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5.3 | Percentage Increase and Decrease |

In these revision notes for Percentage Increase and Decrease, we cover the following key points:

- What is percentage change? How do you calculate it?
- How to express a percentage increase/decrease?
- What is the formula of percentage increase/decrease calculation?
- How do you find the original value after a percentage change?
- What is percentage distribution? Why do we use it?
- What is the rule governing the percentage distribution in a data set?

Knowing the percentage change of a value is very important because it allows us to estimate how a value changes. There are two types of percentage change: percentage increase and decrease. Percentage change is calculated through the formula:

% change = *change in value**/**original value* × 100%

=*∆x**/**x* × 100%

=

where x is the original value while Δx = x_{final} - x_{initial} (initial = original) is the difference between final and initial value of the quantity involved in the study.

The change Δx may be positive or negative. If it is positive, then the final value is greater than the initial one. In this case, we have a **percentage increase**; otherwise, if the final value is smaller than the initial (original) one, we have a **percentage decrease**.

The formula by which we calculate the percentage increase is

Percentage increase = *Actual value - Initial value**/**Initial value* × 100%

If the old and the new percentage (i.e. initial and actual) of an amount are known, the percentage increase is obtained through the formula

Percentage increase = New percentage - Old percentage

or

Percentage increase = Actual percentage - Initial percentage

We can also work out the percentage increase by taking the initial value as 100% and finding the actual value in terms of the initial value as percentage (100%). Then, the initial percentage is subtracted from the actual, in order to obtain the percentage increase.

Percentage decrease on the other hand is the inverse of the percentage increase of an amount. Thus, to obtain the value of a given percentage decrease, we must subtract the new percentage from the old one. Mathematically, we have

Percentage decrease = Old percentage - New percentage

On the other hand, when we know the new and the old values expressed in numbers and not in percentages, we can use the formula

Percentage decrease = *Initial value - Actual value**/**Initial value* × 100%

If we have the percentage change and the new amount given, we can find the original amount by rearranging the formula

Percentage change = *Actual value - Initial value**/**Initial value* × 100%

to obtain

Initial value = *Actual value × 100%**/**100% + Percentage change*

(If the percentage change is positive, we have a percentage increase; otherwise, we have a percentage decrease).

Percentage distribution tells you the number per hundred that is represented by each group in a larger whole. In other words, percentage distribution occurs when we have a data set defined as percentages where each percentage represents different groups of the whole (100%).

If we have the percentage of a number of quantities A, B, C and D involved in a data set, we can express the rule of percentage distribution mathematically as

%A + %B + %C + %D = 100%

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