Introduction to Percentages

What is a Percent?

Percent implies per 100. It is a number or a ratio expressed as a fraction of 100. It is represented by the symbol %.

Example : 20% implies $\frac{20}{100}$.

What does the statement "20% of the students cleared the exam" mean?

=> It means out of every 100 students 20 students have cleared the exam.

To convert x% into fraction we divide x by 100.

To convert a fraction $\frac xy$ into percentage we multiply it with 100.

How to find Percentages?

When solving percentages, identifying the base y in $\frac xy is very important.

For example,

• What percent of B is A $\to \frac AB$ x 100

In this case, we want to express A as a percentage of B. Hence base is B.

• A is what percent more than B $\to \frac {A-B}{B}$ x 100

Here,The difference in A and B gives the value by which A is greater than B.The reference point is B. Hence B is the base. And to find the percentage by which A is greater than B, divide the difference by the base.

• Percentage by which A is less than B  is given by $\frac{B-A}{B}$ x 100
• If A is $x$% more than B, then B is $\frac{x}{100+x}$ % less than A.
• If A is $x$% less than B, then B is $\frac{x}{100-x}$ % more than A.

Percentage Increase and Decrease

• If price increases by $x$% and expenditure remains the same, then the consumption decreases by $\frac{x}{100+x}$ x 100
• If price decreases by $x$% and expenditure remains the same, then the consumption increases by $\frac{x}{100-x}$ x 100

Scaling Factor.

• If A is increased by $x$%, then the resulting value is obtained by multiplying A by (1+$\frac{x}{100}$). (1+$\frac{x}{100}$) is called the scaling factor.

Example:

If A is increased by 20%. Then its value after increase is A(1+$\frac{20}{100}$) => A x 1.2

• If A is decreased by x%, then the resulting value is obtained by multiplying A by (1-$\frac{x}{100}$).

Example:

If A is decreased by 10%. Then its value after increase is A(1-$\frac{10}{100}$) => A x .9