How to calculate a discount, a sale price and the real percentage off

To calculate a discount, multiply the original price by the percentage off as a decimal and subtract the result. An 80 item at 25% off has 80 × 0.25 = 20 taken off, so you pay 60. A quicker route is to multiply by what you keep: 80 × 0.75 = 60. Everything else in this guide, from stacked discounts to coupons, is a variation on that one idea. The discount calculator has a mode for each case, and I wrote it partly to settle arguments about sale signs.

The basic formula

Two equivalent ways to find a sale price:

• Subtract the saving: saving = price × rate; sale price = price − saving. 80 × 0.25 = 20, and 80 − 20 = 60.
• Multiply by what is left: sale price = price × (1 − rate). 80 × 0.75 = 60.

The second form is the one to remember, because it makes harder questions easy. 15% off means you pay 85%, 40% off means you pay 60%, and so on. For mental arithmetic, 10% is the price with the decimal point moved one place left, and most common discounts are multiples or halves of that. How to calculate percentages covers the general percentage toolkit.

What percentage off is this?

Sometimes you see a before and after price and want the real discount. Divide the saving by the original price: a jacket reduced from 80 to 60 has a saving of 20, and 20 ÷ 80 = 0.25, so 25% off. Always divide by the original, not the sale price; 20 ÷ 60 gives 33%, which answers a different question (how much more the full price is than the sale price). The calculator's second tab does this and also tells you what share of the original price you are paying.

Finding the original price

If you only know the sale price and the percentage, divide rather than add back. Something that costs 45 after 25% off did not cost 45 + 25%; it cost 45 ÷ 0.75 = 60. Adding 25% to 45 gives 56.25, which is wrong because the discount was taken from the larger number. The same reverse logic is used to strip tax out of a receipt total.

Stacked discounts

When a shop offers a percentage off and then a further percentage off at the till, the discounts multiply rather than add. 30% off followed by 20% off is not 50% off:

• After 30% off you pay 0.70 of the price.
• After a further 20% off you pay 0.80 of that, so 0.70 × 0.80 = 0.56 of the original.
• That is 44% off in total, not 50%.

Likewise, 50% off then an extra 20% off is 60% off, not 70%. The order makes no difference: 20% then 30% also lands on 0.56. The same arithmetic explains why a 30% price rise followed by a 30% cut leaves you 9% below where you started (1.30 × 0.70 = 0.91).

Discounts, coupons and tax

With a percentage discount and a percentage tax, the order does not change the total, because both are multiplications. Fixed-amount coupons are different. On a 100 item with 10% tax and a 10-off coupon, taking the coupon off first gives (100 − 10) × 1.10 = 99, while taking it off after tax gives 110 − 10 = 100. Which method applies depends on the shop and, for sales tax, on local rules about coupons, so check the receipt if a cent matters. In VAT countries shelf prices normally already include tax, so the discount applies to the tax-inclusive price. The calculator has an optional tax field that it applies after the discount; the sales tax calculator and the VAT calculator handle tax on its own, and How sales tax works and how to calculate it explains the rules.

Common sale-sign traps

• Buy one, get one free is 50% off per item only if you wanted two. Buy two, get one free is three for the price of two, which is 33% off, and second item half price on two equal items is 25% off.
• Up to 70% off means at least one item has that discount; most may have far less.
• Was and now prices are only as honest as the was price. A reduction from a price that barely applied is not much of a saving.
• Bigger discount, worse value: a 20% discount on a large pack can still lose to a smaller pack with no discount once you compare per unit with the unit price calculator.

The calculator handles the arithmetic. Whether you needed the item at all is a question no formula answers.