BIRKBECK MATHS SUPPORT
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Calculations 2
In this section we will look at
- rounding numbers up or down
- decimal places and significant figures
- scientific notation
- using calculators with fractions, rounding, significant figures
and scientific notation
- common mistakes with a calculator
Helping you practice
At the end of the sheet there are some questions for you to practice.
Don’t worry if you can’t do these but do try to think about them. This
practice should help you improve. I find I often make mistakes the
first few times I practice, but after a while I understand better.
Videos
All the examples in this worksheet and all the answers to questions
are available as answer sheets or videos.
Good luck and enjoy!
Videos and more worksheets are available in other formats from
www.mathsupport.wordpress.com
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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1. Rounding numbers
Sometimes numbers are given very precisely, for example,
The population of Wales was estimated to be 2,999,319 in 2010 and we would
say this as 'two million, nine hundred and ninety nine thousand, three hundred
and nineteen' but this is not particularly useful as a week later this number will
be different. It is more useful to say the population of Cardiff in 2010 was about
three million 3,000,000.
Saying 'three million' instead of 'two million, nine hundred and ninety nine, three
hundred and nineteen’ is called rounding and in this case we have rounded to
the nearest million.
Rounding up or rounding down
To decide whether to round up or round down we need to see where the
number is on the numberline, for example
120
121
122
123
124
125
126
127
128
129
130
Here we have drawn £123 pounds on the number line and we can see that it is
nearer the number 120 than the number 130. So if we round to the nearest £10
it becomes £120 because it is nearest to 120 on the number line.
120
121
122
123
124
125
126
127
128
129
130
But if we draw the number 129 we can see that it is nearer to 130 than to 120
and so £129 rounded to the nearest £10 is £130 because it is the closest
multiple of ten on the number line.
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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Rounding to the nearest ten: the less than 5 rule
Here we will start by rounding to the nearest ten only. If we have 0, 1, 2, 3, or 4
pounds then to the nearest £10 we say that we have £0. But if we have 5, 6, 7, 8
or 9 pounds then rounded to the nearest ten we have £10 pounds.
We can see this on the number line.
0
1
2
3
5
4
6
7
8
9
10
The number 4 is nearer 0 than 10 but the number 6 is nearer 10 than 0. Although
5 is in the middle, we round it to 10.
Using the same ideas, if we have 22, its nearest multiple of 10 is 20 and if we
have £27, then to the nearest 10 it is £30.
20
21
22
23
24
25
26
27
28
29
30
Remembering for example, if we have £155 then to the nearest 10 it is £160.
150
151
152
153
154
155
156
157
158
159
160
Easy way to round
However if we draw numberlines each time we round a number it would take a
lot of time, so instead we can round in the following way.
Let's look at the number 27 and round it to the nearest 10. If we are rounding to
ten we put a ring around the number of tens, which in this case is 2.
27
rounded to nearest 10
30
Now we look at the number to the right (which we underline). If this underlined
number is 5 or higher we round the circled number up one to give 3. We then
set numbers to the right of this circled number to zero.
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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Another example:
We want to round 431 to the nearest ten. So we put a ring around the tens and
underline the number to the right
431
rounded to nearest 10
430
Is the underlined number five or more? No, so we don't round the 3 up, we just
leave it as 3, but we change all the numbers to the right to zero to get 430.
Rounding to the nearest 100
If we want to round a number to the nearest 100, then we ring the hundreds
and underline the tens. If there are more than five tens we round up, and if there
are less than five tens we round down. For example
237
rounded to nearest 100
581
rounded to nearest 100
200
600
Rounding to any multiple of ten
If we take a number such as the population of Dundee 141,937 we can round
this to the nearest ten, hundred, thousand etc.
141,937
rounded to nearest 10
141,940
141,937
rounded to nearest 100
141,900
rounded to nearest 1,000
142,000
rounded to nearest 10,000
140,000
rounded to nearest 100,000
100,000
141,937
141,937
141,937
2. Decimal places and significant figures
Dealing with decimal places and significant figures is very similar, so we will
start with decimal places and then move onto significant figures.
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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DECIMAL PLACES
When dealing with decimals we use the words decimal point and decimal
place to help us describe the number. Consider the following number.
