Scope and assumptions
Formulas assume a single percentage discount off a list price. Rates are decimals (20% = 0.20).
The word "discount formula" covers three different calculations that people use interchangeably and then get wrong. This page is the index — it lines up all three so you can grab the one your numbers fit, then points you to the deep dive where it matters. Starting from the wrong formula adds steps or quietly flips the result.
Three inputs, three outputs
| You have | You want | Use |
|---|---|---|
| List + sale | Dollars off | Discount = List − Sale |
| List + rate | Price to pay | Sale = List × (1 − rate) |
| List + sale | Headline percent | (List − Sale) ÷ List × 100 |
Each one, quickly
Dollars off. $100 list, $80 sale → 100 − 80 = $20 off. The simplest of the three, and rarely the one people actually need.
Price to pay. $250 list at 30% off → 250 × 0.70 = $175. If you want to master the speed-calculation at the register or need more step-by-step examples, read our dedicated guide on Sale Price After Percent Off Formula.
Headline percent. $45 list, $38.25 sale → (45 − 38.25) ÷ 45 × 100 = 15% off. The formula for checking whether a tag is telling the truth.
What (1 − rate) is doing
This is the piece that trips people up, and it's worth being precise about. List × rate gives the amount removed. List × (1 − rate) gives what remains — your share.
- $100 × 0.20 = $20 — the discount.
- $100 × 0.80 = $80 — the price.
Both multiplications start identically and end up describing opposite things. Remember the pair: rate = taken away, (1 − rate) = kept.
Running it backwards
The reverse direction — from a sale price back to the original — is where errors cluster. An $80 sale price at 20% off came from 80 ÷ 0.80 = $100, not 80 + 20% = $96. Adding the rate back doesn't undo a discount, because the rate was measured against the larger original, not the smaller sale price.
Where to go deeper
This page is the macro index — the three discount forms and how they relate. For the exact situation you're in, follow the matching spoke in this cluster:
- Price to pay after a percent-off tag — the register-speed method: Sale Price After Percent Off Formula
- Turn two prices into the headline percent — How to Calculate Discount Percentage
- What you kept in your pocket — Calculate Percent Saved
- Recover the pre-sale "was" price — Find the Original Price From a Discount
- The mental method at the shelf — Percent Off Method
- Why 30% is everywhere — What Is 30 Percent Off?
- Stacked promos aren't additive — Stacked Discounts
- Which deal actually saves more — Compare Two Discounts
- Markup vs discount, two different bases — Markup vs Discount Percentage
Where the Three Formulas Get Swapped
Using the rate where (1 − rate) belongs, which returns the discount instead of the price; forgetting to convert 20% into 0.20; picking the wrong member of the trio for the inputs in hand; rounding the discount before computing the sale price and losing a cent or two; treating a percentage as a fixed dollar amount ("20% off" is not "$20 off"); and adding successive rates instead of multiplying the surviving fractions.
FAQ
How do I work backwards from a sale price to the original? Divide by (1 − rate). An $80 item at 20% off came from 80 ÷ 0.8 = $100. This is the reverse of the forward formula and is easy to get wrong by adding 20% to $80 instead.
What's the difference between a discount rate and a markup rate? The base. A discount is a share of the original selling price; a markup is a share of cost. A 25% discount and a 25% markup move the price by different dollar amounts.
Can a discount exceed 100%? Not in retail pricing — you can't pay less than nothing. A figure above 100% usually means a rebate, a trade-in credit, or an arithmetic error.
Why don't two 20% discounts equal 40%? Because the second applies to an already-reduced price. 0.8 × 0.8 = 0.64, so you pay 64% — a 36% total reduction.
Are retailers required to base a discount on a genuine prior price? In the US, the FTC's guides against deceptive pricing say a reference ('was') price must reflect a price actually charged recently; a manufactured anchor can count as a fake discount. The UK's Consumer Protection from Unfair Trading Regulations 2008 enforce the same idea. Rules vary, so check local guidance.
Sources: Khan Academy, Percent word problems; Maths is Fun, Percentages.
Disclaimer: Illustrative examples. Real promotions may stack or condition discounts; confirm at checkout. Informational only.
Sources and references
- Khan Academy — Percent word problemshttps://www.khanacademy.org/math
- Maths is Fun — Percentageshttps://www.mathsisfun.com/percentage.html
- FTC — Guides Against Deceptive Pricinghttps://www.ftc.gov/business-guidance/resources/guides-against-deceptive-pricing-practices
Maintenance and corrections
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