Definition

A percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, "%", or the abbreviation "pct". Since a percent is an amount per 100, percents can be represented as fractions with a denominator of 100. For example, 25% means 25 per 100, 25/100 and 350% means 350 per 100, 350/100.

• A percent can be represented as a decimal. The following relationship characterizes how percents and decimals interact. Percent Form / 100 = Decimal Form

For example: What is 2% represented as a decimal?
Percent Form / 100 = Decimal Form: 2%/100=0.02

Percent change

General formula for percent increase or decrease, (percent change):


\(Percent=\frac{Change}{Original}*100\)

Example: A company received $2 million in royalties on the first $10 million in sales and then $8 million in royalties on the next $100 million in sales. By what percent did the ratio of royalties to sales decrease from the first $10 million in sales to the next $100 million in sales?

Solution: Percent decrease can be calculated by the formula above:
\(Percent=\frac{Change}{Original}*100=\)
\(=\frac{\frac{2}{10}-\frac{8}{100}}{\frac{2}{10}}*100=60%\), so the royalties decreased by 60%.


Simple Interest

Simple interest = principal * interest rate * time, where "principal" is the starting amount and "rate" is the interest rate at which the money grows per a given period of time (note: express the rate as a decimal in the formula). Time must be expressed in the same units used for time in the Rate.

Example: If $15,000 is invested at 10% simple annual interest, how much interest is earned after 9 months?
Solution: $15,000*0.1*9/12 = $1125


Compound Interest

\(Balance(final)=\)
\(=principal*(1+\frac{interest}{C})^{time*C}\), where C = the number of times compounded annually.

If C=1, meaning that interest is compounded once a year, then the formula will be: \(Balance(final)=\)
\(principal*(1+interest)^{time}\), where time is number of years.

Example: If $20,000 is invested at 12% annual interest, compounded quarterly, what is the balance after 2 year?
Solution: \(Balance=20,000*(1+\frac{0.12}{4})^{2*4}=\)
\(=20,000*(1.03)^8=25,335.4\)


Percentile

If someone's grade is in \(x_{th}\) percentile of the \(n\) grades, this means that \(x%\) of people out of \(n\) has the grades less than this person.

Example: Lena’s grade was in the 80th percentile out of 120 grades in her class. In another class of 200 students there were 24 grades higher than Lena’s. If nobody had Lena’s grade, then Lena was what percentile of the two classes combined?

Solution:
Being in 80th percentile out of 120 grades means Lena outscored \(120*0.8=96\) classmates.

In another class she would outscored \(200-24=176\) students.

So, in combined classes she outscored \(96+176=272\). As there are total of \(120+200=320\) students, so Lena is in \(\frac{272}{320}=0.85=85%\), or in 85th percentile.

Percentage

The term percentage means per hundred. Percentages are fractions with a denominator as 100. Percentages are always written with symbol %. Hence, 7% means \(\frac{7}{100}\).

Percentages are ratios, that represent the part of a whole. Meaning, 12% is 12 parts of 100.

Basic Rules of Percents

Compute percentage, given part and the whole

This is done by dividing the part over whole and multiply by 100.

% = \(\frac{part}{whole}\)100

For example,

To find what percent of 120 is 24, we just divide 24 over 120 and multiply the value by 100. Hence, it is 20 %.

In the same way, to find 13 is 24% of what number, we divide 13 over 24%. Hence, the value is, 54.16.

Percentage change

When the value of a product changes from an old to the new value, its percentage change is calculated by, finding the difference between old and new value and dividing the value by old value and multiplying by 100.

percentage change = \(\frac{{old - new }}{old}\) 100

Keep in mind,

If the value of percentage change is negative, it means the value of the product has reduced.

Consecutive Percentage Changes

Till now we have seen how to calculate the percentage, now we will look into how to calculate the final value with two consecutive percentage changes.

In this case, we have a simple formula to calculate the net percentage change. If the successive percentage changes are x% and y% then, the net percentage change is \((x + y + \frac{xy}{100})\).

Keep in mind,

If at all there is a decrease in change in value, the formula is changed accordingly. For example, the value of a product is increased by a % and then decreased by b%, then the updated formula is, \((a - b - \frac{ab}{100})\).

Example, if the value of product is increased by 12% and further increased by 3%, then the net percentage change is, \((12 + 3 + \frac{36}{100})\) that is 15.36%.

In the same way, if the value of a product is increased by 14% and then given a discount of 10%, then the net percentage change is \((14 - 10 - \frac{140}{100})\) that is 2.6%.

Undoing Percentage Changes

The next step in percentage is, how to find the original value. Meaning, if a product is discounted and we know the final value and the percentage of discount, how to find its initial value.

If a is the percentage by which the value is decreased or increased and y is the final amount, then the initial value x is calculated using the following formula,

\(x = y (\frac{100}{{100 + a}})\)

Keep in mind,

If the value is discounted, then change the sign in the formula to negative.

For example,
If a product is being sold for 270 after giving a discount of 15%, then what is the original price of the product?

In order to answer this question, we use the above formula with negative sign as the value is being discounted. Hence, on applying the above formula we get,

\(x = 270 (\frac{100}{{100 - 15}})\)
On solving the above equation we obtain 317.64 as the value.

Population Formula

Example:
The population of a city was 2 years ago 24000, population decreases every year at the rate of 5 %. Find the present population of the city.
Solution :

the population decreased at a particular rate so

Present population of that city is
\(=24000 * ( 1 - 5 %)^2\)
\(=24000 * ( 1 - 5 / 100)^2\)
\(=24000 * ( 19 / 20) * (19 / 20 )\)
\(=21660.\)