Fractions, decimals and percentages:
how to link them
Mathematics Skills Guide
This is one of a series of guides designed to help you increase your confidence in
handling mathematics. In this guide you will find brief theory and exercises designed
to help you bridge the work in Fractions, Decimals and Percentages. It brings out the
essential connections that should help you to work with confidence in all three.
There are often different ways of doing things in mathematics and the methods
suggested in the guides may not be the ones you were taught. If you are successful
and happy with the methods you use it may not be necessary for you to change
them. If you have problems or need help in any part of the work then there are a
number of ways you can get help.
For students at the University of Hull
 Ask your lecturers.
 You can contact a math Skills Adviser from the Skills Team on the email shown
below.
 Access more maths Skills Guides and resources at the website below.
 Look at one of the many textbooks in the library.
Web: www.hull.ac.uk/skills
Email: skills@hull.ac.uk
Most people's experience of fractions, decimals and percentages is of different topics
learned at school at different times between the ages of eight and fourteen. In
consequence many do not realise the way in which these topics can be linked and,
when linked, can be much more powerful as tools for handling number. This
worksheet shows the ways these links can be made.
The important mathematical principle known as the Equivalence of Fractions is the
means we can use to link the three topics. Most people will know about this principle,
though they may not have used this name for it, nor have realised its general
importance to mathematics. One example of it is given by
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1 2 3 4
5
7
8
9 10
50
   



 
....... 
 ......
2 4 6 8 10 12 14 16 18 20
100
and we can use it to facilitate easy conversion from any one of fractions, decimals and
percentages to both the other two.
You will probably recognise the set of equalities and will be used to it either in a similar
form to this or when ‘cancelling’ fractions or ‘reducing them to their lowest terms’.
Specifically, when we say that a half is equal to two quarters we are doing it by
multiplying the top and bottom numbers in the fraction 12 by two; similarly, in the other
cases, we have multiplied top and bottom by other numbers or, diagrammatically we
can see that:
1
2

2
4
Most people will know that a half is 0.5 as a decimal and that 50 over 100 represents
50%. So we have established that 12 is the same as 0.5 and 50%. Let us use this to
make a conversion table before using it to show how to change quickly and easily from
one to the other.
Fractions Decimals
1
0.5
2
1
4
1
8
1
16
1
32
Percentages
50%
0.25
25%
0.125
12.5%
0.0625
6.25%
0.03125
3.125%
Fractions
1
5
1
10
1
20
1
40
1
80
and so on
Decimals
0.2
Percentages
20%
0.1
10%
0.05
5%
0.025
2.5%
0.0125
1.25%
and so on
There are several things to notice about this table:

the process of halving has been used to build up the table; this is often a
useful device for calculating

