Percentage UPSC

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.

Mathematical definition of Percentage:

The concept of percentage mainly applies to ratios, and the percentage value of a ratio is arrived at by multiplying by 100 the decimal value of the ratio.
For example, a student scores 20 marks out of a maximum possible 30 marks. His marks can then be denoted as 20 out of 30 is (20/30) or (20/30) * 100% = 66.66%.

The process for getting this is perfectly illustrated through the unitary method:
If, 20 out of 30 then X out of 100
20 → 30
X → 100
Therefore,
x=20/30×100 = 66.66%

Percentage Change:

If original value is changed to the new value, then % change is given as –

Examples of percentages are:

10% is equal to 1/10 fraction
20% is equivalent to ⅕ fraction
25% is equivalent to ¼ fraction
50% is equivalent to ½ fraction
75% is equivalent to ¾ fraction
90% is equivalent to 9/10 fraction

Percentages have no dimension. Hence it is called a dimensionless number. If we say, 50% of a number, then it means 50 per cent of its whole.

Percentages can also be represented in decimal or fraction form, such as 0.6%, 0.25%, etc. In academics, the marks obtained in any subject are calculated in terms of percentage. Like, Ram has got 78% of marks in his final exam. So, this percentage is calculated on account of the total marks obtained by Ram, in all subjects to the total marks.

Percentage Formula

To determine the percentage, we have to divide the value by the total value and then multiply the resultant by 100.

Percentage formula = (Value/Total value) × 100

Example: 2/5 × 100 = 0.4 × 100 = 40 per cent

How to calculate the percentage of a number?

To calculate the percentage of a number, we need to use a different formula such as:

P% of Number = X

where X is the required percentage.

If we remove the % sign, then we need to express the above formulas as;

P/100 * Number = X

Example: Calculate 10% of 80.

Let 10% of 80 = X

10/100 * 80 = X

X = 8

Percentage Increase and Decrease

The percentage increase is equal to the subtraction of the original number from a new number, divided by the original number and multiplied by 100.

% increase = [(New number – Original number)/Original number] x 100

where,

increase in number = New number – original number

Similarly, a percentage decrease is equal to the subtraction of a new number from the original number, divided by the original number and multiplied by 100.

% decrease = [(Original number – New number)/Original number] x 100

Where decrease in number = Original number – New number

So basically if the answer is negative then there is a percentage decrease.

Percentage Chart

The percentage chart is given here for fractions converted into percentages.

Fractions	Percentage
1/2	50%
1/3	33.33%
1/4	25%
1/5	20%
1/6	16.66%
1/7	14.28%
1/8	12.5%
1/9	11.11%
1/10	10%
1/11	9.09%
1/12	8.33%
1/13	7.69%
1/14	7.14%
1/15	6.66%

Percentage Questions

Q.1: If 16% of 40% of a number is 8, then find the number.

Solution:

Let X be the required number.

Therefore, as per the given question,

(16/100) × (40/100) × X = 8

So, X = (8 × 100 × 100) / (16 × 40)

= 125

Q.2: What percentage of 2/7 is 1/35 ?

Solution:

Let X% of 2/7 is 1/35.

∴ [(2/7) / 100] × X = 1/35

⇒ X = (1/35) × (7/2) × 100

= 10%

Q.3: Which number is 40% less than 90?

Solution:

Required number = 60% of 90

= (90 x 60)/100

= 54

Therefore, the number 54 is 40% less than 90.

Q.4: The sum of (16% of 24.2) and (10% of 2.42) is equal to what value?

Solution:

As per the given question ,

Sum = (16% of 24.2) + (10% of 2.42)

= (24.2 × 16)/100 + (2.42 × 10)/100

= 3.872 + 0.242

= 4.114

Word Problems

Q.1: A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, he had how many apples?

Solution:

Let he had N apples, originally.

Now, as per the given question, we have;

(100 – 40)% of N = 420

⇒ (60/100) × N = 420

⇒ N = (420 × 100/60) = 700

Q.2: Out of two numbers, 40% of the greater number is equal to 60% of the smaller. If the sum of the numbers is 150, then the greater number is?

Solution:

Let X be the greater number.

∴ Smaller number = 150 – X {given that the sum of two numbers is 150}

According to the question,

(40 × X)/100 = 60(150 – X)/100

⇒ 2p = 3 × 150 – 3X

⇒ 5X = 3 × 150

⇒ X = 90

Difference between Percentage and Percent

The words percentage and percent are related closely to each other.

Percent ( or symbol %) is accompanied by a specific number.

E.g., More than 75% of the participants responded with a positive response to abjure.

The percentage is represented without a number.

E.g., The percentage of the population affected by malaria is between 60% and 65%.

Fractions, Ratios, Percents and Decimals are interrelated with each other. Let us look at the conversion of one form to another:

S.no	Ratio	Fraction	Percent(%)	Decimal
1	1:1	1/1	100	1
2	1:2	1/2	50	0.5
3	1:3	1/3	33.333	0.3333
4	1:4	1/4	25	0.25
5	1:5	1/5	20	0.20
6	1:6	1/6	16.667	0.16667
7	1:7	1/7	14.285	0.14285
8	1:8	1/8	12.5	0.125
9	1:9	1/9	11.111	0.11111
10	1:10	1/10	10	0.10
11	1:11	1/11	9.0909	0.0909
12	1:12	1/12	8.333	0.08333
13	1:13	1/13	7.692	0.07692
14	1:14	1/14	7.142	0.07142
15	1:15	1/15	6.66	0.0666

Percentage in Maths

Every percentage problem has three possible unknowns or variables :

Percentage
Part
Base

In order to solve any percentage problem, you must be able to identify these variables.

Look at the following examples. All three variables are known:

Example 1: 70% of 30 is 21

70 is the percentage.

30 is the base.

21 is the part.

Example 2: 25% of 200 is 50

25 is the percent.

200 is the base.

50 is the part.

Example 3: 6 is 50% of 12

6 is the part.

50 is the percent.

12 is the base.

Percentage Tricks

To calculate the percentage, we can use the given below tricks.

x % of y = y % of x

Example- Prove that 10% of 30 is equal to 30% of 10.

Solution- 10% of 30 = 3

30% of 10 = 3

Therefore, they are equal i.e. x % of y = y % of x holds true.

Marks Percentage

Students get marks in exams, usually out of 100. The marks are calculated in terms of per cent. If a student has scored out of total marks, then we have to divide the scored marks by total marks and multiply by 100. Let us see some examples here:

Marks obtained	Out of Total Marks	Percentage
30	100	(30/100)× 100 = 30%
10	20	(10/20) × 100 = 50%
23	50	(23/50) × 100 = 46%
13	40	(13/40) × 100 = 32.5%
90	120	(90/120) × 100 = 75%

Problems and Solutions

Question 1:- Suman has a monthly salary of $1200. She spends $280 per month on food. What percent of her monthly salary does she save?Solution:Suman’s monthly salary = $1200Savings of Suman = $(1200 – 280) = $ 920Fraction of salary she saves = 920/1200Percentage of salary she saves = (920/1200) × 100 = 920/12% = 76.667%