Per cent means ‘for every hundred’. To make comparisons easier, we use the concept of percentage.6% (read as six per cent, which means, 6 out of 100 or 6 for every 100) is same as $\frac{6}{100}$ or 0.006 or 6:100. Any percentage value can also be expressed as
(1) an ordinary fraction
(2) a decimal fraction or
(3) a ratio

Value of Percentage: Value of percentage always depends on the quantity to which it refers.

Consider the Statement: “65% of the students in this class are boys”. From the context, it is understand that boys from 65% of the total number of students in the class. To know the value of 65%, the value of the total number of students should be known.

If the total number of students in 200, then the number of boys = $\frac{200\times 65}{100}$ = 130.
Note that the expressions 6%, 63%, 72%, 155%, etc do not have any value intrinsic to themselves. Their values depend on the quantities to which they refer.

Comparing Percentages: Which of three 25%, 5% and 125% is the larger?

It should be remembered that no comparison can be made about the above percentages, because they do not have intrinsic values. If 25% of 10,000, then value is $0.25\times \, 10,000\, =\, 2,500$ and if 75% refers to 75% of 100, its value is $0.75\times\, 100\, =\, 75$. We can conclude that 25% of 10,000 > 75% of 100.

Ordinary Fractions and Decimal Fractions: It is already noted that any given percentage can be written either in the form of a decimal fraction or an ordinary fraction. If can remember the conversion of a fraction to a percentage and vice-versa his/her speed of calculations can be improved.

Percentage of Increase/Decrease: If the production of rice in 2003 is 800 million tonnes and in 2004 it is 1000 million tonnes, then the increase in the production of rice from 20003 to 2004 is 200 million tonnes, i.e., an increase of 200 million tonnes over the production of rice in.
2003, i.e., $\frac{200}{800}\, =\,\frac{1^{th}}{4}$ increase = (25%).
Percentage increase of decrease of a quantity is the ratio of the amount increase or decrease in a quantity to the original amount of the quantity, expressed to the base of (or with respect to) 100.

Percentage increase = $\frac{Increase\, in \, quantity}{original \, quantity}\times 100$

Percentage decrease = $\frac{Decrease\, in \, quantity}{original \, quantity}\times 100$

Change in quantity = $\frac{Percentage \, change}{100}\times original \, quantity$

Increase and Decreased Values: If Rathod’s income from Rs,2,000 by 20% then the increase in income
= $2000\times 0.2$ = Rs 400

The increase income (i.e., income after increase)
= $(initial\, value)\times (1+ \%\, Increase)$
= 2000(1+0.2) = Rs 2,400

Successive Percentages: Let the population of a town at the beginning of 2004 be 50,000 and during the year it increased by 20% i.e.., at the beginning of 1992, the population was $5000\times (1+0.2)$ = 60,000. Suppose, during 1992 it again increased by 25%. This means that at the beginning of 1993, the population was $60,000\times (1+0.25)$ = 75,000.

To represent situations of the above type, the phrase successive change $(\frac{increase}{decrease})etc.$ is used.

In the case of the above example, we say, “The population of a town which was 50,000 at the beginning of 2004 successively increased by 20% p.a. and 25% p.a. during the next two years”.
In such cases, the final value(1+0.2)(1+0.25)
= $50,000\times 1.2\times 1.25$ = 75,000

In general way, if $p_{1},p_{2},p_{3},$ ……………. are successive percentage of change, then,
Final value = $(1\pm p_{1}\%)(1\pm p_{2}\%)(1\pm p_{3}\%)$ ………..$\times (Initial \,value)$

Where ’+’ is used for increases and ‘-‘ is used for decreases)
Note : Even if the changes are mixed group of some increases and some decreases, the above procedure can be applied.

Percentage Points: Let us consider the agricultural production of a state. Suppose, in the year 2003, quantity of wheat produced was 40% of the total food grains produced. And in the year 2004, wheat produced was 50% of the total food grains produced. Suppose we want to measure the percentage change in the production of wheat, calculating percentage increase from 40 to 50 as $\frac{50-40}{40}\times 100$ and saying it is 25% is not correct.

