Percentages are numbers that we frequently come across in everyday life. Percentages are used to give a common standard. The use of percentages is very common in many aspects of commercial life as well as in engineering. Interest rates, sale reductions, pay rises, exams and VAT are all examples of situation in which percentages are used. The word percent is derived from the Latin word 'Per centum' which means "out of one hundred". Symbolically percent is indicated as %.

If a student scores 75 marks out of a maximum of 100 in an examination, we say that he/she has scored 75% marks.

Percentage can be expressed as a fraction and a decimal also. From the above example we know that 75% is equivalent to $\frac{75}{100}$ .

Hence, 75% = $\frac{75}{100}$
= 0.75
In order to show some applications of percentage, various daily-life problems, particularly problems based on profit and loss and simple interest will be solved.

For example,
1. 70 percent means 70 out of hundred or $\frac{70}{100}$
2. 50% means 50 out of hundred means $\frac{50}{100}$

## What is Percentage?

A percent can be defined as a ratio whose second term is 100.

“Percent is a fraction whose denominator is 100 and numerator indicates the required percent”

Therefore a Percent means parts of 100. The symbol used to denote percent is "%"

### Example on Percentage

Let us see what fraction of the grid is shaded.

Each grid is divided into 100 boxes. For each grid, the ratio of the number shaded boxes to the total number of boxes can be represented as fractions.

The fractions can represented as percents by multiplying each fraction with 100.

$\frac{(87)}{(100)}$ × 100 = 87%.

$\frac{(18)}{(100)}$ × 100 = 18%

$\frac{(40)}{(100)}$ × 100 = 40%

Adding percentage is just like the addition of normal numbers. There are no special rules for adding percentages. For example, ask a child to add up 11% and 22% the answer is a simple 33%. But this sum shows that in adding percentages we don't have to know what the percentage looks like in order to do the sum.

Step 1: Write percentage into fraction form.
Step 2: Add the numerators and put the answer over common denominator.
Step 3: Simplify the fraction.

1. An increase in 200% quality means that the final amount is 400% of the initial amount. (200% of initial + 200% of the increase = 400% of the initial) in other words the quantity is increased four times.
2. An increase in 600% means the final amount is 7 times the original. (600% + 100% = 700% = 7 times larger).

Addition of percentage requires that you just add them up like normal numbers. Add the top numbers when the bottom numbers are the same, and then use the simplest form.

## Subtracting Percentages

Since all the percentages are fractions with a denominator of 100, they can be subtracted without any other manipulations.
Subtraction of a percentage requires that you just subtract them like normal numbers. Subtract the top numbers when the bottom numbers are the same, and then use the simplest form.

Steps to Subtract Percentages
Step 1: Write percentage into fraction form.
Step 2: Subtract the numerators and put the answer over common denominator.
Step 3: Simplify the fraction.

### Examples on Subtracting Percentages

Subtract the given percentage

30% - 15%

Solution:Given 30% - 15%
30% - 15% = ($\frac{30}{100}$) - ($\frac{15}{100}$)

= $\frac{15}{100}$
Try it yourself
1. 45% - 22%
2. 67.8% - 22.7%
3. 50% - 0.75%

## Multiplying Percentages

Steps to Multiply Percentages
Step 1: Write percentage into fraction form.
Step 2: Multiply numerators and denominators.
Step 3: Simplify the fraction.

Take any number and multiply it by a percent, or take a percent and multiply it by another percent.
For example, What is 3 times 25%?
you would simplify that is multiply 25 by 3 and reply 75%.
Suppose if the question is what is 30% of 25%?

Solution
This question has little more work, first we need to change one of the percentage to decimal or a common fraction and then carry out the calculation.

30% = 0.3, 0.3 x 25% = 7.5%
30% = $\frac{30}{100}$, ($\frac{30}{100}$) x 25%
= ($\frac{750}{100}$)% = 7.5%

Try it yourself
Multiply the following
1 20% x 50%
2 100% x 20%
3 0.5% x 25%

## Dividing Percentages

The same technique is used in the multiplication of percentage is use for dividing the percentage also. If the question involves more than one percentage, simply convert one percentage into a decimal or fraction then carry out the operation.

Step 1: Turn the number into a fraction
Step 2: Reduce the first fraction by dividing both parts by a common factor.