if you look at the decimal columns and compare them with the percentage
columns you may be able to notice that the whole number part of the
percentage format is the first two decimal places of the decimal format. This is
not surprising if you remember that percentage means ‘out of a hundred’. But
you can further notice that, where there is a fraction in the percentage, this is
reflected in the third and fourth decimal places in the decimal format.
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
if you look at the forms for a half, a quarter etc and then go to those for a
twentieth, a fortieth etc, you will see another example of the principle
mentioned above because a twentieth is a half divided by 10 and a fortieth is a
quarter divided by 10.
All of these things draw your attention to some useful expedients for calculating of all
kinds and in particular for any connected with fractions, decimals and percentages.
But we can also use similar procedures for some other, initially less simple than the
above: fractions such as a third, a sixth, a ninth etc.
Here is the table for such fractions:
1
3
1
6
1
9
0.3333333 33.3333333 %
0.1666666 16.6666666 %
0.1111111 11.1111111 %
In this case we are going into the idea of a ‘recurring decimal’, ie one in which a figure
or group of figures are repeated indefinitely and each of which can be seen to
represent an exact fraction. (See later section for some notes on recurring decimals.)
To become competent you should try to memorise the first three of each of the parts of
this table and then use the device of dividing to get the others from them.
But it is also possible to derive other equivalencies such as 34 which is made up of
and
1
2
added together. So the decimal form of
3
4
1
4
is 0.5 + 0.25 ie 0.75 which is in turn
75% and could have been derived directly because most people recognise it as being
3.
4
You should try out a few such examples for yourself, using the leaflet on fractions to
help you if you have forgotten how to add fractions but, in all cases, you will see that
the decimal and percentage equivalencies are very easily found from each other and
can often be the way into such a conversion problem.
So let us now look at converting from any one of these to the other two. As might be
expected from the above, decimals and percentages convert most easily into each of
the other two; there is just a little more to converting fractions.
Starting from a decimal
Take any decimal, 0.235 say. Clearly this represents 23.5% because the first two
places always give us the whole number part of the percentage with the other places
bringing in a decimal part of the percentage. Watch people on TV talking about
percentages and you will notice that they are always using this way of talking about
them.
To complete, we can either say that 0.235 is 235 divided by 1000 (from the same
number principle about multiplying and dividing by 10 mentioned above) and so is the
235 , or it is 23.5 divided by 100 (a percentage) and this is the same as 235 if
fraction 1000
1000
we multiply both of the terms 23.5 and 100 by 10.
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Starting from a percentage
This is also simple. Take the percentage 78%; this is 0.78 as a decimal and
78
100
as a
fraction. Note that, in the fraction form here and in the section above, we could reduce
the fractions obtained by dividing top and bottom by the same number.
In the example of
235
1000
78
100
we can divide top and bottom by 2 and get
we can divide top and bottom by 5 and get
39 ;
50
in the example of
47 .
200
Some percentages look as if they might be complicated but in reality are not.
For example, 37 12 % can be split up into 25% and 12 12 %; their decimal and fraction
equivalents are 0.375 and
this); or
3
8
1
4
 81 =
3
8
(see the Fractions leaflet if you are unsure about
can be got directly from 0.375 because this is three times 0.125 which the
conversion table tells us is
1
8
.
Starting from a fraction
Many fractions are easy to convert but some are slightly more complex. In the latter
cases we would recommend using a calculator after you have taken the steps
advocated below.
Some simple examples:
3 : 1 is 0.2, so 3 is 0.2  3  0.6 and 60%
5
5
5
5:
16
If you can remember it,
1 is
16
0.0625 so
5
16
is
1
16
× 10 divided by 2
= 0.625 divided by 2 = 0.3125
or use 5  4  1  1  1  0.25  0.0625  0.3125
16
16
16
4
16
3 , look
If you get an awkward fraction which cannot be dealt with in this way, such as 13
at the numbers in the fraction to see if you can see an easier fraction near to them in
value in order to get an estimate for your answers.
3 is <
For example, 13
1
5
3 which
12
is
1
4
(0.25 and 25%) also
3
13
3 which is
15
3 lies between
13
is >
(0.2 and 20%). So you now know that the decimal form of
0.25 and
0.2 and the percentage between 25% and 20%. It is at this stage that you use your
calculator, keying in 3 divided by 13 to give the answer 0.2307592 which is 23.08% to
2 decimal places, an answer that satisfies the boundaries found above.
This is a somewhat more complex procedure and, to avoid it, you could go straight to
your calculator to get an answer but, as always when you use a calculator, it is
important to have an estimate against which to check the calculator answer so, in
going through this procedure, you both practise your estimating skills and provide an
essential check for your calculator answer.
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Being able to change quickly and easily between fractions, decimals and percentages
will provide you with a very powerful skill in dealing with numbers because it will enable
you to work at will in any of the three media.
Exercise 1
In each example given, change to the other two forms:
1. 35%, 86%, 37%, 87%, 11%, 63.5%, 27%, 55.56%
2. 0.54, 0.64, 0.6125, 0.15, 0.375, 0.245, 0.7777777
3. 4 , 5 , 13 , 2 , 5 , 7 , 7 , 8
5
8
16
7
11
12
13
17
Some notes on recurring decimals
Fractions which are written with a dot over one or more of the digits in them are known
as recurring decimals. For example 0.7 means 0.77777777777...... and goes on to an
infinite number of terms. In expressing it to a specific number of decimal places, two
say, we would express it as 0.78. The 8 is here because, when we correct to a given
number of decimals, we look at the next figure and, if it is <5 we give the existing
figure, while if it is 5 or >5 we correct the next figure to the figure one higher than it is.
This procedure is called rounding up and rounding down and comes into all examples
of numbers where an answer is required to a specific number of places.
0.23 means 0.2323232323 ........... and goes on as before.
0.156 means 0.156156156156 ......and goes on as before. Note that in this case we
only put the dots over the first and last of the recurring figures.
With this in mind, we want now to move to finding out what such recurring decimals
represent in fractional and percentage forms. As in the earlier part of this handout, the
percentage form is easy to write down.
0.777777777 ... represents 77.8% to one decimal place, 77.78% to two places.
0.23232323 ... represents 23.2% to one place of decimals, 23.23% to two places.
0.156156156 ... represents 15.6% to one decimal place and 15.62% to two places
To find the fraction represented by 0.7 recurring, we first multiply the whole decimal by
10 (because it is only one figure which is recurring) and we get to 7.77777777. We
can write this as 7 + 0.77777777 = 7 + 0.7 . We see that 10 times the recurring
fraction has the same value as the number 7 together with itself.
In other words 10  0.7  7  0.7 . This gives 9  0.7  7
Therefore the recurring decimal itself is equivalent to 7 divided by 9 or
7
9
. If you find it
difficult to get your mind round the argument written in this way you may prefer an
algebraic way of expressing the same thing:
Let us give the value x to 0.7 (0.7777777777....); then we have
10x = 7.777777777... = 7 + 0.777777777... = 7 + x
So, using whatever rule you know for solving such an equation we get
9 x  7 leading to x  79 .
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For 0.23 recurring = 0.23 with two figures recurring, we multiply by 100 to give us
23.23232323... which, by the same argument as above is 23 + the recurring decimal
23 .
we started with. So 99 times the decimal is 23 and the decimal is equivalent to 99
Or, done as an equation, we find that 100 x  23  0.23232323...= 23  x
23
so that 99 x  23 giving us x  99
For 0.156156156 .... or 0.156 recurring = 0.156 , because there are three figures
recurring, we multiply the decimal by 1000 and find that this gives us 999 times the
fraction to be 156 so that the decimal is equivalent to 156
.
999
In equation form we have 1000 x  156  x so that 999 x  156 and hence x  156
.
999
Try these for yourself:
Exercise 2
 7
0.8 0.1
0.2 16
 234
0.1
 42857
0.1
0.9
Having tried all of these - and perhaps made up a few for yourself - you might like to
see if you can formulate a rule which you could use for any similar recurring decimals
so that you do not have to go through the whole process every time.
Note: For some specific recurring decimals such as 0.166666666..., it is possible to
derive its value by realising that it comes from either halving 0.3 recurring 13 or

