Percentages
Mathematics Skills Guide
This is one of a series of guides designed to help you increase your confidence in
handling mathematics. This guide contains both theory and exercises which cover:1.
2.
3.
4.
Finding a percentage
Increasing by a percentage
Decreasing by a percentage
Finding a percentage change
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
1. Percentages
The booklets on ‘Estimation and Mental Methods’ and ‘Fractions, Decimals and
Percentages:how to link them’ show how to simplify some calculations by using the
methods they recommend and, in particular, by being able to convert easily between
fractions, decimals and percentages because this underpins the way this handout
approaches the topic of percentages.
In thinking about percentages it is important to realise that they pervade the whole of
our lives but are often misunderstood by many people. Further than that, changes in
percentage, either increases or decreases, occur almost daily in such areas of life as
mortgage rates, bank rates and as ways of expressing changes in population and
other human affairs. So this booklet should enable you to avoid the many mistakes
frequently made in handling percentages in the media and should enable you to
approach psychometric tests and percentages in your ordinary life confident that you
can work accurately with them.
If you have not read the ‘Estimation and Mental Methods’ and ‘Fractions, Decimals
and Percentages:how to link them’ booklets you should do so because methods
developed in it will be used in this one so that the three together provide a
comprehensive whole for handling many of the everyday calculations we need to
make.
Principle for percentages
The first thing to realise about percentages is that there are two quite distinct aspects
of calculations to do with them. These are
- finding a given percentage of something
- finding one thing as a percentage of another
and you will find that, bearing this in mind, the first thing about any question is to
determine which of these two is required for any particular calculation.
2. Finding a given percentage of something
To calculate a particular percentage of a given quantity, you have several methods of
approach at your disposal. By realising that fractions, decimals and percentages are
linked, you will see that finding a percentage can be done by means of either
fractions or decimals. So 37% of something can be seen to be 37 times whatever
100
the quantity was or it can be seen as 0.37 times the quantity.
In the decimal form a calculator is the obvious way forward but you have to
remember that, if you use a calculator, you must do an estimate first to enable you to
check you have pressed the right buttons. So in this case you can see that 37%
must lie between a third and a half, (ie between 33% and 50%) both of which are
easy amounts to check for any quantity.
However, some percentages can easily be calculated by breaking the percentage
into manageable components. For example, 75% can be broken up into 50% and
25% ie a half and a quarter, amounts that can easily be done mentally, 37 % can be
broken up into 25% (a quarter) and 12 1 % (an eighth) while 57% can be broken up
2
into 50% (a half) and 5% (a tenth of a half) - and 2% (easily calculated by dividing by
100 and doubling).
page 1
Example 1 Find 87 1 % of 236.
2
Method 1.
87 1 = 50 + 25 + 12 1
Method 2.
Note 87 1 % is 12 1 % less than 100%.
50% of 236
25%
is 118
is 59
12 1 % is 1 ,
12 1 %
is 29.5
2
2
87 1 % of
2
2
2
2
2
8
1
87 % of 236 = 236 – 236
2
8
= 236 - 29.5 = 206.5
236 is 206.5
Method 3.
Using your calculator 0.875  236  206.5
Example 2. Find 49% of 350.
Method 1.
Notice that 49 is 1 less than 50
50% of 350 is 175 (one half)
1% of 350 is 3.5
49% of 350
Method 2.
Using your calculator
49% of 350 = 0.49 × 350 = 171.5,
a little less than 12 of 350.
is 171.5
Example 3. Find 52 1 % of 376.
2
Method 1.
Notice that 52 1  50  2 1
2
Method 2.
Using your calculator
52 1 % of 376 = 0.525 × 376
2
50% of 376 is 188 (one half)
2
2 1 % of 376 is 9.4 ( 1 of 10%, ie 1 of 37.6)
2
4
4
52 1 % of 350 is 197.4
= 197.4,
a little more than 12 of 350
2
Example 4. Find 57% of 342.
Method 1.
Split 57% into 50% (half) + 5% +2%
50% of 342 is 171
(one half)
5% of 342 is 17.1 (one tenth of one half)
2% of 342 is 6.84 (two hundredths)
57% of 342 is 194.94
Method 2.
Using your calculator
57% of 342 = 0.57 × 342
= 194.94,
If you practise such examples, always using the simple fractions of 100 and of 10
such as 50, 25, 12 1 and 5, 2 12 , 1 14 , you will soon find that you become very efficient
2
in seeing ways to break up any given percentage into manageable fractions. This
can be a useful skill for helping with psychometric tests as well as calculating
percentages you encounter in ordinary life so it is worth persevering with it.
