Percentages
mc-TY-percent-2009-1
In this unit we shall look at the meaning of percentages and carry out calculations involving
percentages. We will also look at the use of the percentage button on calculators.
In order to master the techniques explained here it is vital that you undertake plenty of practice
exercises so that they become second nature.
After reading this text, and/or viewing the video tutorial on this topic, you should be able to:
• calculate a percentage of a given quantity;
• increase or decrease a quantity by a given percentage;
• find the original value of a quantity when it has been increased or decreased by a given
percentage;
• express one quantity as a percentage of another.
Contents
1. Introduction 2
2. Finding percentage amounts 3
3. Finding the original amount before a percentage change 4
4. Expressing a change as a percentage 5
5. Calculating percentages using a calculator 6
www.mathcentre.ac.uk 1 c mathcentre 2009
, 1. Introduction
The word ‘percentage’ is very familiar to us as it is used regularly in the media to describe
anything from changes in the interest rate, to the number of people taking holidays abroad, to
the success rate of the latest medical procedures or exam results. Percentages are a useful way
of making comparisons, apart from being used to calculate the many taxes that we pay such as
VAT, income tax, domestic fuel tax and insurance tax, to name but a few.
So percentages are very much part of our lives. But what does percentage actually mean?
Now ‘per cent’ means ‘out of 100’; and ‘out of’, in mathematical language, means ‘divide by’.
So if you score 85% (using the symbol ‘%’ for percentage) on a test then, if there were a possible
100 marks altogether, you would have achieved 85 marks. So
85
85% = .
100
Let us look at some other common percentage amounts, and their fraction and decimal equiva-
lents.
75
75% = 100 = 34 = 0.75
50 1
50% = 100
= 2
= 0.5
25 1
25% = 100
= 4
= 0.25
10 1
10% = 100
= 10
= 0.1
5 1
5% = 100
= 20
= 0.05 .
It is worth noting that 50% can be found be dividing by 2, and that 10% is easily found by
dividing by 10.
Now let us look at writing fractions as percentages. For example, say you get 18 marks out of
20 in a test. What percentage is this?
18
First, write the information as a fraction. You gained 18 out of 20 marks, so the fraction is 20 .
18
Since a percentage requires a denominator of 100, we can turn 20 into a fraction out of 100 by
multiplying both numerator and denominator by 5:
18 18 × 5 90
= = = 90% .
20 20 × 5 100
Since we are multiplying both the numerator and the denominator by 5, we are not changing the
value of the fraction, merely finding an equivalent fraction.
In that example it was easy to see that, in order to make the denominator 100, we needed to
multiply 20 by 5. But if it is not easy to see this, such as with a score of, say, 53 out of 68, then
we simply write the amount as a fraction and then multiply by 100 100
:
53 100
× = 53 ÷ 68 × 100% = 77.94%
68 100
which is 78% to the nearest whole number. Although it is easier to use a calculator for this type
of calculation, it is advisable not to use the % button at this stage. We shall look at using the
percentage button on a calculator at the end of this unit.
www.mathcentre.ac.uk 2 c mathcentre 2009
mc-TY-percent-2009-1
In this unit we shall look at the meaning of percentages and carry out calculations involving
percentages. We will also look at the use of the percentage button on calculators.
In order to master the techniques explained here it is vital that you undertake plenty of practice
exercises so that they become second nature.
After reading this text, and/or viewing the video tutorial on this topic, you should be able to:
• calculate a percentage of a given quantity;
• increase or decrease a quantity by a given percentage;
• find the original value of a quantity when it has been increased or decreased by a given
percentage;
• express one quantity as a percentage of another.
Contents
1. Introduction 2
2. Finding percentage amounts 3
3. Finding the original amount before a percentage change 4
4. Expressing a change as a percentage 5
5. Calculating percentages using a calculator 6
www.mathcentre.ac.uk 1 c mathcentre 2009
, 1. Introduction
The word ‘percentage’ is very familiar to us as it is used regularly in the media to describe
anything from changes in the interest rate, to the number of people taking holidays abroad, to
the success rate of the latest medical procedures or exam results. Percentages are a useful way
of making comparisons, apart from being used to calculate the many taxes that we pay such as
VAT, income tax, domestic fuel tax and insurance tax, to name but a few.
So percentages are very much part of our lives. But what does percentage actually mean?
Now ‘per cent’ means ‘out of 100’; and ‘out of’, in mathematical language, means ‘divide by’.
So if you score 85% (using the symbol ‘%’ for percentage) on a test then, if there were a possible
100 marks altogether, you would have achieved 85 marks. So
85
85% = .
100
Let us look at some other common percentage amounts, and their fraction and decimal equiva-
lents.
75
75% = 100 = 34 = 0.75
50 1
50% = 100
= 2
= 0.5
25 1
25% = 100
= 4
= 0.25
10 1
10% = 100
= 10
= 0.1
5 1
5% = 100
= 20
= 0.05 .
It is worth noting that 50% can be found be dividing by 2, and that 10% is easily found by
dividing by 10.
Now let us look at writing fractions as percentages. For example, say you get 18 marks out of
20 in a test. What percentage is this?
18
First, write the information as a fraction. You gained 18 out of 20 marks, so the fraction is 20 .
18
Since a percentage requires a denominator of 100, we can turn 20 into a fraction out of 100 by
multiplying both numerator and denominator by 5:
18 18 × 5 90
= = = 90% .
20 20 × 5 100
Since we are multiplying both the numerator and the denominator by 5, we are not changing the
value of the fraction, merely finding an equivalent fraction.
In that example it was easy to see that, in order to make the denominator 100, we needed to
multiply 20 by 5. But if it is not easy to see this, such as with a score of, say, 53 out of 68, then
we simply write the amount as a fraction and then multiply by 100 100
:
53 100
× = 53 ÷ 68 × 100% = 77.94%
68 100
which is 78% to the nearest whole number. Although it is easier to use a calculator for this type
of calculation, it is advisable not to use the % button at this stage. We shall look at using the
percentage button on a calculator at the end of this unit.
www.mathcentre.ac.uk 2 c mathcentre 2009
