In this section, we will discuss more about Percentage 

We get to see very few problems which are on percentage only but the concepts and formulae of Percentage are used in different topics. This makes understanding the concepts very important.

Percent means ‘for every hundred’. Suppose we have a statement, B is 10% of A, this means that for every 100^th part of A, we have 10 part of B or if value of B is 10 then, value of A will be 100.

Let’s understand it with basic formula and some examples on it.

If we have to find y% of x, then

y% of x = (y/100) × x,

y% = y/100

E.g.1: Find: 10% of 500 = ?

By formula,

10% of 500 = (10/100) × 500 = 50 (Ans).

E.g.2: If 40% of m = 110, then find the value of m.

We know that, y% = y/100

40% of m = 110

⇒ 40/100 of m = 110

⇒ m = 110 × 100/40

⇒ m = 110 × 5/2

⇒ m = 55 × 5

⇒ m = 275 (Ans).

Before we begin with specific types let’s learn all the basic concepts with examples.

Conversion:

We can convert percent into fraction and vice versa. We can also convert percent into decimal

• If we have x% and we want to convert it into fraction and decimal, then:

Required fraction = x/100

Required decimal = 0.0x (Move two decimal places from right)

E.g. 3: Convert 25% to fraction.

Required fraction = 25/100 = ¼ (Ans).

E.g. 4: Convert 25% to decimal.

Move two decimal places from right,

Required decimal = 0.25 (Ans). 

If we have a fraction x/y, then:

Required percentage = (x/y) × 100

E.g. 5: Convert ¼ to fraction.

Required percentage = ¼ × 100 = 25% (Ans).

To express one quantity as a percent with respect to other quantity, following formula is used:

Percentage = (Quantity to be expressed in percentage/ Quantity with respect to which percentage is to be calculated) × 100

Note: It is important to convert all the quantities in same unit.

Let’s take examples and try to understand the application of this formula.

E.g. 6: What percent of 1hour is 1 minute and 12 seconds?

Quantity to be expressed in percentage = 1 min 12 seconds

= 1min + 12/60 minutes

= (1 + 1/5) minutes

  = 6/5 minutes

Quantity with respect to which percentage is to be calculated = 1 hour = 60 minutes

∴ Required percentage = (6/5/ 60) × 100

  = 6 × 100/5 × 60 = 20/10

  = 2/1 or 2 (Ans).

E.g. 7: The below table gives rice production in each year. Read and answer the following questions.

Year	Production in tons
2005	325
2006	438
2007	429
2008	258
2009	495

1. What percent is the production in 2008 with respect to production in 2009?

Required percentage = Production in 2008/ Production in 2009

 = 258/495

= 86/165 (Ans).

Percentage of change in quantity

If there is change in a quantity then to calculate percentage change, following formula is used:

Percent of change in quantity = {(Final quantity – Initial quantity)/Initial quantity} ×100

Let’s understand this with previous example from RRB NTPC Quant Questions.

E.g. 7: The below table gives rice production in each year. Read and answer the following questions.

Year	Production in tons
2005	325
2006	438
2007	429
2008	258
2009	495

• What is the increased percentage of production in 2009 compared to year in 2005?

Percent Change = {(Final quantity – Initial quantity)/ Initial quantity} × 100

 = {(495 – 325)/325} × 100

  = (170/325) × 100

  = 0.523 × 100

= 52.3% (Ans).

• What was the decline in production in year 2008 compared to 2007?

Percent Change = {(429 – 258)/429}× 100

  = (171/429) × 100 = 0.398 × 100 = 39.8% (Ans).

From now on we will see how the concepts of percentage are directly applicable in different problems of different topics.

Percentage change:

If there is a percentage change in a quantity, then:

Final quantity = Initial quantity × (100 ± Percentage increase/decrease)/100

Note: Percent change is always calculated in respect of an amount. It is important to pay more attention to the amount in respect of which percent change is asked.

Let’s see some examples Questions on percentage change and try to solve them.

E.g. 9: The monthly income of a person is Rs. 6000. If his income is increased by 40% then what is his monthly income now?

Here, Initial quantity = 6000

Percentage increase = 40%

∴ Final quantity = Initial quantity × (100 ± Percentage increase/decrease)/100

= 6000 × (100 + 40)/100

= 6000 × (140)/100

= 60 × 140

= 8400

Hence, his monthly income now = Rs. 8400 (Ans).

E.g. 10: Mr. Akshar sold a bus for Rs. 20,400 with a loss of 15%. At what price should the bus be sold to get a profit of 15%?

Here, although this question is Profit and loss question, we need this formula of percentage to solve this question.

We know that,

CP = SP + Loss

⇒ CP = 20,400 + 15% of CP

⇒ CP = 20,400 + 15/100 of CP

⇒ CP = 20,400 + 3/20 of CP

⇒ CP – 3/20 CP = 20,400

⇒ 17/20 × CP = 20,400

⇒ CP = 20,400 × 20/17 = 1200 × 20 = Rs. 24000

At 15% profit,

SP = CP + Profit

    = 24,000 + 15% of 24,000

    = 24,000 +15/100 of 2400

    = 24,000 + 3600 = Rs. 27,600 (Ans).

This solution is too long. Lets’ solve it by formula.

Final quantity = Initial quantity × (100 ± Percentage increase/decrease)/100

SP = CP × (100 – loss%)/100                      [Since loss is percentage decrease]

⇒ 20,400 = CP × (100 – 15)/100

⇒ 20,400 = CP × 85/100

⇒ CP = 20,400 × 100/ 85

          = 20,400 × 20/17

          = 24,000

At 15% profit,

SP = CP × (100 + profit %)/100                         [Since loss is percentage decrease]

     = CP × (100 + 15)/100

     = CP × 115/100

     = 24,000 × 23/20

     = 12,000 × 23

     = Rs. 27,600 (Ans).

See, the relevance of percentage in other problems. Almost every calculation is based on our percentage concept. We can minimize this time wasted in calculation by multiplication factor (MF).