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percentage formula

Thursday, 23 March 2023



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.

                                        
                                        13% = 13/100  =0.13 

                                        25% = 25/100  =0.25

                                        18.5% = 18.5/100 = 0.185

                                        50% = 50/100  = 0.5 


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

          

Michel had 100 tiles in all, there were 100 children and 100 shoe pairs. How do we calculate Percentage of an item if the total number of items do not add up to 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?


Solution

(i) Out of 32 students , 8 are absent.

     In percentage = 8/32 * 100
                      
                            = 25%

(ii) Out of 25 ,16 radios are out of order

     In percentage = 16/25 *100

                            = 64%

 (iii) Out of 500 items ,5 are defective

     In percentage = 5/500 *100

                            = 1%

 (iv) Out of 120 voters, 90 of them voted yes

      In percentage = 90/120 *100

                             = 75%

 

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


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