Basic Concepts of Percentages

In this lesson, we cover the absolute basics of Percentages. The purpose of this lesson is to help you answer one simple question: What are Percentages?

Basic Definition:
Percent implies “for every hundred” and the sign % is read as percentage and x % is read as x per cent. In other words, a fraction with denominator 100 is called a per cent. For example, 20 % means 20/100 (i.e. 20 parts from 100). This can also be written as 0.2.

Basic Formula:
In order to calculate p % of q, use the formula:
(p/100) x q = (pxq)/100

Also remember: p % of q = q % of p

Examples:
1. 100% of 60 is 60 x (100/100) = 60
2. 50% of 60 is 50/100 × 60 = 30
3. 5% of 60 is 5/100 × 60 = 3

Example: 60 % of a number is 360. What is 99 % of the same number?
Solution: Let the number be n.
Given (60/100) ×n = 360 => n = 600
99 % of 600 = (99/100) × 600 = 594

Example: 50 % of a number is 360. What is 99 % of the same number?
Solution: Let the number be y.
Given (50/100) x q = 360
=> q = 720
99% of 720 = (99/100) x 720 = 712.80

Expressing One Quantity as a Per Cent with respect to the other:

To express a quantity as a per cent with respect to other quantity, the following formula is used: Example: What percent is 60 of 240?
Solution: First write the given numbers in the fraction form:
60/240 = ¼
Multiply the numerator and denominator with 25 to make the denominator equal to 100
(1×25)/(4×25) = 25/100
25 percent or 25 per 100 is called as 25%

Sample Question for the Basics of Percentage:
Example:A number exceeds 20% of itself by 40. The number is:
(a) 50
(b) 60
(c) 80
(d) 48

Solution: Let the number be p.
20% of itself means => p x (20/100)
Now, according to the question,
p – 20% of p = 40
=> {p – (20 x p)/100} = 40
=> {p-(p/5)} = 40
⇒ 5p – p = 200
∴ p = 50

Alternate Method:
Obviously, it is clear that difference is 80% i.e. 4/5 of number which is equal to 40
4/5p = 40
p = 40 x 5/4= 50.

Tips & Tricks for Percentages:
Basic Tip-1:  If the new value of something is n times the previous given value, then the percentage increase is (n-1)  100%.

Derivation:
Let us consider two values p and q.
Let q be and original value and p be the new value.
According to conditions p= nq
We need to calculate the percentage increase.
You can either use direct formula= {(new value – old value)/old value} x 10
This value becomes= {(p – q)/q} x 100
{(nq – q)/q} x 100
=> (n-1) x 100%

Example: If X= 5.35 Y, then find the percentage increase when the value of something is from Y to X.
Solution:
Use the formula: (n-1)100%
Percentage increase from
Y to X = (5.35 -1)  100= 435%

Basic Tip-2:
When a quantity N is increased by K %, then the:
New quantity = N (1+ K/100 )
Examples:
Increase 150 by 20%= 150 {1+(20/100)} = 150 1.2= 180
Increase 300 by 30%= 300 {1+(30/100)}=  300 1.3= 390
Increase 250 by 27% = 250 {1+(27/100)} = 250 1.27 =317.5

Example: What is the new value when 265 is increased by 15%?
Solution: New quantity = N (1+ K/100)
= 265{1+(15/100)}
New quantity = 1.15 265= 304.75

Basic Tip 3:
When a quantity N is decreased by K %, then the:
New quantity =N (1 – K/100)

Examples:
Decrease 120 by 20%= 120 {1-(20/100)} = 120  0.8= 96
Decrease 150 by 40%=150 {1-(40/100)} = 150  0.6= 90
Decrease 340 by 27%= 340 {1-(27/100)}= 340  0.73= 248.2

Example: If the production in 2015 is 400 units and the decrease from 2014 to 2015 is 13%, find the production in 2014?
Solution:
Remember the formula:
New quantity =N (1 – K/100)
Let the production in 2014 be x.
It has been decreased by 13% , which then becomes 400 in 2015
[X{1-(13/100)}]= 400
Production in 2014= 400 / 0.87= 459.77 units