# Percentage Confusion

#### dani0157

Hi,

Today at work I was trying to understand whether a given amount (revenue) is with or without VAT. (vat is 21%).

Here is the amount with VAT (1980.50)
Here is the amount without VAT (1636.4)

The way I've taught myself to add or subtract a percentage from an amount is to either multiply by 1.21 to add 21% or to multiply by 0.79 to subtract 21%.

Here is what confused me today.
If i do this calculation to add 21% to 1636.4 (1636.4 * 1.21), i get 1980.044 (this is correct)
If i do this calculation to subtract 21% from 1980.5 (1980.5 * 0.79) I get 1564.6 (expected to 1636.4)

Why am I not getting 1636.4 when i subtract 21% from 1980.5?

Also. i was told by my colleague to subtract 21% from 1980.5 I am supposed to divide by 1.21. This way the
calculation makes sense (1980.5 / 1.21 = 1636.77).

Why my colleague's way works and why mine does not?

#### Cervesa

Here is the amount with VAT (1980.50)
Here is the amount without VAT (1636.4)
If $A$ is the amount w/o VAT, then $A(1.21) = \text{ the amount with VAT}$

since $A(1.21) = \text{ amount with VAT}$, then $A = \dfrac{\text{amount with VAT}}{1.21}$

#### dani0157

If $A$ is the amount w/o VAT, then $A(1.21) = \text{ the amount with VAT}$

since $A(1.21) = \text{ amount with VAT}$, then $A = \dfrac{\text{amount with VAT}}{1.21}$
Thank you for trying to help me with my poor math skills.

What you said is the correct way to calculate an amount with/without VAT. What i want to know is how and why the formula works (specifically for calculating an amount without VAT)? If VAT is 21%, then why not just the amount with VAT * 0.79 to subtract 21%? I dont want to just remember a formula without understanding how it works.

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#### Cervesa

It's a common misconception that a fixed percentage increase followed by the same percentage decrease gets one back to the original value.

To make things simple, consider the original amount to be 100. A 21% increase of the value 100 would yield the value 121 … ok with that?

Following that increase with a 21% decrease on the new value of 121 will not yield the original value of 100.

The reasoning is that the follow on decrease of 21% is based on the already increased value of 121, not 100. 21% of 121 is 25.41

(.79)(121) = 95.59 = 121 - 25.41

In plain language, 21% of 121 is greater than 21% of 100, hence the decrease is greater than the initial increase.

Hope this clears it up for you.