Percentages
Luc Hens
23 September 2013
What is a percentage?
A percentage expresses a proportion as a fraction with denominator 100, and
uses the notation % (percent) instead of /100 (“denominator 100”). Percent
means “for every hundred” (centum is Latin for hundred).
A proportion can be expressed in different ways. For instance, suppose that
in a group of 350 students there are 70 female students. The proportion of
female students is:
70
350
(70 out of 350 students are female students). Note that
by dividing both numerator and denominator by 70:
70
350
can be simplified
70
70/70
1
=
=
350
350/70
5
So one can also say: one in every five students is female. The proportion can be
written as a fraction with denominator 100. Divide numeraror and denominator
by 3.5:
70
70/3.5
20
=
=
350
350/3.5
100
So one can also say: twenty out of every hundred students are female, or,
alternatively, twenty percent of the students is female; 20% means the same
thing as:
20
100
The proportion can also be written in decimal form:
70
= 0.20
350
In sum, the same proportion can be written in several ways:
70
1
20
= = 0.20 =
= 20%
350
5
100
To express a proportion as a percentage, multiply the proportion by 100%
70
(by 100%, not by 100). The proportion of females is 350
, and hence the percentage of females is:
70
70 × 100
7000
× 100% =
%=
% = 20%
350
350
350
1
If you use 0.2 or
result:
1
5
instead of
70
350
to compute the percentage, you get the same
70
1
× 100% = 0.2 × 100% = × 100% = 20%
350
5
Growth rate
Suppose you have a shop, and your sales increased by € 18, from € 360 in 2011
to € 378 in 2012. The change in sales is:
new value − old value = € 378 − € 360 = € 18
The growth rate (or rate of change) expresses the change as a proportion of
the old value:
growth rate =
new value − old value
old value
In this case, the growth rate is:
€ 378 − € 360
€ 18
=
= 0.05
€ 360
€ 360
That is, between 2011 and 2012, sales went up by a proportion 0.05 of sales in
2011. You can verify this as follows:
0.05 × € 360 = € 18
which indeed is the observed change.
Percentage growth rate
To convert a (proportional) growth rate to a percentage growth rate, multiply
the growth rate by 100% (by 100%, not by 100):
0.05 × 100% = 5%
That is, between 2011 and 2012, sales went up by 5%. In sum, the formula for
a percentage growth rate is:
percentage growth rate =
new value − old value
× 100%
old value
Computing percentages without a calculator
In order to compute percentages without a calculator, you have to know some
rules of algebra:
a
a×c
=
b
b×c
a÷
(for every c different from zero)
c
d
=a×
d
c
2
As percent means “for every hundred”, 5% is
“5% of € 360” means:
5% × € 360 =
5
100 .
5
5 × € 360
× € 360 =
= € 18
100
100
To divide by a percentage (for instance, to find € 35 ÷ 5%) first write the per5
centage as a fraction (5% = 100
). Then continue:
€ 35 ÷ 5% = € 35 ÷
5
100
= € 35 ×
= € 35 × 20 = € 700
100
5
3