Aren’t we already in a recession?

You provided a response as to how to "back out" sales tax. You used an example of a $20,000 car purchase and backing out 7 percent. Thank you for that, it was a great help. But if that tax was comprised of 4 percent state tax and 3 percent county tax, and I need to back out each to send separate checks, if I use your example, why doesn't it equate to the same as the 7 percent?

I've tried this multiple ways and tried drawing examples using cutting slices from a pie or removing floors from a building. I can never get a combined total of two percentages to equal a single one. I realize that you're backing out a percentage from 100 percent of something and that the second item, while a percentage of the original 100 must be removed from the remainder. It still doesn't calculate. If I "back out" 4 percent from an apple pie, remove that slice, back out another 3 percent from the other side of the pie, those two slices don't quite match my backing out one 7 percent slice?!?

Any help?
— Martin B., Address withheld

Oh. Sorry. We were daydreaming about eating apple pie and watching a building collapse. (Maybe that’s what got you off track.)

Rather than trying to look up a formula on some Web site — with no way of knowing if it’s accurate — let’s just run through the problem and come up with our own.

We’re going to start by assuming that the state and county are taxing the same original (pre-tax) purchase price — and that you’re not given credit by one tax man for taxes paid to another. (In some cases, you can get credit from one jurisdiction for taxes paid to another one. But we’ll leave the question of preparing your tax return for another day.)

Setting that issue aside, the solution goes something like this.

The total amount of money you shelled out included three separate expenses: 1) the pre-tax cost of the item, 2) the tax paid to the state and 3) the tax paid to the county. At this point, we have to switch over to math.

Total Amount Paid = 1 X Pre-tax Purchase Price + .04 X Pre-tax Purchase Price (state tax) + .03 X Pre-tax Purchase Price (county tax)

Then solve for PPP (or find a math teacher and have them do it). You’ll get:

TAP = (1 x PPP) + (.04 x PPP) + (.03 x PPP)
TAP = (1 + .04 + .03) X  PPP
TAP = 1.07 X PPP
PPP = TAP / 1.07

To find out the pre-tax purchase price of the car, divide the total amount paid ($20,000) by 1.07 and you get $18,691.59. That leaves $1,308.41 in taxes.

Since the total tax is 7 percent of $18,691.59, the county tax is 3 percent of $18,691.59 (or $560.75) and the state tax is 4 percent of $18,691.59 (or $747.66.)

To check your work (just like your math teacher told you to), you add back up the pre-tax purchase price ($18,691.59) + the county tax ($560.75) + the state tax ($747.66) and come up with = the total you shelled out ($20,000.)

OK: That's enough math. (Hey, you in the back — you can wake up now.)

Readers with similar accounting puzzles should give them a shot before forwarding along for solutions. We’re really not set up to prepare tax returns at the Answer Desk.

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