Percentage Change

When we want to describe how much a value has increased or decreased, we use percentage change.

Percentage change shows the change relative to the original value, that is, how large the change is in relative terms.

 

Formula for percentage change

The general formula is:

 

$$ \large \text{Percentage Change} = \frac{\text{new value} - \text{old value}}{\text{old value}} \cdot 100\% $$

 

Absolute vs. relative change

If a price increases from 20 to 25 euros, the absolute change is 5 euros. But the relative change is:

 

$$ \large \frac{5}{20} \cdot 100\% = 25\% $$

 

So it is 25% more expensive compared to the original price.

 

Example of an increase

The price of a book rises from 20 euros to 25 euros:

 

$$ \large \frac{25 - 20}{20} \cdot 100\% = 25\% $$

 

Example of a decrease

A product falls in price from 30 euros to 24 euros:

 

$$ \large \frac{24 - 30}{30} \cdot 100\% = -20\% $$

 

The negative sign shows that this is a decrease of 20%.

 

Change in several steps

If a product first increases by 10% and then decreases by 10%, you do not end up at the original price.

 

Example: Starting price 10 euros.

 

After increase: \(10 \cdot 1.10 = 11\) euros

After decrease: \(11 \cdot 0.90 = 9.9\) euros

 

The price ends up 1% lower because the decrease is calculated from the new value.

 

Doubling and halving

  • An increase of 100% means a doubling.
  • A decrease of 50% means a halving.

 

VAT

VAT is an example of percentage change. In the United Kingdom, the standard VAT rate is 20%.

 

  • To add VAT: multiply by \(1.20\).
  • To remove VAT: divide by \(1.20\).

 

Example:

A product costs 800 euros without VAT. With VAT the price becomes:

 

$$ \large 800 \cdot 1.20 = 960\ euros $$

 

If the price with VAT is 960 euros, the price without VAT is found by dividing:

$$ \large \frac{960}{1.20} = 800\ euros $$

 

Summary

  • Percentage change measures change relative to the original value.
  • A positive result means an increase; a negative result means a decrease.
  • Multiple changes in a row must be calculated step by step.
  • VAT example: multiply by \(1.20\) to add VAT and divide by \(1.20\) to remove it.