Always remember that if you use a calculator to determine a percentage of
something, you must find an estimate as well in order to check your answer.
page 2
Exercise 1 Find 37 %, 56%, 62 %, 76% of each of the following: 456, £560, 2345.
For all of these you can use a calculator to do the calculation and this can serve as a
check on your other methods but do not let yourself get into the habit of always using
a calculator because this will tend to erode your developing mental calculating facility
and in ordinary life as well as in the psychometric tests employers set you this is not
necessarily the best way to operate. With other methods at your disposal you will be
better able to operate with figures in ordinary life as well as in the tests.
3. Finding one number as a percentage of another
For this aspect of percentages all you need to do is to make a fraction of the two
3
quantities. For example 45 as a percentage of 60: here 45 reduces to 4 so the
60
percentage is 75%. Some fractions are easily reducible to a fraction to which you
can give the percentage from the table provided in the booklet Linking Fractions,
Decimals and Percentages but some are slightly more cumbersome.
For example if you want to see what percentage 37 is of 49 you make the fraction
37 and use the procedures for linking fractions, decimals and percentages to find out
49
what percentage this represents. Note that you can make a quick estimate by
looking at 40 and seeing that this is four fifths ie 80%. For this one it is easier to us a
50
calculator since the arithmetic is quite heavy and with the estimate already made you
are in a position to check the answer your calculator gives you. Using your calculator
37 = 0.755102 which represents 75.5% (to 1 decimal place).
49
However, many examples of finding one thing as a percentage of another can also
be simplified using the ideas from the Estimation booklet to help you. Thus, while 37
as a percentage of 74 can be seen to be 50% because 37 is half of 74, there are
many other sets of figures that can be calculated as percentages easily from the
fractions they make. For example, 21 as a fraction of 35 is three fifths ie 60%, while
35 as a percentage of 21 is found by putting 35 which reduces to 53 which as a
21
decimal is 1.666... ie 167%. The ‘Fractions, Decimals and Percentages: how to link
them’ booklet will enable you to see ways of simplifying any given fraction where this
is possible
Exercise 2 For the pairs of numbers given, find the smaller as a percentage of the
larger and the larger as a percentage of the smaller. (Notice that in one case the
percentages arrived at will be less than 100% while in the other the percentages
arrived at will be more than 100% because in the first case the fractions will all be
less than 1 and in the second they will be greater than 1. The significance of this will
come out later.)
27, 81
56, 91
350, 490
121, 254
137, 423
page 3
4. Percentage change
As already indicated, percentages in real situations fluctuate sometimes daily, so we
need to be able to look at the two aspects of percentages in terms of possible
changes and, having done so, be in a position to work competently with most of the
situations that can occur relating to such changes. And because so many people are
vague about their purpose and meaning it is easy to be deceived into believing
statements made in the public domain which, on examination, can be seen to be
false.
This can be illustrated by a quote from a statement made by a BT spokesman in
October 1991 about the BT Chairman having recently had an increase of
12 12 % bringing his salary to £469 000. He said that this rise was not excessive
because BT workers had recently received a 7.3% rise.
There is no indication about how this justification was received by those he was
speaking to and there is no means of knowing whether the BT spokesman was
himself ignorant about the way percentages work or whether he was relying on the
ignorance of those to whom he was speaking. The flaws in his argument are brought
out by the following:
10% rise for all; is it fair?
Pay £5000
rise £500
Pay £10,000
rise £1000
Pay £50,000
rise £5000
Pay £100,000
rise £10,000
New pay £5500
New pay £11,000
New pay £55,000
New pay £110,000
It is easy to see from this table that an increase of 10% not only increases the actual
rise received but also the differentials between the lower and higher paid. You can
decide whether this is a fair way of proceeding. Whatever your view turns out to be a
proper understanding about how percentage increases work enables you to see the
flaws in arguments such as that of the BT spokesman. So let us proceed to looking
at increasing by a given percentage.
5. Increasing by a given percentage
For this it is always assumed that the starting figure represents 100%. Let us
consider some simple increases in percentage:
An increase of
50%
on 100% takes us to
150%
 1 1 or  1.5
2
 1 1 or  1.25
4
 1 3 or  1.75
4
25%
125%
75%
175%
100%
200%
125%
225%
2
 2 1 or  2.25
200%
300%
3
4
300%
400%  4
For some examples the fraction form is the easiest to work with, but for others eg
increasing by 73% the decimal form is much easier and for this it is sensible to use a
calculator, always remembering to determine an estimate for the answer, in this case
page 4
that it will be increased by something just less than three quarters. However, some
fractions which do not appear to be easy fractions to handle can in fact be made
simple.
