Chapter 7-Growth Rates and Percentages
7.1
Percentages
The danger with using percentages is that the percentage sign is
easily detached from the percentage value leading to incorrect
statements.
For example we know that 25% is a quarter but writing:
25 = ¼ or 25 = 0.25 is incorrect.
E7.1
A7.1
7.2
E7.2
A7.2
7.3
E7.3
Correct the statements below for 25% is a quarter:
25 = ¼ or 25 = 0.25.
25% = ¼ or 25% = 0.25.
We know that 10% of 100 is 10.
We can write this mathematically as:
10% x 100 = 10. If we replace the % sign the statement
becomes:
10
 100  10 .
100
Note that
10
i.
is the fractional form for 10%.
100
ii. ‘of’ is represented by the x operation.
i.Write the statement, 25% of 100 is 25 in mathematical form
without using the % sign.
ii. what is the fractional form for 25%
25
 100  25
100
25 1
ii.
 is the fractional form of 25%
100 4
i.
We can also write, 10% of 100 is 10, mathematically as:
0.10 100  10 i.e. replace the % form with the decimal form i.e.
0.10 is the decimal form for 10%.
Another example is : 17.5% (the rate of VAT) = 0.175
To convert to decimal form remove the % sign and move the
decimal point two places to the left.
Complete the Table: The first row has been done for you.
1
For 10%
we use
or 0.1
10
78
For 20%
A7.3
7.4
We use
or
19
we use
or
100
For 50%
we use
For 100% we use
For
% we use 0
For
% we use 0.01
1
For 10%
we use
or 0.1
10
20 1
For 20%
We use
 or 0.20
100 5
19
For 19%
we use
or 0.19
100
50 1
For 50%
we use
 or 0.50
100 2
For 100% we use 1
For 0 %
we use 0
For 1 % we use 0.01
Solving % problems using their equation form
Sometimes solving a percentage problem is best done by casting it
into an equation form and then solving it. This may seem a
sledgehammer method in easy cases. It will help with not so easy
problems.
Example: Find 20% of 50.
Write statement: ‘n is 20% of 50’ in equation form using the
decimal form of 20%:
n = 0.20(50) is the equation and its solution is n = 10.
The word is is replaced with the = sign and ‘of’ is the multiplication
operation.
E7.4
A7.4
7.5
So 10 is 20% of 50
Find 17.5% of £350 by casting the statement into an equation in
the decimal form:
0.175(£350) = n
So n = £61.25
Here is a slightly more difficult problem.
36 is 12% of what quantity?
Change the Question to a statement with n replacing ‘what
79
quantity’ :
36 is 12% of n.
Cast this as an equation in decimal form.
36 = 0.12n
Solve:
36  0.12n
 0.12n  36  n 
36
 300
0.12
So 36 is 12% of 300.
E7.5
25 is 2% of what quantity?
A7.5
25 = 2% of n Can be written as 0.2n = 25.
 n = 25/0.2 = 125
7.6
Another problem: 24 is what percentage of 1200?
Statement with n: 24 is n% of 1200.
Equation in fractional form:
n
(1200)
100
24(100)
n 
 20
1200
24 
So 24 is 20% of 1200.
E7.6
36 is what % of 1440?
A7.6
36 = (x%)(1440)
Replacing the % sign and solve:
1440x
36 =
100
36(100)

x
1440
3  x
So 36 is 3% of 1440.
Back percentages
Consider the problem:
80% of nurses interviewed said they did not mind doing night
7.7
80
shifts. If this number is 36 how many nurses were interviewed?
We want to find total number of nurses so call this number n.
Then we get the statement: 80% of n nurses is 36.
Equation in decimal form: 0.80n = 36
36
Solve equation: n 
 45
0.80
E7.7
In a class 24 students received an ‘A’ grade. If this is 30% of the
class what is the size of the class?
A7.7
We want to find total number of students, so call this number n.
Then we get the statement: 30% of n students is 24.
Equation in decimal form: 0.30n = 24
24
Solve equation: n 
 80
0.3
7.8
Percentage growth
In finance we are very interested in percentage growth.
For Example if your normal pay is £30 000 per annum and you get
an increase of 10% what is the new pay level?
We can do this as a 2 stage calculation:
Stage 1: Find the change, we use the ∆ symbol (pronounced
Delta) for the increase.
∆ in £ = 10% of 30k = 0.10(30 000) = 3000
Note the ∆ is +ve implying an increase.
Stage 2: Add ∆ to find new pay level.
