The French word ‘Cent’ or the Latin word ‘Centum’ means hundred.

Therefore the word percentage means per hundred or hundredths.

Any fraction with hundred in the denominator is called percentage.

Percent is indicated using the symbol: %

80% is ; 5% is ; 365% is

A percentage can be represented by a fraction or a decimal.

Percentage

Fraction

Decimal

75%

0.75

30%

0.3

65%

0.65

23%

0.23

66%

0.66

Important points to remember

1. Converting Fractions, Numbers and Decimals to Percentages

5 can also be written as 500%

1.25 can also be written as 125%

3.75 can also be written as 375%

0.575 can also be written as 57.5%

2. A fraction can be converted to percentage by multiplying by 100

´ 100 = 0.8 x100= 80%

´ 100 = 0.1666 x 100 = 16.66%

3. Finding a percent of a given number.

P% of A =

Example:

a) 10% of 50 ==5

b) 25% of 120==30

4. Percentage Comparison.

Given x and a, to find x is what% of a

Multiply the ratio by 100 and add % sign, i.e.,

Example:

a) 4 is what percent of 60? Þ´ 100 = 6.67%

b) 10 is what percent of 100? Þ´ 100 = 10%

c) 207 is what percent of 100? Þ´ 100 = 207%

5. Percent Increase and Decrease

In many cases we are interested in knowing by how much fraction of its origin value a quantity has changed. If the value is given in percentage then:

Percentage change =

6. Percentage Error

Error = Actual value – Value taken (AV>VT)

Error = Value taken – Actual value (VT>AV)

If the value of x is taken as y then error is y – x

, where (y – x) is error and x is the actual value

7. Relative Percentage

In relative percentage, there is percentage comparison between two or more quantities.

Example: If A’s income is 20% less than B’s, then by what percentage is B’s income more than A?

Solution:

Let B’s income be Rs.100

A’s income would be 20% less than B

Hence, A’s income = 80% of Rs.100 = Rs.80

The percentage by which B’s income would be more than A

= % = 25%

Worked Examples

1. Find 5 percent of 400.

Solution:

5% == 0.05

Hence, the required amount = 0.05 x 400 = 20

2. Find 2% of Rs.18.75 to the nearest paise.

Solution:

2% of Rs.18.75 Þ´ 18.75 = 0.02 x 18.75 = 0.375 = 38 paise.

3. In a school the strength of the students is 260 out of which 65% are boys. Find the number of boys and girls in the school.

Solution:

Total number of students = 260

Given that 65% of the students are boys. Hence, 35% of the students are girls.

No. of girls = ´ 260 = 91 and, No. of boys = 260 – 91 = 169

There are 169 boys and 91 girls in the school.

4. A piece of cloth 50 meters long shrinks by 0.6%. How much did it shrink?

Solution:

0.6% of 50 meter = 0.3 meters = 30 cm

5. In a spelling test of 80 words, Ravi spelt 80% of the words correctly. How many words did he spell right?

Solution:

The number of words in the spelling test = 80 words.

Ravi spells 80% words correctly.

No. of words spelt correctly by Ravi = ´ 80 = 64 words.

6. A man spends 25% of his money and then Rs.75 and then 5% of the remainder. If he had Rs.3291.75 left with him, how much money did he possess initially?

Solution:

Let the money before spending 5% be Rs.100.

So, amount after spending 5% is Rs.(100 – 5) = Rs.95

Actual amount of money after spending 5% = Rs.3291.75

So, actual amount of money before spending 5% = Rs.= Rs.3465

Therefore, amount of money left before spending Rs.75 = Rs.3465 + Rs.75 = Rs.3540

If the original money is Rs.100, then the amount of money left after spending 25% of Rs.75.

Money left after spending 10% Original money

Rs.75 Rs.100

Rs.3540 Rs.4720

Therefore his original sum of money is Rs.4720.

7. Find the percentage error in writing 2.54 as 2.5.

Solution:

Actual value=2.5

Value taken=2.54

Error value=2.54-2.5=0.04

8. The price of rice is raised by 30%, by what percent should a house wife reduce consumption so as to not to increase her total expenditure on rice?

Solution:

Let her initial consumption be 100 kg at a price of Rs.10 per kg.

Hence her total expenditure = 100 ´ 10 = Rs.1000

The new price after increase of 30% is 13 per kg

Let the consumption be X kgs

Then,

13X = 1000

Alternative Method

where X is the increase in percentage.

9. Satish’s income is 20% more than that of Sanjana’s. By what percentage is Sanjana income less than that of Satish?

Solution:

Method 1

Let Sanjana income be Rs.100

Satish’s income is

Sanjana income is less than Satish by

= %

Method 2

, where x is percentage difference

10. If the present population of a town is 20000, which is increasing at r rate of 10% every year what will be population of the town after 2 years?

Solution:

(Or Original price – 15% of Rs.40, 000) = 40,000 = Rs.34,000