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 4 percent?
Solution
Let be the price. Then we want to be a whole number of dollars.
That is, , where is an integer.
In general, a whole number n divided by 1.04 is dollars and cents.
Try some numbers, maybe on a spreadsheet.
n
n ÷ 1.04
We discover 12.50 works: 12.50 + .50 = 13.00
And 25 works too: 25 + 1 = 26.00
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. To find the smallest price that works?
Again, check the spreadsheet. You'll see that $12.50 is the smallest price that works for 4 percent.