## 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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