Percentage Change

A percentage is a way to express a change in variable. It represents the relative change between the old value and the new one. A percentage change indicates how much a quantity increases or decreases with respect to the original amount.

    Perecent of Change = $\frac{\text{Amount of Increase or Decrease}}{\text{Original Amount}}$

    $\%$ Change = $\frac{{\text{ New value }} - {\text{ Original value }}}{\text{ Original value }} \times 100$

Say whether the followings are percent of increase or percent of decrease.
🦄 $56$ to $29$      🦌 $23$ to $45$


🦄 $56$ to $29$
$\begin{aligned} \text{ Percent of change }&=\frac{ 29-56 }{ 56 }\times 100\\ &=\frac{-27}{56}\times 100\\ &=-0.482\times 100\\ &= 48.2 \%   \text{ (decrease in percent) }\\ \end{aligned}$

🦌 $23$ to $45$
$\begin{aligned} \text{ Percent of change }&=\frac{ 45-23 }{ 23 }\times 100\\ &=\frac{22}{23}\times 100\\ &=0.957\times 100\\ &= 95.7 \%   \text{ (increase in percent) }\\ \end{aligned}$

Note : original value > new value ⇒ decrease in percent
       original value < new value ⇒ increase in percent


Finding the Original Value after a Percentage Change

🐶 The price of a house has increased by $5\%$ and now costs £ $157,500$. Find out what the original price must have been.


$\begin{aligned} \text{ New price }   £ 157,500 \quad \quad &=\quad 105 \%\\ ↓ ÷ 105 \quad &=\quad\quad ↓ ÷ 105 \\ £ 1,500 \quad\quad &=\quad 1\%\\ \quad ↓ × 100\quad &= \quad\quad ↓ × 100\\ £ 150,000 \quad\quad&= \quad 100\%\\ \end{aligned}$
The original price must have been £ $150,000$.