__Objectives:__

At the end of this lesson, the students should be able to:

- Know the meaning of percentage and percentage composition.
- Express a quantity as a percentage of another.
- Compare data with different sample sizes.

A percentage is shown using the symbol **% **and it **means a fraction of 100**. Percent is derived from **per** cent.., the abbreviation in English of **per centum**, a phrase borrowed directly from Latin. Percentages are used universally to contrast and compare quantities and they are used in the businesses world and all works of life. As percentages play a very important role in our society, it is important

therefore to understand the concept.

**15%** means ** 15 parts out of 100** and is the same as the fraction 15/100.

**87%** means ** 87 parts out of 100** and is the same as the fraction 87/100.

A further way of expressing parts out of 100 is the decimal form and so percentages can also be expressed as decimals:

**15%** is the same as **0.15**.

**87%** is the same as **0.87**.

__Percentage Composition__

Often you will need to figure out what percentage of one quantity is another. For instance, if a class has 26 students and 14 are female, what percentage of the students are females? It is 14 out of 26, or ^{14}/_{26} = 0.538461538462…, or about 54%.

Have you ever wanted to know what your mark of 36 out of 40 in a math test is as a percentage? *You express one quantity as a percentage of another by first setting up the first quantity as a fraction of the second, making sure that the units of both quantities are the same. You can convert the fraction into a percentage by multiplying it by 100%.*

__Example 1__

You scored 36 out of 40 marks in your mathematics exam. Express your mark as a percentage.

__Solution __

__Example 2__

Express $6 as a percentage of $40.

__Solution __

Set up the fraction and multiply it by 100% since both amounts are in the same unit ($).

= **15%**

This means that $6 is 15% of $40.

__Example 3__

Express 75cm as a percentage of 2.5m.

__Solution__

First change 2.5m to 250cm to get a common unit.

^{75}/_{250} × 100%

75cm is 30% of 2.5m.

__Using Percentages to Compare Information__

Whilst researching for an essay or dissertation you may come across many sources of data in tables, graphs or reports which you would like to incorporate into your work. However, this can be difficult if they do not share a common base line. Percentages are useful for comparing information where the sample sizes or totals are different. By converting different data to percentages you can readily compare them.

__Example 1__

Consider the marks obtained by two students. Florence scored 700 marks out of a maximum of 800 marks and Jane scored 720 marks out of 900 marks. Who scored higher?

__Solution__

Looking at the raw scores it appears that Jane scored more marks than Florence. But we cannot compare the marks of these students as the maximum marks are not the same in the both cases. We can compare only if the maximum marks are equal. Percentage comes of help here. By converting these marks into percentages, we are actually making the maximum marks equal, to 100.

**Marks scored by Florence = 700 out of 800
= 700/800
= (700/800) x 100%
= 87.5%**

**Marks scored by Jane = 720 out of 900**

** = 720/900**

** = ( 720/900 ) x 100%**

** = 80%**

**Comparing the percentages above, we can conclude that, Florence performed better than Jane.**

__Example 2__

Which is greater, 15% of 380 or 25% of 210?

__Solution__

Hence 15% of 380 is greater than 25% of 210.

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