# Fractional cents in price that includes tax

As an example, if I say that a price is \$15 including 5% tax, does that make the price \$14.285714... + \$0.714285... tax or \$14.29 + \$0.71 tax? This can add up to whole dollars worth of tax difference if a lot of items are purchased at once.

I also notice an issue at certain values, where if I say that a price is \$14.80 including 5% tax, that makes the price \$14.095238... + \$0.704761904... tax, which rounds to \$14.10 + \$0.70. Except that \$14.10 * 0.05 = 0.705, which rounds to 0.71. Does this mean that it is not possible for a price to be \$14.80 including 5% tax? Either it is \$14.10 + \$0.71 tax = \$14.81, or it is \$14.09 + \$0.70 tax = \$14.79.

Perhaps it is not actually meaningful to say "including tax" in this way except descriptively, and one must choose a pre-tax price, that then has tax applied to it, not working backwards computationally from a price that already includes tax?

Maybe it is not even valid to say that a base price can be extended to 6 or more digits (fractional cents)?

• If a price contains tax it's 105 %. You should be able to calculate 100 %. Commented Sep 25, 2021 at 10:07
• @BernhardDöbler yes, but is it \$14.29 or \$14.285714... Commented Sep 25, 2021 at 19:09
• @BobBaerker the ellipses were important since these are repeating decimals Commented Sep 25, 2021 at 19:10
• 6 decimals is 1/10,000th of a cent. Carrying the number further might be important to NASA but in this question, I doubt it. Commented Sep 25, 2021 at 19:27
• @BobBaerker yeah but that's the value I was representing. you've changed the meaning of what I said, not just the formatting Commented Sep 25, 2021 at 20:13

Does this mean that it is not possible for a price to be \$14.80 including 5% tax?

It is possible - if the pre-tax price of the item is \$14.095 you'll end up with an after-tax price of \$14.80 (rounded of course). There's no law in general that says item prices must be in whole cents (virtually all gas stations add 9/10 of a cent in smaller print). Some retailers (notably concessions at sporting events) only advertise an "after tax" price to keep change-making much simpler.

But, note that sales tax is not computed per-item. If you buy multiple items, the total pre-tax amount (for taxable items, of course) is totalled and sales tax applied to that total. Plus, when a business pays sales tax, it reports it's total gross taxable sales and pays sales tax on that amount. In other words, it doesn't necessarily account for "sales tax" on each sale and just pay the total of all sales tax collected to the state/city/etc. So it's perfectly reasonable for a business to only charge "after-tax" prices, back out an equivalent pre-tax price (possibly not rounded) and pay the appropriate amount of sales tax.

• so is the base price of something that is "\$15 including 5% tax" then \$14.285714...? How should this be displayed on a receipt or UI? I think I will go with the convention of using an elipsis or tilde or something so it would show ~\$14.29 or \$14.29.. to indicate an approximate price. Otherwise it would not necessarily add up to what is shown. Commented Sep 25, 2021 at 18:25
• actually I think I will show it to three decimal places, so \$14.285... which should help indicate which direction it will round Commented Sep 25, 2021 at 19:11

You didn't specify the country/state, but in the places I have lived the tax is always rounded up.

So using an example from excel:

• if the price is \$14.09 the tax is "=roundup(14.09*0.05,2)"
• which is "=roundup(0.7045,2)"
• or \$0.71
• for a total of \$14.80

Also in the places I have lived when a price is quoted pre-tax, then a sub-total of all the items is calculated before determining the tax. That minimizes the extra tax that would be paid if each line of the bill had the tax due rounded up.

Also prices to the fractional cent aren't illegal. Every place I have bought gas quotes it to a number such as \$3.049 a gallon.

• was trying to keep it neutral since I'm more concerned with correctness of the computation, (including being able to display it clearly which I didn't mention) but yes, this is Canada, and we round to the nearest cent. Commented Sep 25, 2021 at 19:13