40%, in the year 2003 refers to 40% of total agricultural production of that year.
Similarly, 50% in the year 2004, refers to 50% the total agricultural production of 2004.

Neither the values of agricultural production of 2003 and 2004 are known, nor is any relation between the two quantities given.

Hence, 40% of the year 2003 and 50% of the 2004 cannot be compared, until the total agricultural production ‘relations is known’.
With the available data we cannot find the percentage $\frac{increase}{decrease}$ in the production of wheat from 2003 to 2004. In such situations, the phrase, ‘percentage points’ is used. “Wheat production (as a percentage of the total agricultural produce) in the year 2004 is 10 percentage points more than that wheat production (as percentage of the total agricultural produce) of the year 2003.

Important Notes on Percentage

• Percentage increase = $\frac{Actual\, increase}{original \,quantity} \times 10$
• Percentage decrease = $\frac{Actual\, decrease}{original\, quantity} \times 100$
• Change in value = $\left ( \frac{Percentage \,change}{100} \right )\times Initial\, value$
• If percentage increase is p percentage then new value = $\left ( 1+\frac{p}{100} \right )\times initial\, value$
• If the value of an item goes up/down by x%, the percentage reduction/increment to be now made to bring it back to the original level is $\frac{100x}{(100\pm x)}$ %
• If A is x % $\frac{more}{less}$ than B, Then B is $\frac{100x}{(100\pm x)}\%\frac{less}{more}$ than A.
• If the price of an item goes up/down by x % then the quantity consumed should be $\frac{reduced}{increased}\, by\, \frac{100}{(100\pm x)}$ % so that the total expenditure remains the same.
• If $p_{1},p_{2},p_{3}$ …………. are successive percentages of change then. Final Value = $((1\pm p_{1}\%)(1\pm p_{2}\%)(1\pm p_{3}\%).............)\times \,initial\, value$
• If a certain is first increased by x% and then increased by another y%, then the effective percentage increase in the quantity will be $\left ( x+y-\frac{xy}{100} \right )$ %.

Solved Examples on Percentage

Example: Find $26\frac{1}{2}$ % of 2000.

Solution. of 2,000 = $\frac{53}{2}\times \frac{1}{100}\times 2000$ = 530

Example: Two numbers are 25% and 40% more than a third number. What is the first number as a percentage of the second number ?

Solution. If the third number is 100 then the first two numbers are 125 and 140.
$\therefore$ The first number as a percentage of the second $\Rightarrow \frac{125}{140}\times 100\,=\,\frac{625}{7}\,=\,89\frac{2}{7}$ %

Example: Two number are 40% and 60% more than a third number. By what percentage is the first number less than the second number?

Solution. If the third number is 100 then the numbers are 140 and 160.
$\therefore$ first number is 20 less then the second number.
$\therefore$ Required percentage = $\frac{20}{160}\times 100$ = 12.5%

Example: 50% of 20% of a number is what percent of 50% of 5% of the same number ?

Solution: $\frac{50}{100}\times \frac{20}{100}\,=\,\frac{x}{100}\times \frac{50}{100}\times \frac{5}{100}$
$\therefore x\Rightarrow 400$

Example: Manoj gets 40% more than Suresh. How much percentage does Suresh get less than Manoj ?

Solution. Let Suresh’s salary be Rs 100.
$\therefore$ Manoj’s will be =100+40% of 100= Rs 140.
$\therefore$ Suresh gets Rs 40 less than Manoj’s salary.
$\therefore$ Required percentage = $\frac{40}{140}\times 100\,=\,28\frac{4}{7}$ %

Alternate method : The problem can be done by using the formula $\frac{100x}{100+x}\%\,=\,\frac{100\times 46}{140}\,=\,28\frac{4}{7} \%$

Percentage Questions from Previous Year Exam