maths > commercial-arithmetics

Introduction to Percentages

what you'll learn...

overview

A ratio is represented with two numbers, eg: 3:4$3 : 4$. Comparing two different ratios involve some arithmetics. eg: 4:5$4 : 5$ and 3:4$3 : 4$, which one is larger in ratio?

To simplify such comparison, the second term is standardized to a value, say 100$100$, then the given ratios are 80:100$80 : 100$ and 75:100$75 : 100$. It is easier to compare these ratios.

Such standard representation is simplified by dropping the known number $100$. and the simplified form is "percentage". eg: and .

problem with ratios

Consider a fruit basket with apples, oranges, and bananas

•  The ratio of apples to bananas is $1 : 4$

•  The ratio of oranges to bananas is $2 : 9$

It is not evident to figure out if apples or oranges are more in the basket from the given two ratios.

The ratios are to be converted to similar ratios.

Consider a fruit basket with apples, oranges, and bananas

•  The ratio of apples to bananas is $1 : 4$. This means that, there is $1$ apple for every $4$ bananas. Or $2$ apples for every $8$ bananas.

•  The ratio of oranges to bananas is $2 : 9$. This means that, there are $2$ oranges for every $9$ bananas. Or $4$ oranges for every $18$ bananas.

The two ratios can be converted as
Ratio of apples to bananas is $9 : 36$ and ratio of oranges to bananas is $8 : 36$. In this, $9$ is greater than $8$, so number of apples is greater than number of oranges.

The comparison of $1 : 4$ and $2 : 9$ is harder as the second term is not identical. These are converted to $9 : 36$ and $8 : 36$.

To simplify such comparison, the second term can be standardized to one value like $100$ and all ratios are specified as $x : 100$.

standardize

To simplify understanding and using comparison of quantities, the second term is standardized to $100$ and the ratio is given as "percentage"

Ratio $1 : 4$ equals $25 : 100$ which is given as a percentage .

Ratio $2 : 9$ equals $22.22 : 100$, which is given as a percentage

The word "percent" means: for every hundred.

"per-cent" meaning "for every-hundred"

Number of apples is of number of oranges. The is pronounced as percent.

context

Consider $4$ apples and $16$ oranges.

•  Number of apples as percent of number of oranges is $\frac{4}{16} \times 100$ which is equivalently given as Number of apples is of number of oranges.

•  Number of apples as Percent of the number of fruits is $\frac{4}{4 + 16} \times 100$ which is equivalently given as number of apples is of the number of fruits.

Note that the percentage is given in two forms. First is the percentage of one quantity with reference to another quantity. Second is the percentage of one quantity with reference to the whole.

Students are reminded to note the context in which a percentage is defined.

summary

Percentage : Comparison of one quantity to another OR specification of one quantity in the whole given as a number for every hundred.

Outline