Calculate This!

We needed to tackle the math:

Chain Lakes Motel

How can I calculate the original amount of a sale transaction if all I have is the sales tax rate and the sale final total?

We have wrestled with this calculation for months. I could not find a college math book and we searched the web, and could not find even a hint. Finally, we bit the bullet and sat down for 3 1/2 hours to find the solution. This is what we found:

Let's assume we have a total owed price of 105.50 and a sales tax rate of 5.5%:

P - sale final total, $105.50
R - sales tax rate, 5.5% (.055)
X - orignal pre-tax sale amount

X = (P-(P * R)) + ((P-(P * R)) * R) - (((P-(P * R)) * R) * R)
X = (105.50-(105.50 * .055)) + ((105.50 * .055) * .055) - (((105.50 * .055) * .055) * .055)
X = (105.50-(5.8025)) + ((5.8025) * .055) - (((5.8025) * .055) * .055)
X = (99.6975) + (.3191375) - (.0175525)
X = 99.9999 (Rounding to the penny would be $100)

The calculation is accurate to the 10000th digit. If more accuracy is needed with larger numbers, simply add 1 more iteration: ((((P-(P * R)) * R) * R) * R)

For the tax amount, simply reverse the 2nd and 3rd iteration.