Percents in Context: Discounts, Taxes, and Tips

A 20 percent off sale and a 20 percent tip use the same arithmetic because both are just scaling a number by a factor. Taking 20 percent off means keeping 80 percent, so you multiply by 0.80. Adding a 20 percent tip means keeping the original 100 percent and adding 20 more, so you multiply by 1.20. In both cases you convert the percent to a decimal and either subtract it from 1 (money leaving) or add it to 1 (money being added), then multiply once instead of finding the percent separately and then adding or subtracting it by hand.

This matters because real purchases often stack several percent changes in a row, and each one acts on the result of the step before it, not on the original number. A discount shrinks the listed price, then tax is charged on that already-shrunk price, not on the sticker price. That's why you multiply by two separate factors in sequence: price times (1 minus discount rate) times (1 plus tax rate). To undo a percent change and recover an original price, divide by the multiplier instead of subtracting the percent back out — if 68 dollars is the price after a 15 percent discount, the original price is 68 divided by 0.85.

This idea of a single multiplying factor is the seed for percent change as a growth factor elsewhere in this unit, where repeated or compounding percent changes build on exactly this same multiplication logic.