For example, consider the following division

$\frac{25%}{5}$ = 5%
$\frac{20%}{10%}$ = $\frac{20%}{0.1}$ = 200%
[or $\frac{20%}{(\frac{1}{10})}$ = 20% x ($\frac{10}{1}$) = 200%]

Making simplifier

If dividing one percent by another you can put them in the form of a fraction and cancel the percentages.

$\frac{5%}{10%}$ = $\frac{5}{10}$

= $\frac{1}{2}$
= 50%

Try it yourself

1. $\frac{25%}{5%}$
2. $\frac{10%}{0.5%}$
3. $\frac{48%}{2%}$
4. $\frac{78%}{39%}$
5. $\frac{36%}{18%}$

## Converting Percentages

There are two conversions possible in conversions of percentage.

1. Conversion of percentage to decimal
2. Conversion of percentage to fraction

### Converting Percentage to Decimal

To convert a percentage into a decimal simply reverse the process and move the decimal point two places to the left. In some cases it may be necessary to add a zero.

Rule:
• 1) The rule for going from percentage to decimals is simple.
• 2) Divide the percentage by 100. The result is a decimal number because the denominator in a percent is always 100. So, for example, 67% becomes $\frac{67}{100}$ or 0.67.

### Example on Converting Percentages

1) Convert 98%, 5%, 32.2% to decimal

Solution:

1) 98% = $\frac{98}{100}$ = 0.98
2) 5% = $\frac{5}{100}$ = 0.5
3) 32.2% = $\frac{32.2}{100}$ = 0.322

2) Write 45% as a decimal number.

Solution:

To write 45% as a decimal number, we have to remove the percentage symbol and place a decimal point such that there will be two digits to the right of the decimal point.

Hence, 45% = 0.45

3) Represent 123% as a decimal number.

Solution:
Following the method of writing a percent as a decimal number, we get,

123% = 1.23

We can see from the above examples that, if the given percent is less than 100, the whole number part of the decimal number will be 0 and if the percent is more than 100, the whole number part will be a nonzero number.

4) Write 2.3% as a decimal number.

2.3% = 0.023, by shifting the decimal point two places towards left.

### Converting Percentage to Fraction

Converting percentages into fraction is so fast. The rule is to drop the percent sign and put the number over 100 in a fraction. The result is a fraction.

Rule:
• 1) Drop the percent sign and use the numerical portion of the term
2) Set the percentage number as a numerator over the denominator 100.
3) Simplify the fraction.
Example

Convert 26% to a fraction

Solution:

Given 26%
Step-1: 26% just becomes 26
Step-2: Since the denominator is 100 for a percentage it becomes $\frac{26}{100}$
Step-3: On simplification $\frac{26}{100}$ = $\frac{13}{50}$.

Fraction Decimal: Fraction decimal or decimal fraction is a proper fraction with a denominator (bottom number) is a power of 10 (that is 10, 100, 1000, 10000.....etc).

Example: $\frac{3}{10}$, $\frac{12}{100}$, $\frac{64}{1000}$

Fraction Decimal can also be write as decimal point that is with out a denominator to do the operations like addition, subtraction, multiplication and division.
Example: $\frac{51}{100}$ is a fraction decimal, which can be written as 0.51.

### Converting Decimal Fraction to Percent

To convert decimal fraction to percent, first convert decimal fraction to decimal and then multiply it by 100 and add a percent symbol “%” at the end.

Example 1: Convert $\frac{4}{10}$ to a percent.
Solution:

• Step1: Convert $\frac{4}{10}$ to decimal $\Rightarrow$ 0.4
• Step 2: Multiply 0.4 by 100 $\Rightarrow$ 0.4 x 100 = 40
• Step3: Add percentage sign (%) at the end $\Rightarrow$ 40%

Example 2: Convert $\frac{55}{100}$ to a percent.
Solution:
• Step1: Convert $\frac{55}{100}$ to decimal $\Rightarrow$ 0.55
• Step 2: Multiply 0.55 by 100 $\Rightarrow$ 0.55 x 100 = 55
• Step3: Add percentage sign (%) at the end $\Rightarrow$ 55%

Example 3: Convert $\frac{31}{1000}$ to a percent.
Solution:

• Step1: Convert $\frac{31}{1000}$ to decimal $\Rightarrow$ 0.031
• Step 2: Multiply 0.031 by 100 $\Rightarrow$ 0.031 x 100 = 3.1
• Step3: Add percentage sign (%) at the end $\Rightarrow$ 3.1%

## Percentage Table

Here is a table of commonly occuring values coming in Percent, Decimal and Fraction forms:

## Percentage Difference

Percentage difference is the difference between any two quantities over the average of those two quantities which is expressed in percentage. It is used when both the quantities mean same kind of thing.

