I'm visiting an unfamiliar state. This state has a sales tax of 6.1% but I don't know this. All I know is the tax rate is of the form
a.b% where
a and
b are single digits.
I buy an item at a store and am able to deduce the tax rate from the item's price and the tax amount as listed on the receipt.
What is the minimum cost of the item that would allow me to make such a deduction?
Note: The tax amount listed on the receipt is rounded to the nearest cent.
I assume that taxes are rounded to the nearest penny.
Let p be the smallest qualifying purchase.
Then p*.060, p*.061 and p*.062 must round to different amounts.
This can only b the case (for the minimum p)
if p*.060 < (p.061 - .05)
and p*.062 >= (p.061 - .05)
and p*.061 is close to a whole penny before rounding
Solving p*.062 >= (p.061 - .05) gives p > $5.00
The tax on $5 is 30.5 cents, so we want to try taxes of $.31 and $.32 etc.
$.31 / .061 = $5.082, but neither $5.08 or $5.09 works
5.08 gives taxes of 31 cents at both the 6.1 and 6.2% rates.
5.09 gives taxes of 31 cents at both the 6.0 and 6.1% rates
$.32 / .061 = $5.245, but neither $5.24 or $5.25 works
5.24 gives taxes of 32 cents at both the 6.1 and 6.2% rates.
5.25 gives taxes of 32 cents at both the 6.0 and 6.1% rates
$.33 / .061 = $5.410, which is our answer because
the tax is 32 cents at 6.0%,
33 cents at 6.1%
and 34 cents at 6.2% $5.41
FINAL ANSWER: $5.41
Analytical Solution (spoiler)
