How a discount is calculated
A discount removes a share of the original price. "25% off" means you keep 75% of the price, so the arithmetic is a single multiplication rather than a subtraction:
sale price = original price × (1 − discount ÷ 100)
A €80 jacket at 25% off costs 80 × 0.75 = €60, and the amount saved is the €20 difference. Enter 80 as the base and −25 as the percentage above to see it worked through line by line.
The multiplication form is worth preferring over "work out 25% of 80, then subtract it" even though both give €60. It is one operation instead of two, it extends straight to stacked offers, and it makes the reverse calculation obvious: if multiplying by 0.75 gets you there, dividing by 0.75 gets you back.
Working backwards from the sale price
If you know both prices and want the discount, switch to the percentage change mode:
discount = ((original − sale) ÷ original) × 100
A jacket reduced from €80 to €60 is a 25% discount. This is the useful direction when a shop advertises "was/now" prices without stating a percentage — and the direction that catches out inflated "was" prices, since a large percentage off a price nobody ever paid is not a large saving.
Recovering the original price
The third direction is the one people ask for most and get wrong most: you know the sale price and the discount, and you want the price before the reduction.
original price = sale price ÷ (1 − discount ÷ 100)
A coat on sale at £91 after 30% off was £91 ÷ 0.70 = £130 before. The instinct to add 30% back is wrong: £91 + 30% is £118.30, nearly £12 short, because the 30% was taken off the larger original price, not the smaller sale price.
The rule generalises. To undo any percentage change you divide by the multiplier you would have multiplied by. Adding the same percentage back never returns you to the starting point.
| Discount | You pay (multiplier) | To recover the original, divide by | Adding the % back gives |
|---|---|---|---|
| 10% | 0.90 | 0.90 | 99.0% of the original |
| 20% | 0.80 | 0.80 | 96.0% of the original |
| 25% | 0.75 | 0.75 | 93.8% of the original |
| 30% | 0.70 | 0.70 | 91.0% of the original |
| 40% | 0.60 | 0.60 | 84.0% of the original |
| 50% | 0.50 | 0.50 | 75.0% of the original |
| 70% | 0.30 | 0.30 | 51.0% of the original |
The last column is the size of the error if you add the percentage back instead of dividing, and it grows fast: on a 50%-off item the shortcut leaves you a quarter short of the real original price.
Stacked discounts and coupons
Discounts applied one after another multiply — they never simply add up. A 30% seasonal reduction followed by a 20% coupon leaves you paying 0.70 × 0.80 = 0.56 of the original, a 44% total discount rather than 50%.
| First discount | Second discount | You pay | True total discount |
|---|---|---|---|
| 10% | 10% | 81% | 19% |
| 20% | 10% | 72% | 28% |
| 30% | 20% | 56% | 44% |
| 50% | 20% | 40% | 60% |
| 50% | 50% | 25% | 75% |
The order of stacked discounts never changes the final price — multiplication is commutative. What does change it is whether tax is applied before or after the discount.
Two 50% discounts are the clearest case: they leave you paying a quarter of the price, not nothing. Any claim that stacked offers add to 100% off is arithmetically impossible, because each multiplier is greater than zero.
Discounts, tax and the order of operations
Where tax sits relative to the discount changes what you pay, which is why receipts sometimes disagree with a mental estimate.
Under a VAT system the shelf price already includes tax, so a percentage off the shelf price reduces the tax proportionally and the order is invisible to you. Under a US-style sales tax the price is quoted before tax, and the discount is applied to that pre-tax figure before tax is added to the reduced amount.
you pay = price × (1 − discount ÷ 100) × (1 + tax ÷ 100)
A $200 item with 25% off and 8% sales tax is 200 × 0.75 × 1.08 = $162. Applying the tax first and the discount afterwards gives the identical $162 — multiplication does not care about order. What does matter is a discount that is legally applied only to part of the bill, such as a coupon that excludes taxes and fees, in which case the two operations are no longer applied to the same base.
Percentage off versus amount off
A "$20 off" voucher and a "20% off" voucher are the same only at a price of $100. Below that the fixed amount is worth more, above it the percentage is. If you are choosing between two vouchers on a $160 basket, $20 off beats 10% off but loses to 20% off.
How to calculate a discounted price
- Enter the original price. Type the full price into the base field.
- Enter the discount. Type the discount as a negative percentage, for example −25 for 25% off.
- Read the sale price. The result panel shows the price you will pay, and the difference from the original is your saving.
- Check the working. The steps under the result show the multiplier used and the multiplication itself, so you can confirm the figure rather than take it on trust.
Frequently Asked Questions
- How do I calculate 20% off a price?
- Multiply the price by 0.80. A $45 item at 20% off costs 45 × 0.8 = $36, saving $9.
- How do I find the original price from a discounted price?
- Divide the sale price by 1 minus the discount. A $60 item after 25% off was 60 ÷ 0.75 = $80 originally. Adding 25% back to $60 gives $75, which is wrong.
- Do a 30% and a 20% discount make 50% off?
- No. They multiply to 0.7 × 0.8 = 0.56, which is 44% off in total.
- Does it matter which discount is applied first?
- No. Multiplication is commutative, so 30% then 20% and 20% then 30% both leave you paying 56% of the original. Order only matters when a discount applies to part of the bill rather than all of it.
- Is a $20 voucher better than 20% off?
- Only below $100. At exactly $100 they are equal, and above it the percentage saves more. On a $160 basket, 20% off saves $32 against the voucher's $20.
- How much do I save at 33% off?
- One third of the price. Divide by 3 for the saving and multiply by 0.67 for what you pay — a $90 item costs about $60.30 and saves $29.70.
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