How do you reverse a percentage change? Is there any simple way to know how to reverse a percentage?
For example if I have 100 and it goes down by 10% I end up with 90. If I then add to it by 10% I end up with 99, not the 100 that you would think of. Is there a simple trick to quickly work out the reverse of a percentage change (even if it only works for some subset of all).
 A: Yes, add $90$ divided by $90$ and multiply the result by $10$.
As another example: Let us say you subtract $5\%$ from $120$. That is, $\frac{120}{100} \cdot 5 = 6$. This gives $120 - 6 = 114$. Then to get back to $120$ you add $\frac{114}{95} \cdot 5 = 6$.
The way to think about this is the following. What you are doing is you are dividing the number into 100th pieces of equal length. Then, you are subtracting let us say $5\%$ or in other words $5$ pieces of equal length. Now, you are left with only $95$ pieces $(95\%)$ of equal length. To get back you divide the pieces by $95$ since you want them to be of equal length. And then multiply again by how many pieces you want to get back, in this case $5$.
Working with percentages is just working with ratios.
A: You may too consider all the percentage operations as multiplications by a factor :


*

*taking $10$% of $x\;$ is computing $\,\displaystyle \frac {10}{100}\cdot x=0.10\;x\quad$ ( replace '%' with $\displaystyle \frac 1{100}\;$)

*subtracting $10$% to $x\;$ is computing $\;\displaystyle x-\frac {10}{100}x=\left(1-\frac {10}{100}\right)x=(1-0.10)\;x=0.90\;x$

*of course adding $12$% to $x\;$ is simply computing $\;(1+0.12)\;x=1.12\;x$.


(note that subtracting  $10$% and adding $12$% becomes multiplying by $\;0.90\cdot 1.12=1.008\;$ and allows us to observe that the order of the % operations doesn't matter!)
Reverting these operations will be done by dividing by the multiplicative factor :


*

*divide $0.10\;x\,$ by $\,0.10\,$ to get $x$ back

*to subtract $10$% to $\,x\;$ you computed $\;0.90\;x\;$ so divide this by $\,0.90\,$ to get the initial $\,x$.

