Math: what discount will exactly cancel the tax?

Tutoring high school math, you see financial word problems. The tutor gives an example.

Problem: At 12% tax, what discount will lead to paying exactly the sticker price after tax?

Solution

The final price, f, is as follows:

f = (1-discount)(1+tax)p

Where p is original price, and discount and tax are in decimals, not percent (eg, tax is 0.12 rather than 12%)

We need f=p, at tax=0.12:

p=(1-discount)(1+0.12)p

Dividing both sides by p we get

1=(1-discount)(1.12)

Then, dividing by 1.12,

0.8929 = 1-discount

Rearranging we get

discount = 1-0.8929 = 0.1071 or 10.71%.

Apparently, at 12% tax, a discount of 10.71% is needed to equate the after-tax price to the sticker price.

Cheers.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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