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## Rounding

### Introduction

As examples, you might be asked to

 round to the nearest number round to the nearest thousand round to the nearest centimeter

In everyday life, distances are commonly quoted to the nearest kilometre, or whatever - they are not usually quoted precisely, except in rare cases.

For example, if the distance between two points is stated as 9 kilometres, then you might possibly expect that the real distance could be anywhere between 8.5 kilometres and 9.5 kilometres.

 Margot Is egg production going OK. How many did you have today ? Tom About two dozen Barbara I collected 22 this morning Tom That's about two dozen, isn't it ? Barbara Well, no it isn't if you want to be precise, no it isn't. Tom I wasn't being precise. Remember, I said about two dozen. Barbara Then why are you making such a big issue about it, then ?

 A pedant writes Sir, In Rotten Boroughs in Eye 1048 you included an article about Age Concern charity in Doncaster, in which the charity was criticized for receiving £ 50,000 for a " shopping for oldies" scheme. "Alas only 204 shopping visits were completed in 18 months - at a cost of some £ 245 per trip". Maybe it is because I am studying to be an astrophysicist and I have to do maths every day at uni, but I appear to be the only one to have noticed that 245 x 204 = 49,980. I wish you luck with your maths in the future, Faithfully yours ANDREW BOUDA Czech Republic

### Rounding to the Nearest Whole Number

For example,

• the number 29.345 will be rounded to 29
• the number 29.654 will be rounded to 30

The important feature is the first decimal place.

 If this 5 or greater, then the number will be rounded up       If the number is 4 or less, then the number will be rounded down

So all the following numbers will become 25

• 24.5
• 24.513
• 24.769
• 24.976543
• 25.034
• 25.3
• 25.476

Remember - the first decimal place is all that needs to be considered.

 Quick Quiz     Round these numbers to the nearest whole number   a) 34.65   b) 234.789   c) 12.3469   d) 12.3   e) 345.5

### Rounding to the Nearest Ten

This, as for all rounding, employs a procedure analogous to that used for rounding to whole numbers.

For example,

 the number 67 will be rounded to 70 to the nearest ten the number 42 will be rounded to 40 to the nearest ten the number 85 will be rounded to 90 to the nearest ten

The important feature in this case is the first "ignored" number.

 If this 5 or greater, then the number will be rounded up If the number is 4 or less, then the number will be rounded down

 Quick Quiz     Round these numbers to the nearest ten   a) 34.65   b) 2349   c) 124.69   d) 12.3   e) 3467   f) 23452

### Rounding to the Nearest Hundred, Thousand.. etc.

For example,

 the number 329.345 will be rounded to 300 to the nearest hundred the number 367 will be rounded to 400 to the nearest hundred the number 4329.654 will be rounded to 4000 to the nearest thousand the number 6829.4 will be rounded to 7000 to the nearest thousand

The important feature in this case is the first "ignored" number.

 If this 5 or greater, then the number will be rounded up If the number is 4 or less, then the number will be rounded down

 Quick Quiz     Round these numbers to the nearest hundred   a) 346.5   b) 2349   c) 1473.9   d) 1234.3   e) 3467   f) 23821 and these to the nearest thousand   g) 3895   h) 2426   i) 5473.9   j) 52343   k) 32766   l) 345821 and these to the nearest million   m) 8 378 723 n) 45 983 233

 Quick Quiz   The crowd at a fotball match is given as 34,700 to the nearest hundred. What is the largest number and lowest number of spectators that could actually have attended ?

### Rounding to the Nearest 0.1 (Rounding to One Decimal Place)

For example,

 the number 329.345 will be rounded to 329.3 to the nearest 0.1 ( or nearest tenth ) the number 36.78 will be rounded to 36.8 to the nearest 0.1 ( or nearest tenth )

Note the similarity with the previous methods of rounding.

 Quick Quiz     Round these numbers to the nearest 0.1   a) 32.3492   b) 2034.234   c) 526.682   d) 12.034   e) 46.873   f) 23.52

### Rounding to a number of Decimal Places

Rounding so that there is one figure to the right of the decimal point is called

rounding to one decimal place

Rounding so that there are two figures to the right of the decimal point is called

rounding to two decimal places

and so on, by analogy.

