# Percentage Change, Percentage Increase, Percentage Decrease

In this section we learn about percentage changes.
Percentage changes can be either percentage increases or percentage decreases.
When a quantity varies, we'll often be inerested in calculating its variation (another word for change) as a percentage of its initial value. A more "mathematical way" of saying this is that we're interested in finding its "relative variation" (relative to its initial value).
This percentage is known as a percentage change.

Percentage changes are important as they allow us to evaluate how much things have varied without paying attention to the actual amount by which they have varied. Consequently, percentage changes are often used to measure performance and compare two or more variations, without giving any "unfair advantage".

For instance, imagine two people decide to go on a diet, Sarah and John. When they start, Sarah weighs $$62$$kg and John weighs $$84$$kg.
At the end of their diet: Sarah weighs $$55$$kg and John weighs $$76$$kg.
It's clear that John lost more kilos but relative to their initial weight Sarah lost more (a little more than $$10\%$$) so one could argue that Sarah's diet was more successful than John's.

## Percentage Change Formula

Given a quantity $$A$$ whose value has changed (like the price of a product that has increased, or decreased), we'll often be required to calculate its percentage change.
Calling the initial amount of the quantity $$A_{\text{initial}}$$ and the final amount $$A_{\text{final}}$$, to find the percentage by which the quantity has varied (the percentage change) we use the formula: $\text{Percentage Change} = \frac{A_{\text{final}} - A_{\text{initial}}}{A_{\text{initial}}}$ Where:

• if the result is positive, it is a percentage increase

• if the result is negative, it is a percentage decrease.

### Example

A brand of phones saw its main product price change from $$\ 800$$ to $$\900$$.
What was the percentage change? Was it a percenatage increase or decrease?

### Solution

We call the initial price of the phone $$A_{\text{initial}}=800$$ and the new price $$A_{\text{final}}=900$$.
We now use the percentage change formula: \begin{aligned} \text{Percentage Change} &= \frac{A_{\text{final}} - A_{\text{initial}}}{A_{\text{initial}}} \\ & = \frac{900 - 800}{800}\\ &= \frac{100}{800} \\ \text{Percentage Change} &= 0.125 \end{aligned} The percentage change was $$12.5\%$$.
This was a $$12.5\%$$ increase since the result is positive. We're dealing with a percentage increase.

## Example

Following a strict diet, James' weight went from $$105$$kg to $$77kg$$.
Rounding your answer to 3 significant figures, find the pecentage change of James' weight.

### Solution

Letting James' weight before the diet be $$A_{\text{initial}}=105$$ and weight once his diet was over $$A_{\text{final}}=77$$.
We use the percentage change formula: \begin{aligned} \text{Percentage Change} &= \frac{A_{\text{final}} - A_{\text{initial}}}{A_{\text{initial}}} \\ &=\frac{77-105}{105} \\ &= \frac{-28}{105} \\ \text{Percentage Change} &= -0.26667 \end{aligned} Rounding our result to three significant figures, we can state that James lost $$26.7\%$$ of his initial weight.
This percentage changeis a percentage decrease/em>, we can tell from the fact that the result is negative.