So the new pay level = £30 000 + £3000 = £33 000.
E7.8
Increase a salary of £50,000 by 7% using a 2 stage calculation.
Working in £
Stage 1: ∆ in £ =
Stage 2: Add ∆ to find new pay level.
So the new pay level in £ = .
A7.8
Stage 1: ∆ in £ = 7% of 50k = 0.07(50 000) = 3500
Stage 2: Add ∆ to find new pay level.
So the new pay level = £50 000 + £3500 = £53 500.
7.9
Because in finance we are dealing with percentage growths over
81
many periods we need a more efficient (shorter) way of dealing
with growth problems.
Suppose 1000 is increased by 10%. What is the new value?
New value = original value + ∆ = 1000 + 0.10(1000)
Now this expression is made up of two terms. Factorising we get
New value = 1000(1 + 0.10) = 1000(1.01) = 1010.
So we can go straight to the new value in one step by multiplying
the original value by 1.10.
A growth rate of 10% has been transformed to a growth factor
of 1.10.
E7.9
A7.9
7.10
Find the new value in one step if 250 is increased by 10%.
Growth rate = 10% ; growth factor = 1.10
New value = 250(
)=
Growth rate = 10% ; growth factor = 1.10
New value = 250(1.10) =275
So if we want to increase by a growth rate of 10% we multiply the
original value by the growth factor, 1.10.
Now if we wanted to increase the original value by 20% we would
multiply by the growth factor of 1.20
Example: A plant of 320cm grows by 7% in a week. What is its
height at the end of the week?
E7.10
A7.10
7.11
The growth rate of 7%  growth factor of 1.07 so:
New height = 320(1.07) = 342.40cm
A plant of height 120cm grows by 11% in a week. What is its
height at the end of the week?
growth rate = 11%, growth factor =
so:
height at end of week =120(
)cm =
A plant of height 120cm grows by 11% in a week. What is its
height at the end of the week?
growth rate = 11%, growth factor = 1.11
so:
height at end of week =120(1.11) cm = 13.32 cm.
In general if the growth rate is r% then the growth factor is
(1 + r per cent in decimal form)
Examples:
Normal pay
100pd
%growth growth factor
new pay
50%
1+0.50 = 1.50 100(1.50)=150p
82
E7.11
d
new share price
75(1.25)p =
93.75p
price after Vat
Share Price
75p
%growth growth factor
25%
1.25
Price (£) before
Vat
25.00
Vat rate
growth factor
17.5%
1.175
Normal pay
100pd
500pw
10000pa
Share Price
5.25
3.50
6.50
S
Price before Vat
25.00
X
%growth growth factor
50%
1+0.50 = 1.50
1.25
5%
%growth growth factor
8%
1.10
1.50
10%
Vat rate growth factor
17.5%
17.5%
new pay
100(1.50)=150
Normal pay
100pd
500pw
10 000pa
%growth
50%
25%
5%
Share Price
5.25
%growth growth factor
8%
1.08
3.50
6.50
S
Price before Vat
25.00
10%
50%
10%
Vat rate
17.5%
1.10
1.50
1.1
growth factor
1.175
X
17.5%
1.175
new pay
100(1.50)=150
500(1.25)=625
10 000(1.05)=
10 500
new share price
5.25(1.08) =
5.67
3.5(1.1)=3.85
6.5(1.5)=7.35
S(1.1) = 1.1S
price after Vat
25(1.175) =
29.375
X(1.175)
=1.175X
25(1.175)
=29.38
Complete the Table: The first row has been done for you.
new share price
price after Vat
A7.11
7.12
growth factor
1+0.50 = 1.50
1.25
1.05
Back Percentages using growth factors
Now consider this problem.
A plant grows by 20% to a height of 120cm at the end of the
week. What was the height at the beginning of the week?
Using h for the original height and (20% giving) a growth factor of
83
1.20 we can write the equation:
h(1.20)  120cm
h 
E7.12
120
cm  100cm
1.20
The population of Woodgreen grew by 12% to 6048 in a decade.
What was the population at the beginning of the decade?
Growth rate = 12% growth factor =
Equation using P for original population:
A7.12
The population of Woodgreen grew by 12% to 6048 in a decade.
What was the population at the beginning of the decade?
Growth rate = 12% growth factor = 1.12
Equation using P for original population:
1.12P = 6048
6048
P =

1.12
 P = 5400
7.13
Multiperiod growth
An investment of £1000 grows by 5% in the first year and by 4%
in the second year. What is the value of the investment at the end
of the two years?
