What is a percent?

The word ‘percentage’ is derived from the Latin phrase ‘per centum’, which means ‘by the hundred’.  Percentages are fractions with a denominator of $$100$$. Putting in another way, it is the relationship between the portion and the whole in which the whole is always valued at $$100$$. A percentage is a fraction of a ratio in which the full value is always $$100%$$.  For example, if Sam received a $$60%$$ on his math test, it implies that he received a $$30$$ mark out of a possible $$100$$. In fraction form, it is $$\frac{60}{100}$$ and in ratio terms, is $$60:100$$. 

Percentage definition

A part or an amount in every hundred is defined as a percentage. It is a fraction having a denominator $$100$$ and is denoted by the sign $$\%$$.

E1.12: Calculate the given percentage of a quantity         

There are three possible unknowns or variables in any percentage problem:

• Percentage

• Part

• Base

You must be able to recognise these factors to solve any percentage problem. Take a look at the examples below.  Each of the three variables is known:

For example, if $$20%$$ of $$200$$ is $$40$$, $$20$$ is the percentage, $$200$$ is the base and $$40$$ is the part.              

The term ‘percentage’ is another term for ‘hundredths’.  As a result, $$1%$$ equals on hundredth, i.e., $$1\%=\frac{1}{100}=0.01$$.

Percentage formula

To find the percentage of a whole in terms of $$100$$, use the percentage formula.  You can represent a number as a fraction of $$100$$ using this formula. 

$$\text{Percentage}=\frac{Value}{Total value}\times 100$$

For example, out of $$40$$ pens, the number of red cells is $$8$$. Hence, out of $$100$$, the number of red cells will be $$\frac{8}{40}\times 100=20%$$.

Worked examples

Example 1: Find the percentage of the following:

1. What is the percentage of $$5$$ out of $$50$$?

2. What is the percentage of $$20$$ out of $$200$$?

3. What is the percentage of $$40$$ out of $$50$$?

Question 1

Step 1: Write the formula.

$$\text{Percentage}=\frac{Value}{Total value}\times 100$$ 

Step 2: Write the given information.

$$\text{Value}=5$$ and $$\text {Total value}=50$$.

Step 3: Substitute the given value in the formula.

$$\frac{5}{50}\times 100=10\%$$

So, the percentage is $$10\%$$.

Question 2

Step 1: Write the formula.

$$\text{Percentage}=\frac{Value}{Total value}\times 100$$ 

Step 2: Write the given information.

$$\text{Value}=20$$ and $$\text{Total value}=200$$

Step 3: Substitute the given value in the formula.

$$\frac{20}{200}\times 100=10\%$$

So, the percentage is $$10\%$$.

Question 3

Step 1: Write the formula.

$$\text{Percentage}=\frac{Value}{Total value}\times 100$$ 

Step 2: Write the given information.

$$\text {value}=40$$ and $$\text {total value}=50$$.

Step 3: Substitute the given value in the formula.

$$\frac{40}{50}\times 100=80\%$$.

So, the percentage is $$80\%$$.

E1.12A: Calculate the given percentage of a quantity

To calculate the percentage, divide the value by the total value and multiply the result by $$100$$. It is represented as:

$$\text{Percentage} =\frac{Value}{Total value} \times 100$$

For example: $$\frac{2}{5}\times 100=0.4\times 100 = 40\%$$.

How to calculate the percentage of a quantity

We must use a separate formula to determine the percentage of a number, that is:

$$P\%$$ is equal to $$X$$, where $$X$$ is the required percentage.

If we want to remove the $$\%$$ sign, we need to express the above formula as:

$$\frac {P}{100}\times \text{Number} = X$$

For example: Calculate $$10\%$$ of $$80$$.

Let $$10\%$$ of $$80$$ is equal to $$X$$

$$\frac {10}{100}\times 80 =X$$

$$X=8$$

Worked examples

Example 1: Find the percentage of the following. 

1. If $$10%$$ of $$40\%$$ of a number is $$10$$, what is the number?

2. If $$40%$$ of $$60\%$$ of a number is $$20$$, what is the number?

Question 1

Step 1: Write the formula.

$$P\%$$ of a number is equal to $$X$$.

Step 2: Write the given information.

Here, $$P=\frac{10}{100}\times \frac{40}{100}$$

$$X=10$$

Step 3: Substitute the given value in the formula.

$$\frac{10}{100}\times \frac{40}{100}\times\text{Number}=10$$

$$\text{Number}=\frac{10\times 100\times 100}{10\times 40}$$

$$\text{Number}=250$$ 

So, the number is $$250$$.

Question 2

Step 1: Write the formula.

$$P\%$$ of a number is equal to $$X$$.

Step 2: Write the given information.

Here, $$P=\frac{40}{100} \times \frac{60}{100}$$

$$X=20$$

Step 3: Substitute the given value in the formula.

$$\frac{40}{100}\times \frac{60}{100}\times \text{Number}=20$$

$$\text{Number}=\frac{20\times 100\times 100}{40\times 60}$$

$$\text{Number}=8.33$$ 

So, the number is $$8.33$$.

So, the number is $$8.33$$.

E1.12B: Express one quantity as the percentage of another and how to calculate the percentage of a quantity

$$P\%$$ of a number corresponds to $$X$$.

Here, $$X$$ is the required percentage.

If you remove the $$%$$ sign, you need to write the above formula as:

$$\frac{P}{100} \times \text{Number}=X$$

For example: Calculate $$20%$$ 0f $$80$$.

Let $$20%$$ of $$80$$ be $$X$$.

$$\frac {20}{100}\times 80$$ is $$X$$.

So, $$X=16$$.

Percentage chart

The percentage chart for fractions converted into percentages is given here:

Worked examples

Example 1: Find the percentage of the following quantities.

1. Find $$30%$$ of $$150$$.

2. Find $$20%$$ of $$2\text{hours}$$.

3. Find $$15%$$ of $$150$$.

Question 1

Step 1: Write the formula.

$$P\%$$ of a number is equal to $$X$$.

Step 2: Write the given information.

We need to find $$30\%$$ of $$150$$.

$$P=30$$

$$\text{Number}=150$$

Step 3: Substitute the given value in the formula.

$$\frac{30}{100}\times 150=X$$

$$X=3\times 15$$

$$X=45$$ 

Therefore, $$X=45$$.

Question 2

Step 1: Write the formula.

$$P\%$$ of a number is equal to $$X$$.

Step 2: Write the given information.

We need to find $$20\%$$ of $$2\text{hours}$$.

$$P=30$$

Since, $$2\text{hours}=120\text{minutes}$$.

$$\text{Number}=120$$

Step 3: Substitute the given value in the formula.

$$\frac{20}{100}\times 120=X$$

$$X=2\times 12$$

$$X=24$$

Therefore, $$X=24\text {minutes}$$.

Question 3

Step 1: Write the formula.

$$P\%$$ of a number is equal to $$X$$.

Step 2: Write the given information.

$$P=15$$

$$\text{Number}=150$$

Step 3: Substitute the given value in the formula.

$$\frac{15}{100}\times 150=X$$

$$X=1.5\times 15$$

$$X=22.5$$

Therefore, $$X=22.5$$.