ABC Tools

The Discount Calculator, Explained

After reading this you will know how to turn any "% off" sticker into a real price, why stacked discounts multiply instead of add, and how to spot the marketing tricks that make a deal look bigger than it is.

What a discount actually is

A discount is a fraction of a price that you do not pay. A "20% off" tag on a 100 item means you keep 20 and hand over 80. That single case is easy. The interesting part starts when a shop advertises two or three reductions at once, because the second discount does not apply to the original price. It applies to whatever price is left after the first one.

Here is the hook. A sign reads "50% off, then an extra 30% off". Your instinct adds the numbers and expects 80% off. The true saving is 65%. The extra 30% comes off the already-halved price, not off the full price. This article shows why, and gives you a formula you can check by hand.

When to use it, and when not

Reach for a discount calculation whenever a price is reduced by a percentage and you want the amount you will actually pay. That covers seasonal sales, clearance racks, coupon codes stacked at checkout, and staff or member discounts applied on top of a promotion.

It is the wrong tool for a few nearby situations. A discount is not a tax: sales tax and VAT are added to a price, so for those use the VAT / Sales Tax Calculator. A discount is not a tip, which is also an addition; see the Tip Calculator. And a percentage off tells you nothing about value per unit. A "40% off" 2-litre bottle can still cost more per litre than a full-price 5-litre one. To compare sizes, use the Unit Price Comparison.

The formula and the intuition

Start with one discount. If the original price is P and the discount is d percent, you pay the fraction that remains:

P_{\text{final}} = P \times \left(1 - \frac{d}{100}\right)

Here P is the original price, d is the discount percentage, and 1 - d/100 is the keep fraction: the share of the price you still pay. A 20% discount has a keep fraction of 0.8.

Now stack discounts. Each one acts on the price left by the one before, so you multiply the keep fractions together:

P_{\text{final}} = P \times \left(1 - \frac{d_1}{100}\right)\left(1 - \frac{d_2}{100}\right)\left(1 - \frac{d_3}{100}\right)

The symbols d_1, d_2 and d_3 are the three discount percentages. Any discount you do not use is 0, and its keep fraction is 1, so it changes nothing.

The single effective discount, the one number that would produce the same final price on its own, is:

d_{\text{eff}} = 100 \times \left(1 - \frac{P_{\text{final}}}{P}\right)

Because you multiply keep fractions rather than adding discounts, d_{\text{eff}} is always smaller than the plain sum of the discounts (unless one of them is zero).

The order does not matter. Multiplication commutes, so "50% then 30%" and "30% then 50%" both leave a keep fraction of 0.5 × 0.7 = 0.35. Shops sometimes present the bigger number first because it looks more generous, but the total is identical.

A worked example with the demo numbers

20% off, reproduced by hand

The demo button uses the field defaults: an original price of 100, a first discount of 20%, and no second or third discount. Follow the arithmetic:

  1. Keep fraction of the first discount: 1 - 20/100 = 0.8.
  2. The other two discounts are 0, so their keep fractions are both 1.
  3. Final price: 100 \times 0.8 \times 1 \times 1 = 80.
  4. You save 100 - 80 = 20.
  5. Effective discount: 100 \times (1 - 80/100) = 20\%. With one discount, the effective rate equals the discount itself.

Now stack a second discount

Keep the price at 100, the first discount at 50%, and add a second discount of 30%. This is the case in the footnote on the tool page.

  1. First keep fraction: 1 - 50/100 = 0.5. Price drops to 50.
  2. Second keep fraction: 1 - 30/100 = 0.7. The 30% comes off the 50, removing 15.
  3. Final price: 100 \times 0.5 \times 0.7 = 35.
  4. Total saving: 100 - 35 = 65, so the effective discount is 65%, not 80%.

Watching the gap grow

The gap between the "sum" a shopper imagines and the true effective discount widens as the discounts get bigger. The chart below plots both for a fixed first discount of 50% while the second discount rises from 0% to 50%.

The added-up figure climbs faster than reality. At "50 + 50" the sum says 100% (free), but the true saving is only 75%, so you still pay 25.

Notice the last bar. Two 50% discounts do not make something free. The keep fraction is 0.5 \times 0.5 = 0.25, so a 100 item costs 25.

With an original price of 100, a first discount of 40% leaves 60. A second discount of 25% on top leaves 60 × 0.75 = 45, an effective discount of 55%, not 65%. Raising either discount lowers the final price, but two discounts never combine to more than their multiplied keep fractions allow.

Reading and interpreting the result

Three numbers matter after any discount calculation.

Final price
The amount you hand over. This is the figure to compare against other shops and against your budget.
Savings
Original price minus final price. Useful, but only against the price you would otherwise have paid. A saving off an inflated "was" price is not a real saving.
Effective discount
The single percentage that captures the whole promotion. Use it to compare deals with different structures. "50% + 30%" (65% effective) beats a flat "60% off".

The effective discount is the honest headline. If a store advertises "up to 80% off" but the checkout maths gives you 65%, the effective figure is what you actually got.

Common mistakes

The biggest error is adding stacked discounts. It always overstates the saving, and the tool exists mainly to correct it. Three others come up often.

  • Confusing discount and markup. A price cut of 50% followed by a 50% price rise does not return you to the start. 100 \times 0.5 \times 1.5 = 75. You end up 25% lower.
  • Ignoring the base price. A discount off a raised "recommended" price can be no bargain at all. Compare final prices across shops, not discount percentages.
  • Forgetting tax comes after. In regions where tax is added at the till, the discount lowers the pre-tax price and the tax is then applied to the reduced figure.

A "buy one get one 50% off" offer is not 50% off. You buy two items and save 50% on one of them, so the effective discount across both is 25%. Work out the total for the quantity you actually intend to buy.

Related tools

Discounts rarely travel alone. If a discounted price is charged in another currency, the Currency Converter puts it in your own. To judge whether a discounted annual subscription beats the monthly plan, use the Subscription Cost Calculator. A cash-back card can stack another few percent on top of a sale; the Cash Back Calculator checks whether its fee is worth it. And for a big-ticket discounted purchase, the Price Per Use Calculator tells you the cost spread over its whole life. If a "saving" tempts you into a recurring purchase, the Cost of a Habit Calculator shows the yearly total, and the Group Expense Splitter and Inflation Calculator cover shared bills and long-run value.

Frequently asked questions

Why is 50% plus 30% off only 65% and not 80%?

The 30% comes off the price after the 50% cut, not off the original. A 100 item drops to 50, then loses 30% of 50 (which is 15), landing at 35. That is a 65% total saving.

Does the order of the discounts change the price?

No. You multiply the keep fractions, and multiplication does not care about order. "30% then 50%" gives the same 35 as "50% then 30%".

Can stacked discounts ever reach 100%?

Only if one single discount is 100%. Two discounts under 100% each leave a positive keep fraction, and any positive number times another positive number stays positive. Two 60% discounts leave 0.4 \times 0.4 = 0.16, so you still pay 16%.

How do I add sales tax after a discount?

Apply the discount first to get the net price, then add the tax to that reduced figure. The VAT / Sales Tax Calculator handles the addition.

Is a bigger discount percentage always the better deal?

Not if the starting prices differ. Compare the final prices you would actually pay, or the effective discount off the same base. A 70% cut on a padded price can cost more than a 40% cut on an honest one.

These figures are educational estimates, not financial advice. For decisions with real money at stake, talk to a qualified adviser.

Try it yourself in the tool

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