I need a formula to back calculate a starting number when I have an ending number after fee & taxes

Sorry if this will be too easy a question for most of you but my algebra classes were thirty-some years ago. I recently bought a couple things at an auction. I didn't think to factor in auction fee and taxes to my "max bid" so I ended up paying a good sum more than I planned. They add a 16% auction fee to your winning bid, then the tax (local and state together) is 9.25% of your bid + fee. Your total is the bid + fee + tax. I put together an Excel spreadsheet in which I can enter up to ten different items and it will calculate the fee, tax and total cost for each item as well as for all items together. That was easy.
Total cost = bid + fee + tax
Total cost = bid + (16% of bid) + (9.25% of (bid + fee))
Total cost = bid + (16% of bid) + (9.25% of (bid + (16% of bid)))
Sorry if I got carried away with parenthesis.

Then I thought, what about making a spreadsheet calculator that would let me put in the maximum cost I would be willing to pay for an item so it would deduct the taxes and fee to give me the maximum bid amount I could enter to wind up at the maximum total cost. I've been puzzling over how to flip that equation around to solve for the bid amount for a couple hours now and concluded that I can't remember enough of my old algebra classes to make it happen.


If someone here could help me out with that I'd sure appreciate it. Once I know what to do mathematically I can sort out how to put the math functions into the spreadsheet.

I tried to upload the spreadsheet file in case that would be useful but the upload function is not set up for receiving spreadsheets so I've uploaded a couple pictures of the spreadsheet, one normal view and one with the cell formulas showing.

Thanks in advance.

 

so combine the percentages into a single percent.
bid+16/100 *bid +9.25/100 *(bid +16/100 *bid)
100/100 *bid +16/100*bid +9.25/100*bid +148 /10000 *bid
125.25/100 *bid +1.48/100 *bid
126.73/100.
so the total cost is bid *126.73%
so if you want the highest bid given a total cost, divide it by 126.73/100
so for example if the total cost is say 25 dollar, the highest bid is 19.72

 

Many thanks! Clearly I was overthinking it. Your explanation helped me realize the way that I could have figured it out at the beginning. I had a $25 bid that resulted in a total cost of $31.68 after fee and taxes. All I needed to do was divide the total cost by my bid amount to arrive at the ratio of the two. (31.68/25=1.2673) (I'm probably misusing the term "ratio".) Now I can divide my budget for future purchases by that ratio to arrive at the max bid.
Thanks again!