Examples of this are increasing by 12 1 %, 37 1 %, 87 1 % and the VAT rate of 17 1 %.
2
2
2
2
1
1
For these think about 12 as 10 + 2 , for which you could get a tenth and a quarter
2
2
of a tenth, quite easy to do in your head. Similarly 37 1 is 25 + 12 1 , ie a quarter and
2
2
a half of a quarter.
Others like 57% can also be done as 50% (a half) plus 5% (a tenth of that) plus 2%
which is easy (either a fifth of a tenth or twice a hundredth)
So, when asked to increase by a given percentage, look at the numbers involved to
see if you can make them into easy fractions or split them into easy fractions; if not
then use the decimal form and use your calculator, always remembering that, with
calculator use for anything, you should always first make an estimate of the answer
in order to be able to judge whether the display answer is a reasonable one.
Example 1 The price of something before VAT is added is £236; what will its price
be including VAT? This can be done in two ways:
Method 1:
17 1 % = 10% + 5% + 2 1 % , so the VAT will be £23.60 + £11.80 + £5.90, by getting
2
2
10% (a tenth) and then successively halving the amount to get 5% and 2 1 % and, by
2
adding all of these to the original price, the total price will be:
£236 + £23.60 + £11.80 + £5.90 = £277.30
Method 2:
To add VAT of 17 1 % find 1.175 x 236 using your calculator in the same way as
2
above for simple percentages of a given quantity. Notice that the answer will be
something less than £236 + a fifth of £236 ie £236 + twice £23.60 (ie £47.20). That
is it will be a little less than £283.20 which is the result of adding a fifth of £236 to
£236. The calculator gives £277.30 when the answer is given to two decimal places
because
1.175 × 236 = 277.3
Example 2 The population of a certain country is 270,000. Trends suggest a
population growth of 13% will occur during the next 10 years. What would you
expect the population be at the end of those ten years?
Method 1:
13 = 10 + 1 + 2 so, after ten years, the population should be 270,000 + 27,000 +
2,700 + 5,400 = 305,100
Method 2:
After ten years the population should be 113% of what it is now: 270,000 ×1.13
which, using a calculator, is 305,100. Increasing by 13% represents an increase of
something under 15% which would be 27,000 + 13,500 which is 40,500; 40,500
more would be 310,000, a bit more than the actual answer.
page 5
Example 3 Increase 357 by 37%
Method 1
37% = 25% (a quarter) + 10% (a tenth) + 2% (1% × 2)
so 357 increased by 37% = 357 + 89.25 + 35.7 + 7.14 = 489.09
Method 2
357 increased by 37% = 357 × 1.37 = 489.09 using a calculator
Note: Some people find it quite difficult to adopt this way of working out increased
percentages; they prefer to find the amount of the increase first and then add it onto
the original. There is nothing at all wrong with this except that you would have to do
two calculations rather than being able to encompass the calculation within a single
process, besides the fact that the way given above builds directly on the work on
changing between fractions, decimals and percentages and so, in the long term will
be a more economical way of working.
Exercise 3
Find the result of increasing the following by the percentages indicated:
1. Increase 3624 by 56%, 37%, 72%, 84%, 82%
2. Find the cost including VAT at 17 1 % on: £360, £2376, £23.50, £445.75
2
6. Decreasing by a given percentage
Decreasing by a given percentage has some features which are radically different
from that of increasing by a given percentage and understanding these differences is
crucial to competent operation of percentages.
As with increasing by a given percentage, the starting point is always 100%.
If we decrease something by 50% we reduce it to 50% (a half or 0.5)
If we decrease it by 25% we reduce it to 75% ( 34 or 0.75)
by 75%
to 25% ( 14 or 0.25)
by 100%
to zero.
So the crucial difference here is that it is impossible to reduce anything by more than
100% unless the negative has meaning. So we cannot reduce a population or a
weight or height by more than 100% but we can reduce an amount of money by
more than 100% because it will bring us into a position of debt as many students
know only too well. In this case we have to think rather differently about what this
means.
If reducing by 100% brings us to zero, then reducing by 200% brings us to a debt of
the amount we had in the first place ie it is -1 times what we started from. Reducing
by 300% brings us to a debt of twice what we started from ie we have -2 times what
we started from.
page 6
So we could calculate any intermediate percentages by seeing what the resulting
negative would be as either a negative fraction or decimal of what we started with. In
practice, as was demonstrated above, the fraction forms are more cumbersome to
handle than are the decimal forms which, in turn, are more easily done using a
calculator.
