PERCENTAGES
Widely used in the educational, economic,
social and political activities. Schools use it to evaluate students’
performance, businesses use it to check sales as well make discounts, and
politicians use it get electoral results. . Percentages are the plural form
while percent is the singular form.
Definition. Percent means “per hundred” or hundredths. The
word “cent” comes from the Latin centum, meaning hundred. Hence a hundred years
is called a century. ‘Per hundred’ means
out of one hundred. In daily
conversations we talk about percentages meaning the above.
The mathematical symbol for percent is %. ’15 percent’ is written as 15% meaning 15 out
of 100.
A
PERCENTAGE IS A WHOLE OF A FRACTION.
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From the diagram
above, 15 blocks are shaded. It means 15% ( 15 percent ) of the blocks are
shaded. The 15% can be written as a fraction and as decimal.
Ø
As fraction : 15/100
Ø
As decimal:
0.15.
CONVERTING FRACTIONS TO PERCENTAGES
As
seen above, a percentage is a fraction with denominator 100. Therefore to
convert any fraction to a percentage, use any of the following methods:
1)
Convert given fraction to an equivalent fraction
with a denominator of 100.
2)
Multiply the fraction given by 100/1.
3)
Change given fraction to decimal and multiply by
100/1 then attach % to your result.
EXAMPLE 1.
Convert 2 to
a percentage.
5
METHOD
1- Convert given fraction to an equivalent fraction with a denominator of 100
Solution
2 = x
5 100
Since the
denominator 5 of the given fraction is multiplied by the 20 to give 100 the
denominator of the equivalent fraction, multiply the numerator of the given
fraction by 20. That is 2× 20=40.
2 = 40 .
Therefore, 2 = 40% answer.
5 100 5
The question here
is how did we get 20? Answer: Divide the
denominator of the equivalent fraction by that of the given fraction. (100÷5=20).
Now you multiply the denominator of the given fraction by 20 to the numerator
of the expected equivalent fraction.
METHOD 2- Multiply the given fraction by 100.
1
Solution.
2 = 2
× 100 %
5 5 1
2 = 40%
5
METHOD 3- Change fraction to decimal and multiply by 100 %
1
2 = 0.4
5
2 = 0.4× 100 %
5 1
2 = 40% answer.
5
EXERCISE 1 . Change
the following fractions as percentages using any of the methods above.
i)
40
, ii) 19 ,
iii) 17 , iv)
8
100 100 50 25
SOLUTION
i)
40 = 40% , ii) 19
= 19%, iii) 17= 34 = 34%, iv) 8 = 32 =32% .
100 100 50 100 25 100
EXERCISE 2-Express these percentages as
fractions, simplifying each answer to its lowest term.
i)
10%, ii) 16%
, iii) 25% , iv) 56%, v) 33% vi) .75% , vii) 190%
SOLUTION
i)
10%= 10 = 1 , ii)
16% = 16 = 4 , iii) 25% =
25 = 1 , iv) 56% = 56
=
13 , 100 10 100 25 100 4 100 4 .
EXERCISE 3-Convert the following
decimals into percentages
a)
0.25 ,
b) .04 , c) .831, d) .6734
SOLUTION
a)
.25= .25× 100= 25%, b) .04= .04×100= 40%, c)
.831= .831×100=83.1%, d) .6734=.6734×100= 67.34%
EXERCISE 4- Write these percentages as
decimals.
i)
8% , ii) 19%, iii) 72%, iv 0.5% , v) 0.7%.
SOLUTION
i)
8%= 8 = .08 ,
ii) 19%= 19 = 0.19 , iii)
72%= 72 = 0.72 , iv) 0.5% = 0.5 = 0.005, v) 0.7%= 100 100 100 100
EXERCISE 5- Consider the diagram below;
a)
How many small blocks make up
100% 0f this diagram?
b)
Write 100% as a fraction.
c)
Write 100% as a decimal.
d)
Is 100% the same as: i) 1 whole? , ii) 10
wholes?, iii) 100 wholes?
e)
What percentage is the same as 5 wholes?
f)
What percentage is the same as 50 wholes?
SOLUTION
a)
100, b) 100 , c) 1.00,
d) i. Yes, ii. Yes , iii) Yes , e)
100%, f) 100%
100
COMPARING FRACTIONS,
DECIMALS AND PERCENTAGES.
To compare fractions, decimals and
percentages, write them in the same form- either as fractions, as decimals or
as percentages.
EXAMPLE 1. Arrange
1 , 0.4, 60% in increasing order.
2
SOLUTION
a)
If we write all of them as percentages:
1
= 50%, 0.4 = 40%, 60%= 60%
2
Arranging
the numbers in increasing order gives 40%, 50% and 60%, hence 0.4, 1 ,
60%
2
b)
If we write the numbers as fractions with a
common denominator:
1=
50
2 100
0.4=
40
100
60%= 60
100
Arranging the numbers
in increasing order gives 40 ,
50 , 60 ,
100 100 100
Hence 0.4, 1 60% in increasing order.
2
Hence 0.4, 1 60% in increasing order.
2
c)
If we write all of the as decimals:
1
= 0.5
2
0.4
=0.4
60%
= 60 = 0.6
100
Arranging
in increasing order gives 0.4, 0.5, and 0.6, hence, 0.4, 1, 60%
2
EXERCISE. 1 . Arrange each of the
following group of numbers in ascending order;
i)
3
, 30.76, 70% . ii) 0.87, 85%, 5 ,
iii) 25%, 1 , 0.23
4 6 5
SOLUTION
i)
3 , 30.76 , 70%. Writing all as decimals give:
4
70%=
0.7, ¾=.75, 30.76=30.76. Arranging in
increasing order gives 0.7, 0.75, and 30.76. Hence in increasing, it will be 70%, 3 , 30.76
4
ii)
.87, 85%, 5 . writing all as decimals
give:
6
0.87
=0.87
85%= 85 =0.85
100
5
= 0.83
6
From
the decimals above, in increasing order gives
5 , 85%, 0.87
6
iii)
25% , 1 , 0.23 . Writing all as decimals
give:
5
25%
= 25 =0.25
100
1
=0.20
5
0.23=
0.23
From
the decimals above , in increasing order gives
0.20, 0.23, 0.25 hence 1 , 0.23, 25%
5
NB.
You have choice to write the numbers either as decimals, fractions or
percentages before rearranging. Any direction will give you the correct result
if properly applied.
EXERCISE 2- Arrange each of the
following groups of numbers in descending order.
i)
7 , 12%, 0.38,
ii) .375, 37%, 2
20 5
SOLUTION
In
descending order means moving down starting from the largest number to the
smallest.
i)
7 , 12%,
0.38 . write all as decimals
20
7
=0.35
20
12%=
12 = 0.12
100
In descending
order gives 0.38, 7 , 12%
20
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