Value at end of year 1 = £1000(1.05) = £1050
Value at end of year 2 = £1050(1.04) = £1092.
E7.13
However we can do this in one stage:
Value at end of two years by ‘linking’ the growth factors.
=£1000(1.05)(1.04) = £1092.
The linking is by multiplying the growth factors.
An investment of £500 grows by 5% in the first year and by 7% in
the second year. Obtain the value of the investment at the end of
the two years in one stage?
A7.13
Value at end of two years by ‘linking’ the growth factors.
=£500(1.05)(1.07) = £561.75.
7.14
An investment grows by 5% in the first year and by 6% in the
second year. What is the whole period growth factor and growth
84
rate?
The yearly growth rates are 5% and 6% so the yearly growth
factors are 1.05 and 1.06.
The whole period growth factor = (1.05)(1.06) = 1.113.
So the whole period growth rate
= 1.113 – 1 = 0.113 = 11.3 %
E7.14
An investment grows by 6% in the first year, 6% in the second
year and remains unchanged in the third year. What is the whole
period growth rate?
The yearly growth rates are ….% , …..% and …..% so the yearly
growth factors are ….., …… and …...
The whole period growth factor = ( )( )(
)=
So the whole period growth rate =
A7.14
=
%
The yearly growth rates are 6% , 6% and 0% so the yearly growth
factors are 1.06,1.06 and 1.
The whole period growth factor = (1.06)(1.06)(1) = 1.1236
So the whole period growth rate = 0.1236 = 12.36%
7.15
Percentage decrease
We will do this directly using growth factors.
An investment of £1000 falls by 5% in a year. Find the value of the
investment at the end of the year in one stage (without finding ∆)?
The growth is now negative = -5%
So the growth factor is (1 – 0.05) = 0.95
So value at end of the year = £1000(0.95) = £950.
E7.15
Note the terminology ‘growth factor’ was used even though the
value fell. The fall was reflected in the growth factor being below
1.
Complete the Table: The first row has been done for you.
Change
Fall by 5%
Growth rate
-5%
-10%
growth factor
1-0.05 = 0. 5
0.88
Increase by 2%
4%
85
1.03
Drop by 12%
Increase by 1
basis point
A7.15
Change
Fall by 5%
Fall by 10%
Fall by 12%
Increase by 2%
Increase by 4%
Increase by 3%
Drop by 12%
Increase by 1
basis point
7.16
Growth rate
-5%
-10%
-12%
2%
4%
3%
-12%
0.01%
growth factor
1-0.05 = 0. 5
1-0.10 = 0.90
0.88
1.02
1.04
1.03
0.88
1.0001
Multiperiod growth including negative growth
An investment of £25000 decreases in value by 5% in the first
year and decreases by a further 4% in the second year. What is
the end period value of the investment? What is the whole period
percentage change in value?
We will do this directly without the use of deltas.
The period of investment is 2 years with yearly growth rates of
-5% and -4%. The corresponding growth factors are 0.95 and
0.96 (below 1).
End period value = £25 000(0.95)(0.96) = £22 800.
The whole period percentage change is obtained as follows:
Whole period growth factor = (0.95)(0.96)
So whole period growth rate = (0.95)(0.96) – 1 = -0.088 = -8.8%
E7.16
An investment of £2000 decreases in value by 5% in the first year
and decreases by a further 3% in the second year. What is the end
period value of the investment? What is the whole period
percentage change in value?
86
A7.16
An investment of £2000 decreases in value by 5% in the first year
and decreases by a further 3% in the second year. What is the end
period value of the investment? What is the whole period
percentage change in value?
The period of investment is 2 years with yearly growth rates of
-5% and -3%. The corresponding growth factors are 0.95 and
0.97 (below 1).
End period value = £2 000(0.95)(0.97) = £1843.
The whole period percentage change is obtained as follows:
Whole period growth factor = (0.95)(0.97)
So whole period growth rate = (0.95)(0.97) – 1
= -0.0785 = -7.85%
87
Exercise 7
1. A football team wins 18 of its 42 games in the season. What
is the percentage of the games won?
1. 20 students, which is 80% of a class, passed their exam. What is the size
of the class?
3.
(i) A plant of height 10cm increases by 5%. What is the new height?
(ii)
A plant increases in height by 4% in a week and grows to a height
of 108 cm. What was the height at the beginning of the week?
(iii)
A investor invests £10000 for 20 years at a compounded rate of 7%
per annum. What is the value of the investment after the 20 years?
88