Given a % of change and the resulting value after the change, how do you calculate the original value?

Question

Given a % of change and the resulting value after the change, how do you calculate the original value?

For example, X increased by 50% = 150. In this case we can easily use guess and check to see that X = 100, but how do you calculate this given any percentage of change and any resulting value?

I though this formula should work, but it does not. As you can see my algebra is quite rusty, so maybe you can help.

start = x
percentage of change = p
resulting value = r

x * (p / 100) + x = r

Which can be simplified to

2(p / 100)x = r

which is equivalent to

x = r / 2(p / 100)

But when I check my work, this is wrong:

100 = 150 / 2(50 / 100)
100 = 150 / 2(.5)
100 = 150 / 1
100 = 150

Answer 1

Hint: If a number $x$ changes by $p$ percent, the resulting value is $\left(1+\frac{p}{100}\right)x$.

So the basic equation you have to solve is $\left(1+\frac{p}{100}\right)x =r$.

Your expression "2(p/100)x" is incorrect.