**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

*is 50/100 × 60 = 30*

**60**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**

### Percentages: The Complete Lesson

#### Table of Contents

**Concept Articles**

**Exercises**

**Tests**