What is a percentage?
The term "percentage" was derived from the Latin word "per centum", which means "by the hundred".
Percentages are fractions with 100 as the denominator. It is the relation between part and whole where the value of "whole" is always taken as 100.
Per cent is represented by the symbol % and means hundredths too. That is 1% means 1 out of hundred or one hundredth. It can be written as: 1% = 1/ 100 = 0.01
Percentage Formula
Percentage = value × 100
Total value
To understand this, let us consider the following example.
Example,
If the marks of a student in math are 25 out of 50 then the corresponding percentage can be calculated by expressing "marks obtained" as a fraction of "total marks" and multiplying the result by 100
. i.e., percentage of marks = 25/ 50 × 100 = 50%.
Learn more about percentages and how to convert them into fractions and decimals.
Example
How to calculate the percentage of marks of a student who scored 15 out of 40 in math?
Solution:
Here, the marks of the student are 15/40.
But here, the denominator is not a factor of 100.
Thus, finding the percentage in the unitary method is helpful here.
Percentage of marks = 15/40 × 100
= 27.5%.
How to find percentage when total is not 100 ?
Percentages when total is not hundred . In such cases, we need to convert the fraction to an equivalent fraction with denominator 100
Consider the following example. You have a necklace with twenty beads in two colors.
We see that these three methods can be used to find the Percentage when the total does not add to give 100.
In the method shown in the table, we multiply the fraction by 100 /100 . This does not change the value of the fraction. Subsequently, only 100 remains in the denominator. Mary has used the unitary method.
Asha has multiplied by 5 /5 to get 100 in the denominator. You can use whichever method you find suitable. May be, you can make your own method too
Converting Fractional Numbers to Percentage
Fractional numbers can have different denominator. To compare fractional numbers, we need a common denominator and we have seen that it is more convenient to compare if our denominator is 100. That is, we are converting the fractions to Percentages. Let us try converting different fractional numbers to Percentages.
Example
Write 1 /3 as per cent.
Solution
1/3 = 1/3 *100/100
=1/3 *100%
= 100/3%
= 33 1/3%
Example
Out of 25 children in a class, 15 are girls. What is the percentage of girls?
Solution
Out of 25 children, there are 15 girls.
Therefore, percentage of girls = 15/ 25 ×100 = 60.
There are 60% girls in the class.
Example
Convert 5/4 to per cent.
Solution
We have, 5 /4 =5 /4 × 100 %
=125 %
From these examples, we find that the percentages related to proper fractions
are less than 100 whereas percentages related to improper fractions are more than
100
Converting Decimals to Percentage
We have seen how fractions can be converted to per cents. Let us now find how
decimals can be converted to per cents.
Example
Convert the given decimals to per cents:
(a) 0.75
(b) 0.09
(c) 0.2
Solution
(a) 0.75 = 0.75 × 100 %
= 75/ 100 × 100 % = 75%
(b) 0.09 = 9 /100 = 9 %
(c) 0.2 = 2 /10 × 100% = 20 %
Example
(i) Out of 32 students, 8 are absent. What per cent of the students are absent?
(ii) There are 25 radios, 16 of them are out of order. What per cent of radios are out of order?
(iii) A shop has 500 items, out of which 5 are defective. What per cent are defective?
(iv) There are 120 voters, 90 of them voted yes. What per cent voted yes?
Converting Percentages to Fractions or Decimals
We have so far converted fractions and decimals to percentages. We can also do the reverse. That is, given per cents, we can convert them to decimals or fractions.
Parts always add to give a whole
In the examples for coloured tiles, for the heights of children and for gases in the air, we find that when we add the Percentages we get 100. All the parts that form the whole when added together gives the whole or 100%. So, if we are given one part, we can always find out the other part.
Example
Suppose, 30% of a given number of students are boys.
This means that if there were 100 students, 30 out of them would be boys and the remaining would be girls.
Then girls would obviously be (100 – 30)% = 70%
Example
1. 35% + _______% = 100%,
2. 64% + 20% +________ % = 100%
3. 45% = 100% – _________ %,
4. 70% = ______% – 30%
solution
1. 65%
2. 16%
3. 55%
4. 100%
Example
If 65% of students in a class have a bicycle, what per cent of the student do not have bicycles?
Solution
Given 65% of students have bicycle
Total percentage =100
Therefore , percent of students do not have bicycle = 100 - 65 =35%
Example
We have a basket full of apples, oranges and mangoes. If 50% are apples, 30% are oranges, then what per cent are mangoes?
Solution
Given 50% are apples and 30 % are oranges
Total percentage =100
Therefore percent of mangoes = 100- (50+30)
= 100 - 80
= 20
How percentage helps for estimating the parts?
Percentages help us to estimate the parts of an area.
Example What per cent of the adjoining figure is shaded?
Solution
We first find the fraction of the figure that is shaded.
You will find that 4/6 of the figure is shaded.
And, = 4/6 ×100 %
= 66.66%
Thus, 66.6 % of the figure is shaded.
Uses of percentage
- The percentage is used to compare numerical data and compare data by corporations, firms, governments, schools and colleges.
