2410.13 – Bertha's Tax

Some years back, the sales tax around here was 4%. At that time Bertha priced a cat toy so that when she sold it in her store, which is cleverly named Bertha's Pet Store, the price plus the tax came out to an even number of dollars without rounding. What is the smallest possible price for which this works? (Note: Assume the tax does not round to the nearest penny, i.e. 4% of 96 cents is not 4¢.)

Bonus: Generalize the problem. What is the smallest such price for a tax rate of $X$ percent?

Solution

Let $x$ be the price. Then we want $x + 0.04x$ to be a whole number of dollars. That is, $1.04x = n$ , where $n$ is an integer.

You may notice that $25 \times 4 = 100$ , so $25 \times .04 = 1$ . So, if $x = \$25$ , then $1.04x = \$26$ .

You may also notice that $26$ is even, and while $25$ is not even, we can split $\$25$ by half into $\$12.50$ . Then, we have $1.04 \times \$12.50 = \$13$ . You'll find that $12.50 is the smallest price that works for a 4% sales tax.

If you've got this on a spreadsheet and you fiddle around with different interest rates, it becomes evident that the only interest rates that will work at all are those with factors of 2 and 5 (same as the factors of 100). You can do it for 4 percent, 10 percent, 16 percent, etc., but not for 3 percent or 7 percent.