Some years back, when sales tax was exactly 4%, Bertha priced a cat toy in dollars and cents 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 exact number of dollars, with no rounding at all. What is the smallest possible price for which this works? (Note: 96 cents for a price is no good because you have to round the tax to the nearest penny. 4% of $0.96 isn’t 4 cents.)
We want x + 0.04x to be a whole number of dollars.
That is, 1.04x = n, a whole number of dollars.
That is, 1.04x = n, a whole number of dollars.
In general, a whole number n divided by 1.04 is dollars and cents.
Try some numbers, maybe on a spreadsheet.
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
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%, 10%, 16%, etc., but not for 3% or 7%. 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%.