Percentage (%)
Introduction:-
In mathematics, a percentage is a number or simply a ratio expressed as a fraction of 100.
"Percent" comes from the Latin "Per Centum". The Latin word Centum means 100 When we say "Percent" we are actually saying "per 100 or out of 100".
Percentage is usually denoted by using ' % 'sign. % sign is an easy way to write any number divided by 100.
A percentage is a pure number. (Dimensionless Number)
Ex. :- Instead of saying " I got 70 marks out of 100 in English, one can say " I got 70 % marks in English.
" The percent value is computed by multiplying the numeric value of the ratio by 100."
NOTE :- 1) X % of Y = Y % of X.
Sol. ⇒ 10 % of 50 = 5 as well as 50 % of 10 is 5.
Representation of Percentage :-
Percent can be represented or expressed in the form of decimals and fractions.
1. The percent value is calculated by multiplying the numeric value ratio by 100.
For E.g. 450 students of class IX & X forms what percentage of total number of students in school which is 1800.
Sol. ⇒ 450/1800 = 1/4
⇒ 1/4 × 100 = 25 %
2. Percentage of percentage is calculated by converting both percentage to fractions of 100
For E.g. 50 % of 60 % = ?
Sol. ⇒ 50/100 × 60/100 = 30/100 = 30 %
3-Ways to ask a problem:-
1. What is 30 % of 200 ?
2. 60 is what percent of 200 ?
3. 60 is 30 % of what ?.
In such questions students get confused as what should exactly be done ?
Hence, to tackle this we can use a simple formula for calculation by which confusion will get removed and calculation become easier.
Formula :-
Now let's try to answer all the 3 questions.
1) What is 30 % of 200 ?
Sol. ⇒ 30/100 × 200 = 30 × 2 = 60
2. 60 is what percent of 200 ?
Sol. ⇒ Here, we will put 60 in place of ' is ' and 200 in place of ' of '
hence we get, 60/200 × 100 = 60/2 = 30 %
3. 60 is 30 % of what ?.
Sol. ⇒ Here, we will put 60 in place of ' is ' and an unknown variable 'x' in place of ' of '
hence we get, 60/x × 100 = 30 %
6000/x = 30 %
x = 6000/30 = 200
Conversions :-
1) Percent to Decimal :-
⇒ Divide by 100 & remove % sign
(or you can simply shift the decimal point to 2 places to the left)
2) Decimal to Percent :-
⇒ Multiply by 100 & add % sign
(or you can simply shift decimal point to 2 places to the right)
3) Fraction to Decimal :-
⇒ Simply divide numerator by denominator.
4) Decimal to Fraction :-
⇒ Remove the decimal point by dividing it by power of 10 equals to decimal place and then simplify the fraction.
5) Fraction to Percentage :-
⇒ Simply multiply the fraction by 100.
6) Percentage to Fraction :-
⇒ Simply remove % sign (i.e. divide by 100) and simplify..
Percentage increase & decrease / Percent Change :-
When value of a quantity is increases or decreases, we compare the increased or decreased value to the initial (original) value of the quantity.
The difference thus obtained is further divided by the initial (original) value and the fraction thus obtained is multiplied by 100 to get value of Percentage increase/decrease in the value.
Percentage Difference :-
" Percentage difference is calculated as the difference between two values divided by the average of two values. "
Percent Error :-
" Percent error is the difference between False/ approximate / Estimated value and the True/Actual value in comparison to the actual value and is expressed as a percentage. "Ex. Instead of writing his age 16, Rohit wrote it as 14 in application form. Find percentage Error.
Sol.⇒ | 16 - 14 |
___________________ × 100 = 1/8 × 100 = 25 %
16 %
Percentage Points :-
News :- SBI increases Home Loan interest rates from 10 % to 12 %.Whether it is a 2 % rise or 20 % rise ?
⇒ Let's calculate,
12 % - 10 %
____________________ × 100 = 20 % rise.
10 %
" We can also say that it was a rise of 2 percentage points."
When we subtract one percentage from another you get "Percentage Point"
1 Percentage Point = 1 % as a simple difference.NOTE :- It does not mean a relative change it is difference between two percentage.
In the next article we will discuss word problems based on Percetage.
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