When a number is multiplied by 1, it remains unchanged. Correspondingly, when expressed as a percentage, the price of a product remains the same if [[$ 100 \: \% $]]​ of the price has to be paid.

In order to calculate increases and reductions that are expressed as percentages, you must first examine how the total value changes when expressed as a percentage. When the resulting percentage is converted to a percentage factor and multiplied by the original numerical value, the percentage change is applied. If the percentage factor is greater than one, the numerical value will increase, whereas if the percentage factor is less than one, the numerical value will be reduced. 

Calculating percentage increase and decrease

When the base value a increases by p %, the result is...



 When the base value a decreases by p %, the result is...

NB! [[$ \left ( \displaystyle\frac {100 + p} {100} \right ) \cdot a $]]​ may be presented in a simplified form[[$ \left ( 1 + \displaystyle\frac {p} {100} \right ) \cdot a $]]​.

Example 1

At a Finnish ice cream bar, the tax-free price of a portion of ice cream is [[$ 4\: \text {€} $]]​. Value-added tax (VAT) must be added to the price of the ice cream portion. How much is the taxed sales price of an ice cream serving?

Solution:

Method I

The taxable price of an ice cream serving is [[$ 100 \: \% + 22 \: \% = 122 \: \% $]]​ compared to the tax-free price. Thus, the price will increase [[$ 1,22 $]]​ times. The taxable price of a portion of ice cream is obtained by multiplying the percentage factor by the original price:

Method II

The VAT rate is [[$ 22 \: \% $]]​.

First, calculate how many euros the amount of VAT to be added is from the tax-free price:

[[$ \displaystyle\frac {22} {100} \cdot 4 \: \text {€} = 0,22 \cdot 4 \: \text {€} = 0,88 \: \text {€} $]]​

Then, add this to the tax-free price:

[[$ 4 \: \text {€} + 0,88 \: \text {€} = 4,88 \: \text {€} $]]​

Answer: The taxable price for a portion of ice cream is [[$ 4,90\: \text {€} $]]​.

Example 2

You are buying a snowboard while on a holiday in Lapland. The original price of a snowboard is [[$ 400\: \text {€} $]]​. There is a discount sale in the sports equipment store where the buyer can get a [[$ 20 \: \% $]]​ discount on all products. How much is the reduced price of a snowboard?

Solution:

Method I

[[$ 100 \:\% \: – 20 \:\% = 80 \:\% $]]​ of the original price remains to be paid. This means that the reduced price of the board is [[$ 0,8 $]]​ times its original price. The reduced price is obtained by multiplying the percentage by the original price:

Method II

First, calculate the amount of the discount in euros:
[[$ \displaystyle\frac {20} {100} \cdot 400 \: \text {€} = 0,22 \cdot 400 \: \text {€} = 80 \: \text {€} $]]​​

Then, subtract the discount from the original price:

[[$ 400 \: \text {€} \: - 80 \: \text {€} = 320 \: \text {€} $]]​.

Answer: The new price of a snowboard is [[$ 320 \: \text {€} $]]​.