- The percentage is used by shopkeepers and companies to calculate the profit/loss percentage on the goods sold.
- The percentage is used by Banks and financial institutions to calculate interest % on loans, fixed deposits, and savings accounts.
- Per cent is also used by economists to calculate the growth rate, inflation rate etc.
- It is also used for many other rates like depreciation rate on cars, trucks, and other vehicles.
Converting Percentages to “How Many”
Consider the following examples:
Example
A survey of 40 children showed that 25% liked playing football. How many children liked playing football?
Solution
Here, the total number of children are 40.
Out of these, 25% like playing football.
so 25 % of 40
= 25/100 * 40
= 10
so 10 children like playing football
Example
Zara bought a sweater and saved $200 when a discount of 25% was given. What was the price of the sweater before the discount?
Solution
zara has saved ` 200 when price of sweater is reduced by 25%.
This means that 25% reduction in price is the amount saved by Zara
25% of the original price = 200
or
25/100* price =200
1/4 * price = 200
price = 200* 4
=800
Thus the original price of the sweater is 800
Ratios to Percentage
Sometimes, parts are given to us in the form of ratios and we need to convert those to percentages.
Example
Reena’s mother said, to make Idlis, you must take two parts rice and one part urad dal. What percentage of such a mixture would be rice and what percentage would be urad dal?
Solution
In terms of ratio we would write this as
Rice : Urad dal = 2 : 1.
Now, 2 + 1=3 is the total of all parts.
This means 2/ 3 part is rice and 1 /3 part is urad dal.
Then, percentage of rice would be 2 /3 *100 %
=200 /3
= 66 2 /3 %.
Percentage of urad dal would be 1/3 * 100 %
=100 /3
= 33 1/ 3 %
Example
If $250 is to be divided amongst Rash, Ruweiyda and Sayid , so that Rash gets two parts, Ruweiyda three parts and Sayid five parts. How much money will each get? What will it be in percentages?
Solution
The parts which the three boys are getting can be written in terms of
ratios as 2 : 3 : 5.
Total of the parts is 2 + 3 + 5 = 10.
Amounts received by each
2 /10 × 250 = 50
3/10 × 250 =75
5/10 × 250 = 125
Percentages of money for each
Ravi gets 2/10 ×100 % = 20%
Raju gets 3/10 ×100 % = 30 %
Roy gets 5/10 ×100 % = 50 %
Increase or Decrease as Per Cent
There are times when we need to know the increase or decrease in a certain quantity as percentage.
For example, if the population of a state increased from 5,50,000 to 6,05,000.
Then the increase in population can be understood better if we say, the population increased by 10 %.
Percentage increase = amount of change × 100
original amount or base
Percentage of decrease = Amount of change × 100
original amount
How do we convert the increase or decrease in a quantity as a percentage of the initial amount? Consider the following example
Example
A school team won 6 games this year against 4 games won last year. What is the per cent increase?
Solution
The increase in the number of wins (or amount of change) = 6 – 4 = 2.
Percentage increase = amount of change × 100
original amount or base
= Increase in the number of wins ×100
original number of wins
= 2 /4 × 100
= 50
Example
The number of illiterate persons in a country decreased from 150 lakhs to 100 lakhs in 10 years. What is the percentage of decrease?
Solution
Original amount = the number of illiterate persons initially
= 150 lakhs.
Amount of change = decrease in the number of illiterate persons
= 150 – 100 = 50 lakhs
Therefore, the percentage of decrease
= Amount of change × 100
original amount
= 50 /150*100
= 33 1/3 %
Profit or Loss as a Percentage
The profit or loss can be converted to a percentage. It is always calculated on the CP. For the above examples, we can find the profit % or loss % by using this formula;
Profit % = Profit × 100
C P
Loss % = Loss × 100
C P
Example
The cost of a flower vase is $120. If the shopkeeper sells it at a loss of 10%, find the price at which it is sold.
Solution
First Method
We are given that CP = $120 and Loss per cent = 10.
We have to find the SP.
loss of 10% means if cp is 100 then Loss is $10
Therefore S. p = $ (100 -10) =$ 90
when C.P is 100 then S.P is 90
Therefore if C.P were 120 then
S.P = 90/100 *120
= $108
Second Method;
Loss is 10 % of the cost price
= 10 % of 120
= 10/100 *120
= $12
Therefore
S.P = C. P - Loss
= $120 -12
= $ 108
Example
Selling price of a toy car is $ 540. If the profit made by shopkeeper is 20%, what is the cost price of this toy?
Solution
We are given that
SP = $ 540
Profit = 20%.
We need to find the CP.
First Method
20% profit will mean if CP is $100
profit is $ 20
Therefore, SP = 100 + 20
= 120
Now, when SP is $120, then CP is $ 100.
Therefore, when SP is $ 540,
C.P =100 /120*540
= $ 450
Second Method
Profit = 20% of CP and
SP = CP + Profit
So, 540 = CP + 20% of CP
= CP + 20 /100 × CP
= [1 +1/5] CP
=6 /5 CP .
Therefore, 540*5/ 6 = C P
C.P = 450
or $ 450 = CP
