Why ‘percentage gain required to recover from percentage fall calculator’ need?

Suppose a stock or fund falls by 50% then half of its value will remain. Now if this remaining value increases by 100% then the old value will be recovered. Thus, a calculator is needed to find out how much percentage of gain is required for any percentage of fall.

Example- Calculator’s value:

Calculation steps:

What is the ‘percentage gain required to recover from percentage fall calculator’ formula?

a= percentage fall in percentage
let percentage gain required to recover, b in percentage
Now we suppose that, any value is x and after fall is y
then, y= x-x*a*0.01 …..eq1
then, x= y+y*b*0.01 …..eq2
put the value of x in …..eq1
y= (y+y*b*0.01)*(1-a*0.01)
1= (1+b*0.01)*(1-a*0.01)
1= 1-a*0.01+b*0.01-a*b*0.01*0.01
1-1+a*0.01= b*0.01*(1-a*0.01)
b= a/(1-a*0.01)
b= 100*a/(100-a)%