tip

If you have a scientific electronic calculator, you will almost certainly have a button that will calculate time. Technically, it’s for degrees but as there are 60 minutes in a degree, and 60 seconds in a minute, you can also use it to calculate time.

The button will have a symbol like this:

At the very least, you should be able to use that to convert time to decimals and back again, making calculations much easier.

The other way of calculating how much time has passed between Time A and Time B is:

- Work out how many minutes from Time A to the next hour (or minute, if in minutes and seconds).
- Work out how many hours between that next hour, and the last whole hour before Time B.
- Work out how many minutes from the whole hour until time B.
- Add these three numbers together.

Worked Examples

Pat was due to catch a train at 11.44am, which she has just missed. The next train is not until 1.17pm. How long will she have to wait?

Before you start, convert all the numbers to 24-hour clock for ease. 11.44am becomes 1144 and 1.17pm becomes 1317.

*The number of minutes from 11:44 to 12:00 (the next hour) is 16.**The number of hours from 12:00 to 13:00 is one.**The number of minutes from 13:00 to 13:17 is 17.*

The total time that Pat will have to wait is 1 hour plus (16 + 17) minutes = 1 hour 33 minutes.

I am organising a series of canoe races, and have synchronised clocks for the start and the finish. The plan is for the first race to start at about 30 minutes after synchronisation, then four more races to start at 2-minute intervals after that, but it never works exactly. The starter keeps a record of the exact times, and it turns out that on this occasion, races started at 28:02, 30:00, 32:15, 34:40 and 37:00.

The finish line has a clock, which is exactly synchronised with the start clock, and the finisher records the time on the clock when the racers cross the finish line.

For each of the following finish times, work out how long the competitor took (click on the + icons to see the working and answers):

James started at 28:02 and finished at 59:02. How convenient, there are no additional seconds! You can simply subtract 28 from 59, and discover that he took **31 minutes** exactly.

Simon started at 34:40 and finished at 1:10:34. It’s probably easier to call that (60+10) = 70:34. From 34:40 to 35:00 is 20 seconds. From 35:00 to 70:00 is 35 minutes. From 70:00 to 70:34 is 34 seconds. Simon therefore took 35 minutes plus 34 seconds plus 20 seconds = **35:54 minutes**

Mary started at 30:00 exactly, and finished at 1:15:02, or 75:02. Again, this is quite straightforward. There are no seconds before the next minute, but there are 45 minutes from 30:00 to 75:00, and 2 seconds from 75:00 to 75:02. She therefore took **45:02 minutes**.

### Calculating the total amount of time spent

**Sometimes you might need to add up how much time you have spent on a task that you have completed over several days or sessions. Adding a list of hours and minutes isn’t as straightforward as doing a normal addition calculation, because (as above) we are not working with a decimal system.**

For example, you may be doing a gardening job for a neighbour, who has agreed to pay you an hourly rate of £10 per hour. On the first day you spend 3 hours and 25 minutes on the task, on the second day you spend 2 hours and 50 minutes and on the third day you finish it off in an hour and a quarter.

You need to add together the three amounts of time 3:25, 2:50 and 1:15. The first step is to add together all of the minutes: 25 + 50 + 15 = 90 minutes. If you need to remind yourself how to perform an addition calculation, have a look at our page on **addition**.

As there are only 60 minutes in an hour, the next step is to convert 90 minutes into hours and minutes:

90 ÷ 60 = 1.5 hours

The answer is a decimal, so we can convert it back into hours and minutes. You can use a scientific calculator with a degrees function as described above, or you can do a bit more maths.

We can convert 0.5 hours into minutes.

0.5 × 60 = 30

1.5 hours is therefore 1 hour and 30 minutes.

Next we need to add together the whole hours we have spent, which was 3 on the first day, 2 on the second day and 1 on the third day: 3 + 2 + 1 = 6 hours.

Add the total hours to the total minutes (in hours and minutes):

6h + 1h 30m = 7 hours and 30 minutes, or 7:30 spent gardening. In decimal form, this is 7.5 hours.

In order to calculate how much you have earned, we multiply the time (in decimals) by the hourly rate:

7.5 × 10 = 75. You have therefore earned $75.

### Conclusion

**The most important thing when calculating with time is to check whether the answer looks about right?**

**If your two numbers are less than an hour apart, is your answer less than an hour? If they are about 2 hours apart when you look at them, is your answer? If not, you may have lost or gained some time somewhere! **

**If you want to know more about ‘about right’ take a look at our page on Estimation, Approximation and Rounding.**

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Measurement Systems

Probability