Example

234.5854

 rounded to 1 decimal figure :   234.6 rounded to 2 decimal figures :   234.59 rounded to 3 decimal figures :   234.585

Example

5.0134

rounded to one decimal place would be

5.0

### Significant Figures

A number which has been rounded to the first non-zero figure has been rounded to one significant figure. For example

 132 100 0.00249 0.002

By analogy, rounding to 2 significant figures produces

 132 130 0.00249 0.0025

Note that when displaying numbers to a certain number of significant figures, any leading zeros are not taken into account. For example,

0.00923

rounded to 2 significant figures is

0.0092

However, trailing zeroes are taken into account. For example,

20.43

rounded to 3 significant figures is

20.4

and

102

rounded to 2 significant figures is

100

Important When doing calculations, your answer should not be given to a greater number of significant figures than the least number of significant figures used in your input data

For example, if your input data was 20.3, 9.2 and 3098, your answer could only be quoted to two significant figures (becasue of the two significant figures in 9.2)

 Make sure that you understand the difference between a number stated to a number of significant figures a number stated to a number of decimal places

### Necessary and Unnecessary Zeros

For numbers which contain non-zero figures to the right of the decimal point, any trailing zeros are superfluous, unless they are significant figures.

For example

2.500 = 2.5

3.970 = 3.97

but to four significant figures

2.500 must be quoted as 2.500

and

3.970 must be quoted as 3.970

Zeros obviously cannot be dispensed with when they are needed to 'keep a place'.

For example, zeros cannot be dispensed with in the following numbers

7.205

5000

When a number is less than 1 and greater than -1, you have a choice as to how to present it. A number like

0.5674
could also be written as

.5674

It is a matter of personal preference, although the former is probably more common.

### Examples

 Wednesday, 15 November, 2000, 16:00 GMT Students march against fees The marchers formed a mile-long procession Fifteen thousand people took part in a march through the streets of London in protest at university fees. The National Union of Students rally was calling for the abolition of the £1,050 annual tuition fee and opposing the introduction of "top-up" fees which could cost up to £4,500 a year. The march - part of the union's Winning for Students campaign - started out from the University of London and finished in Kennington Park, South London.
 If you fancied a quick journey to the English capital on Sunday, forget it. It's the London Marathon 30,000 people are expected to run in a race which has become a fixture in the nation's calendar since 1981. Such is the status of the event that half a million people will be in attendance to cheer on the competitors, while it is expected that the race will be televised to over 100 countries around the world. The first London Marathon was run on March 29 1981. Incredibly, the number of applicants for the race - 20,000 - was smaller than the number that will actually run this Sunday. With the amount of competitors increasing every year, more money will be raised this Sunday than during previous races. It is estimated that in the 18 year history of the event, £ 80m has been raised. At 42 kilometers, starting at Greenwich Park and finishing by The Mall, the race itself is a gruelling test of endurance and stamina. 15,000 runners pull out during the months preceding the race for a variety of reasons, including lack of appropriate fitness and health concerns. Additional events now form the day that is the London Marathon. A mini marathon is run for 2,000 children between the ages of 11-17.
 History of the Naval Dockyard ......by 1626 there were 4,000 soldiers quartered at Portsmouth. ...... one thousand dockyard men emigrated to Canada during the period of 1869-70. ......bombing in World War II virtually destroyed some areas of the City. The Dockyard was devastated. In 1945 there was a workforce of 45,000. ......there was further decline in the 1980s resulting from Defence Reviews. In July 1981 Parliament confirmed the run-down of the Dockyard by reducing the labour force from 7,200 to 1,200. The current workforce is 1700.
 Distances * Paris - Alençon 195 km * Caen - Cherbourg 120 km * Paris - Caen 240 km * Deauville - Alençon 120 km * Paris - Cherbourg 360 km * Deauville - Rouen 90 km * Paris - Dieppe 195 km * Rouen - Dieppe 60 km * Paris - Mt St Michel 325 km * Rouen - Le Havre 85 km * Paris - Rouen 140 km * Rouen - Mt St Michel 255 km

## Past Exam Questions

#### 1. A pop concert has an audience of 8 496 people. Two thirds of the people at the concert pay a £15 entrance fee while the remainder pay a reduced rate of £7. What is the income from the concert to the nearest thousand pounds?

• A    £80 000
• B    £82 000
• C    £104 800
• D    £105 000