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Steve

How to Calculate Reverse Percentages

A reversed percentage sign on a red balloon being held by a woman's hand

What exactly is a reverse percentage?


Reverse percentages allow you to answer the question of "what was the original amount BEFORE a percentage amount was added/subtracted". They are very useful because percentages by themselves are dependent on where you start.


Here is an example of a scenario where you'd need a reverse percentage:


Emma is getting paid £60,000 after a 10% raise. What was she getting paid before?

Many people are tempted to work out what 10% of £60,000 is and subtract that to work out what she was getting paid before. So, 10% of £60,000 is £6,000, which means that she was getting paid £54,000. The problem is that this is wrong.


Percentages are dependent on where you're starting from. In this case, Emma was not starting on £60,000 when she got the 10% raise...that's where she ended up after the increase.


Let me demonstrate why this answer is wrong:

If Emma was indeed on £54,000, then after a 10% raise she would be earning an extra £5,400, which means she'd be earning £59,400. This is NOT £60,000 and so £54,000 must be the incorrect answer.


 

So what is the correct way to calculate reverse percentages?


starting point = ending point / (1 + percentage change)


So in this case:

  • ending point = 60,000

  • percentage change = 10%


Therefore, starting point = 60000/(1 + 10%) = 60000/(1 + 0.1) = 60000/1.1 = 54545.45


Emma was previously getting paid £54,545.45 before her 10% salary increase that put her on £60,000.



Another Example


Perhaps you need another example to make the idea stick.


A shiny new smartphone is discounted 20% and now costs £800. What did it cost before this fabulous discount?

Let's apply the formula:

  • ending point? In this case it will be 800 because that's what the phone costs now

  • percentage change? It will be -20% because there has been a reduction in the price of the item. It costs 20% less, and so the percentage change is -20%


Therefore starting point = 800/(1 + (-20%)) = 800/(1 - 20%) = 800/(1 - 0.2) = 800/0.8 = 1000


The smartphone was originally £1000 before the 20% discount that reduced its cost to £800.


 

Tips to remember


  • Percentages depend on where you start

  • So, normal percentages help you work out where you end up, given where you start

  • Reverse percentages help you work out where you started, given where you ended up

  • They "reverse" the effect of percentage changes



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