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Professor's Disease

Recently, a new store owner asked for help in reverse-calculating a price so that the final amount charged was a "nice" number.

I wrote: "Suppose X is the cost of the item, Y is the 'revised' amount you will sell the item for, and the tax rate is R. Then, Y = X/(1+R)...rounded down to nearest cent.

For example, if the tax rate is 0.085 and the cost of the item is $100, then y = 100/(1+.085) = 92.16589..., which rounded down is $92.16.

She wrote back: "I'll try this at work. I don't have a calculator here...... Thank you...... So, you take 1 + .085?"

I responded: "If .085 is your current tax rate... and then divide the item's price by that sum (i.e. 1+.085). To check it out, try your hand at these amounts...If you price an item at $5 or $25, the resultant prices charged are $4.60 or $23.04 respectively."

Two months later she responded: "I need help!!! My friend here says you have the "professor's disease" because you don't explain how you got the answers (4.60 and 23.04)....I'm still trying to figure this out....."

So, I responded: "Not sure how I could be much clearer, as the explanation was there if it was read (an assumption on my part).... Thus, have your friend re-read the first line and follow instructions....which tells you to 'divide the item's price by that sum (i.e. 1+0.085)'.... So, for the given the amounts of $5 and $25, 5/1.085 = 4.6083 and 25/1.085 = 23.0415 which rounded down to two-decimal places becomes $4.60 and $23.04."

So, while I go ask my family doctor to run tests on my "professor's disease," perhaps I understand better why items end up being priced at 0.99.