I can’t remember how to do this but I think maybe I need to solve an integral. I know how to solve it by iterating until it converges within a given error but I am wondering if it can be solved exactly with an equation.

I am going to describe this as a hypothetical tax problem. I do not think there is any tax like in this universe but bear with me as this is an intellectual exercise.

I have a certain amount of income x. I must pay taxes on my after-tax income at a rate of r. That is, the rate is applied to my income after the tax is subtracted. But that means I need to know the amount of tax before I can calculate it, sort of a “this sentence is false” tax paradox. Is there an equation for that?

(This same kind of calculation may actually apply in the case where an employee receives a company benefit that is taxable, and the company does not want the employee to have to bear the tax burden so does a “gross up” of the benefit so that it will equal some number net taxes. When I started my job, my new employer paid me to cover the educational expenses that my prior employer required me to reimburse upon leaving. That was paid as a taxable bonus, so they increased it to cover the taxes.)

I’m guessing that the following is what you mean: You have received an amount of money X. What is the amount Y such that r% of Y, plus Y, equals X. The answer is that (1 + r%) times Y equals X. So say that you have received $300. You want to know what the amount X is such that 1 plus 50% equals $300. The answer is $200, since (1 + .50) times $200 equals $300. How do you than calculate that the answer is $200? You divide $300 by (1 + .50). $300 divided by 1.50 equals $200. Another example is that you have received $487. What is the number X such that X times (1 + 5%) equals $487? The answer to this is found by dividing $487 by 1.05, which equals approximately $463.81.