2.340587
1
decimal point
The decimal point is between the number 2 and 3. If we count the numbers
after the decimal point, this tells us the number of decimal places. In this case
we have 6 decimal places .
The number 3 is in the first decimal place
The number 4 is in the second decimal place
The number 0 is in the third decimal place
The number 5 is in the fourth decimal place
The number 8 is in the fifth decimal place
The number 7 is in the sixth decimal place
Try to work out how many decimal places the following numbers have:
1) 2.351
4) -6.2
2) 9.01356
5) 130.01
3) 14.2
6) 200.1002
The answers are: three, five, one, one, two and four.
Rounding to decimal places
Rounding to a number of decimal places works in exactly the same way as
rounding to 10 or 100.
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Consider the number 2.340587, if we want to round this to one decimal place
we are asking if the number is nearer 2.3 or 2.4. We could draw a number line
to see which is nearer
2.3
2.31 2.32
2.33
2.34
2.35
2.36
2.37 2.38
2.39
2.4
2.340587
Or we could look at the number directly, circle the decimal place we are
rounding to and use the number to the right to tell us whether to round up or
round down. We use d.p. to mean decimal place. So if we round to 3 decimal
places we write 3 d.p.
2.340587
rounded to 5 d.p.
2.34059
2.340587
rounded to 4 d.p.
2.3406
2.340587
2.340587
2.340587
rounded to 3 d.p.
rounded to 2 d.p.
rounded to 1 d.p.
2.341
2.34
2.3
Notice that if we round to a d.p. the extra zeros to the right of the that decimal
place can be left out. But it is important to remember we can’t do this when
rounding to tens, hundreds etc. Consider if we have £2,315 then rounded to the
nearest thousand is definitely not £2 but is £2,000.
SIGNIFICANT FIGURES
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In the same way as decimal places counts the numbers after the decimal point,
‘significant figures’ count the numbers from the first number in the expression.
Consider the population of Wales again,
second significant figure
seventh significant figure
2,999,319
first significant figure
sixth significant figure
It is usual to write Significant Figure is usually written as s.f. so the
3rd s.f = 9 and the 4th s.f. = 9 and the 5th s.f. = 3
To round this number to any significant figure we could use the numberline as
before but this time we will just round by looking directly at the number.
2,999,319
rounded to 6 s.f.
2,999,320
2,999,319
rounded to 5 s.f.
2,999,300
2,999,319
rounded to 4 s.f.
2,999,000
2,999,319
rounded to 3 s.f.
3,000,000
2,999,319
rounded to 2 s.f.
3,000,000
2,999,319
rounded to 1 s.f.
3,000,000
Notice that we have a slightly special case here. Since there are three nines in
a row when we round to 3 s.f. from 7 s.f. we have to round up from 2,999,319 to
3,000,000. The same is true for rounding to 2 s.f. and 1 s.f.
The questions at the end of this worksheet include plenty of practice at
rounding significant figures and decimal places.
3. Scientific notation
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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Scientific notation uses the ideas we started to explore in indices to express
large or small numbers in a convenient way that prevents scientists from
making mistakes in calculations by missing out zeros when writing numbers.
To write numbers in scientific notation
- the number is always re-written as a number bigger than or equal to 1 and
smaller than 10, so that there is only one number to the left of the decimal point
- the magnitude of the number is always written as a power of ten
For example
Where the number in bold is in scientific notation.
A quick trick
An easy way to write any number in scientific notation is to count the number of
places the decimal point has to move. For example if we write the number
338,100,000 we have to move the decimal point 8 places to the left to make
sure that only one number is to the left of the decimal point, giving a power -8:
8 7 6 5 4 3 2 1
and if we now consider 0.00003193 we see that the decimal point has to move
five places to the right, and so the power is -5.
-1 -2 -3 -4 -5
Once we are able to put any number into scientific notation we can just do the
opposite to get a number from scientific notation into decimal format.
