When a company reimburses an employee for out-of-pocket expenses, it’ll typically also add in a “gross-up” for income tax, since the reimbursement is reported as W2 income. Otherwise, the employee would have not been fully reimbursed–they would be out the taxes withheld. Since the gross-up amount itself is also subject to the same taxes, how do you arrive at what the correct gross-up is?

My brute-force approach had me doing some middle-school algebra:

\[u = (r + u) \times t\]

where \(u\) is the gross-up amount, \(r\) is the original reimbursement and \(t\) is the tax rate. Solve for \(u\):

\[\begin{aligned} u - ut &= rt \\ u(1 - t) &= rt \\ u &= \frac{rt}{1 - t} \end{aligned}\]

What’s interesting is the right side can be arrived at in procedural terms by looking at the problem in a different light: what amount would result in the taxes on the original reimbursement (\(rt\)), when you took out the taxes owed on it (\(u (1 - t)\))?

Or even more clearly if we think in terms of \(u + r\), i.e. the total amount paid by the company to cover the reimbursement. Starting from the result above, find \(u + r\):

\[\begin{aligned} u + r &= \frac{rt}{1 - t} + r \\ u + r &= \frac{rt + r(1 - t)}{1 - t} \\ u + r &= \frac{r}{1 - t} \end{aligned}\]

In other words, to figure \(r + u\), ask what amount would result in \(r\) after taxes:

\[(u + r)(1 - t) = r\]

The latter is a much easier way of thinking about the problem.