dividing 0.5 by 3.
Alternatively, using a similar procedure as for other recurring decimals and multiplying
by 10 we get 1.6666666666666... = 1 + 0.66666666666... = 1 + 23  53 and from this,
dividing by 10 again we have 0.1666666666... = 53 divided by 10 which is
5
30
or
1
6
.
If you can become fluent in changing between fractions, decimals and percentages
and can remember the procedures and the rule you abstracted from these procedures,
you will find that you can approach examples of all of them with more confidence and
competence.
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ANSWERS
Exercise 1
7 = 0.35
1.
35% = 20
27
0.54 = 54% = 50
2.
43
86% = 50
= 0.86
0.64 = 64% = 16
25
37
37% = 100
= 0.37
49
0.6125 = 61.25% = 80
87
87% = 100
= 0.87
3
0.15 = 15% = 20
11 = 0.11
11% = 100
0.375 = 37.5% = 38
63.5% = 127
= 0.635
200
27 = 0.27
27% = 100
49
0.245 = 24.5% = 200
7777777
0.7777777 = 77.77777% = 10000000
55.56% = 1389
 0.5556
2500
3.
4
5
5
8
13
16
2
7
= 0.8 = 80%
= 0.625 = 62.5%
= 0.8125 = 81.25%
= 0.2857 = 28.57% (to 2 decimal places)
Exercise 2
0.8 = 88.89% (to 2 dec places)= 89
0.1 7 = 17.17% (to 2 dec places)= 17
99
216
0.2 16 = 21.61% (to 2 dec places)= 999

0.1 42857 = 14.29% (to 2 dec places)=
5 = 0.4545 = 45.45% (2 d.p.)
11
7 = 0.5833 = 58.33 ( 2 d.p.)
12
7 = 0.5385 = 53.85% (2 d.p.)
13
8 = 0.4706 = 47.06% ( 2 d.p.)
17
8
0.1 234 = 12.34% (to
37
142857
= 71
999999
2 dec places)= 1234
9999
0.9 = 100.00% (to 2 dec places)= 1
We would appreciate your comments on this worksheet, especially if
you’ve found any errors, so that we can improve it for future use. Please
contact the Maths Skills Adviser by email at skills@hull.ac.uk
The information in this leaflet can be made available in an alternative format on
request using the email above.
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