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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For example
1)
2)
3)
4)
Rounding with scientific notation
Rounding when the number is in scientific notation is easy if we are happy
rounding with numbers written in their usual format since we apply exactly the
same rules but just to the decimal and not to the power, for example
1)
2)
3)
4)
5)
And some trickier ones
1)
2)
3)
Notice how we need to include the significant figures even if they are zero. For
example if I say I have £200 pound to 3 s.f. then I have exactly £200, and not
£201 and not £199. But if I have £200 to 2 s.f. then I could have £204 or £195.
4. Using calculators
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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Here we continue from the work in 02 Calculations 1 (so we assume that you
are using a scientific algebraic calculator for example any of the Casio FX
range). We first look at using rounding and significant figures and then how to
type numbers in scientific notation in calculators
Significant figures and rounding
When using calculators for complicated calculations it is usual to get an
answer that has as many significant figures as a calculator will allow.
For example 1 inch = 2.54 cm, so 1 cm = 1 ÷ 2.54 inches and if we enter this into
a calculator we get an answer as 0.3937007874015..... and the number will be
longer if your calculator allows more numbers to be displayed. This reason is
one of the reasons it is important to be able to correctly round numbers.
As a rule of thumb 3 significant figures are good enough for most calculations
(so in this case we would write 1 cm = 0.394 inches) but check or ask to see
how many significant figures a particular calculation needs.
Scientific notation
You can test how your calculator displays numbers in scientific notation by
entering 1,000,000,000 x 1,000,000,000, which should give an answer of
To enter a number in scientific notation you should use the button [EXP] or
[
]. For example to enter the number
type
Common Mistakes: There are easy mistakes to make with calculators
involving missing out brackets. Try the following calculations. If you get different
answers making sure you are using brackets correctly. All answers are to 3 s.f.
1)
4)
2)
5)
3)
6)
5. Now your turn
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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Generally the more maths you practice the easier it gets. If you make mistakes
don’t worry. I generally find that if I make lots of mistakes I understand the
subject better when I have finished. If you want to see videos explaining these
ideas and showing the answers visit www.mathsupport.wordpress.com
A) Rounding including decimal places and significant figures
1) round 1.38972 to 1 d.p.
5) round 500,419 to 3 s.f.
2) round 14.90012 to 6 s.f.
6) round 0.099501 to 3 d.p.
3) round 0.056443 to 2 s.f.
7) round 10101 to 4 s.f.
4) round 0.01507 to 2 d.p.
8) round 0.992 to 1 d.p.
B) Write the following numbers in scientific notation
1) 230
5) 0.219
2) 1,579
6) 0.00005007
3) 0.00546
7) 999.9
4) 11
8) 0.000100012
C) Write the following numbers as one number in decimal format
1)
5)
2)
6)
3)
7)
4)
8)
D) Carry out the following calculations using your calculator
1)
2)
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3)
6)
4)
7)
5)
=
8)
E) General word problems involving calculations, indices and rounding
1) The cost of a second hand Skoda car is £5975. What is this amount to the
nearest 10, to the nearest 100 and to the nearest 1,000?
2) The GDP (Gross Domestic Product) is the market value of all final goods and
services from a country in one year. In millions of US Dollars the GDP of the UK
is 2,258,565 and the GDP of Japan is 5,390,897. Using the exchange rate of 1 US
Dollar = 0.6430 UK pounds, give both these numbers in pounds sterling in
scientific notation to 3 significant figures.
3) In Berlin in 2009 the 100m world record was broken by Usain Bolt. His time
was 9.58 seconds. This beat the previous world record of 9.69 seconds which
was help by Tyson Gay. How much faster was Usain than Tyson and write this
number in scientific notation.
4) Most bacteria species are spherical, and have a radius of about a
micrometer. We can write 1 micrometer in scientific notation as
volume of a sphere can be calculate from the formula Volume =
. If the
, where
the symbol r=radius, and =3.14, calculate the volume of a single bacteria in
and give your answer to 3 s.f in scientific notation
5) For certain drugs the dosage for children will depend on their weight and
their temperature. For a fever less than
weight of the child. Above
the dose is 5mg per kilogram of
the dose is 10mg per kilogram of weight of the
child. The dose should be taken every 8 hours. If a child weighs 11kg and has a
temperature of
how many milligrams of the drug do they take in one day.
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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