Formula: $\frac{First\ value - Second\ value}{(\frac{First\ value + Second\ value}{2})}$ $\times$ 100%

Here, “|” symbol indicates absolute value such that negative values also become a positive value.

### Steps to Calculate the Percentage Difference

Step 1: Find the Difference, that is subtract 1 quantity from other quantity and ignore the minus or negative (-) sign.
Step 2: Find the average of those two quantities, that is add the quantities and then divide it by 2.
Step 3: Now, divide the difference of the quantities by it's average and then convert that to a percentage.

### Examples on Percentage Difference

Example 1: If milk costs 5dollars in one shop and $7 in another shop, then calculate the percentage difference? Solution:Step 1: The difference = 5 - 7 = - 2, here we have to ignore the negative sign. So difference = 2 Step 2: The average =$\frac{(5 + 7)}{2}$=$\frac{12}{2}$=$6
Step 3: $\frac{2}{6}$ x 100% = 33.33%

Example 2: Calculate the percentage difference between 28 and 16?

Solution:
Step 1: The difference = 28 - 16 = 12
Step 2: The average = $\frac{(28 + 16)}{2}$ = $\frac{44}{2}$ = 22
Step 3: $\frac{12}{44}$ x 100% = 27.27%

Example 3: If black jeans pant costs 35.60 dollars and blue jeans pant costs 38.40 dollars, then calculate the percentage difference?

Solution:
Step 1: The difference = 35.60 - 38.40 = - 2.8, here we have to ignore the negative sign. So difference = 2.8
Step 2: The average = $\frac{(35.60 + 38.40)}{2}$ = $\frac{74}{2}$ = $37 Step 3:$\frac{(2.8)}{37}$x 100% = 7.568% ## Percentage Change Percentage change is the ratio of subtracting the old value from the new value to the old value which is expressed in percentage. It is used when comparing an old value with a new value. Formula:$\frac{New\ value – Old\ value}{|Old\ value|}\times$100 % ### Steps to Calculate the Percentage Change Step 1: Find the change, that is subtract the old quantity from the new quantity. Step 2: Divide the step 1 by the old value and then convert that to percentage. ### Examples on Percentage Change Example 1: If a pair of pen costs went from 7dollars to 8dollars, then calculate the percentage change? Solution: Step 1: Change = 8 - 7 = 1 Step 2: Divide by old value and multiply with 100 =$\frac{1}{7}$x 100% = 14.29% Example 2: If the cost of a pant increased from 30dollars to 35dollars, then calculate the percentage change? Solution: Step 1: Change = 35 - 30 = 5 Step 2: Divide by old value and multiply with 100 =$\frac{5}{30}$x 100% = 16.67% ## Percentage Error It is the ratio of the difference between the approximate and the exact quantities to a exact quantity which is expressed in percentage. It is used to compare approximate quantity with an exact quantity. Formula:$\frac{|Approximate\ Value – Exact\ Value|}{|Exact\ Value|}\times$100% ### Steps to Calculate the Percentage Error Step 1: Find the error, that is subtract 1 quantity from the other quantity and ignore the minus or negative (-) sign. Step 2: Now, divide the error of the quantities by the exact value and then convert that to percentage. ### Examples on Percentage Error Example 1: If you estimate there are 45 chocolates in the box when there are 73 actually, then calculate the percentage error . Solution: Given Estimate value or Approximate value = 45 Actual Value or Exact value = 73. Step 1: Error = |Approximate Value – Exact Value| = |45 – 73| = 28 Step 2:$\frac{28}{73}\times$100% = 38.36% Example 2: If John expected to get amount 40 dollars for his birthday, but he got only 35 dollars, then calculate the percentage error? Solution: Given Estimate amount or Approximate amount =$40

Actual amount or Exact amount = 35 dollars.

Ratio = 1: 1