Exercise 4 Decrease the following numbers by each of the given percentages:
480 2376 5372 868 by 50% 75% 33 13 % 49%
7. Finding a percentage increase or decrease
Numbers can be increased or decreased and it is useful to be able to calculate the
percentage increase or decrease these changes represent. For example, when
price increases are announced, it is useful to be able to see what percentage
increase they represent so as to compare this with the current rate of inflation.
Similarly when there is a sale and prices are reduced it is useful to be able to
calculate the percentage reduction this represents. Although sales are usually
announced by claiming there are particular percentage decreases in operation in
such claims as 20%, 50% or even, sometimes, as much as 75% off, they can just be
given as discounts of a particular amount so it is useful to be able to calculate the
percentage discount.
To do this we need to return to the idea of expressing one thing as a percentage of
another, sometimes finding the percentage represented by the fraction made from
putting the smaller over the larger, sometimes by putting the larger over the smaller.
In this context we use the expedient of always putting the new amount over the old
amount. In the case of an increase this will give a fraction greater than 1 from which
the percentage increase can be read; in the case of a decrease it will give a fraction
less than 1. We look at examples of both.
Increasing
Example 1
Find the percentage increase when a population of a town increases from 150,000 to
200,000.
200,000
 20  4  1.333....
150,000
15
3
so the percentage increase is 33.3% to one
decimal place.
Example 2
Find the percentage increase when the price of a game of squash is increased from
70p per session to 80p per session. 80  8 . Here the figures do not work out so
70
7
easily so it’s best to use a calculator for this one: 87 on a calculator gives 1.1428571
so the percentage increase is 14.3% correct to one decimal place.
Example 3
If the policy at the Sports Centre is always to increase by units of 10p the next
increase is likely to be to 90p. 90  9  1.125 (remembering that 12 1 % is an eighth)
80
8
2
so the percentage increase that time will be 12 1 %.
2
page 7
Decreasing
Example 1
Find the percentage decrease when the population of a village falls from 1200 to
900. Here, following the given procedure, 900  9  3 so the population has been
1200
12
4
reduced by a quarter or 25%.
Example 2
Find the percentage discount when the price of an article is reduced from £450 to
£400. 400  8 so the reduction is 1  0.1111... (either with a calculator or by dividing)
450
9
9
so the reduction is 11.1% to one decimal place.
Some people prefer to look at the amount of the change over the original rather than
as presented here. There is nothing wrong with this in practice, particularly in the
case of a decrease, because this gives the percentage decrease directly instead of
requiring a further small calculation but, for the sake of consistency with the earlier
section on expressing one thing as a percentage of another, we use the expedient of
making a fraction of the given numbers in all cases. The eventual choice is up to
you.
Exercise 5. Find the % increase or decrease represented by changes from one to
the other in each of the following pairs of numbers:
250, 275
275, 250
1251, 2139
2139, 1251
1256, 3016
3016, 1256
8. Some comments on commercial practice
While in mathematics it is customary to make the starting amount 100% when
dealing with increases in percentage, in the commercial world this is not the custom.
There, since takings are the crucial amount rather than costs to the retailer, it is
customary to make the selling price 100%. Otherwise the methods of calculation are
the same. There are also some other different conventions in commerce and other
real world situations so it is advisable always to check on the practice in particular
circumstances before proceeding.
Sustained practice of percentage examples and constant noting of percentages
quoted in the media should bring you to a confidence about their use which you did
not have before.
page 8
ANSWERS
Exercise 1
37%
168.72
£207.20
867.65
456
£560
2345
Exercise 2
1
33.33%
300%
2
61.54%
162.5%
Exercise 3
1 (to 2 dec places)
2 (to nearest penny)
Exercise 4
1
2
3
4
240
1188
2686
434
56%
255.36
£313.60
1313.2
3
71.43%
140%
5653.44
£423
Exercise 5
1
10% increase
2
9.09% decrease
3
70.98% increase
4
47.64%
209.92%
4964.88
£2791.80
120
594
1343
217
320
1584
3581.33
576.67
4
5
6
62%
282.72
£347.20
1453.9
76%
346.56
£425.60
1782.2
5
32.39%
308.76%
6233.28
£27.61
6668.16
£523.76
6595.68
244.8
1211.76
2739.72
442.68
41.51% decrease
140.13% increase
58.36% decrease
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.
page 9