🏷️ Discount calculator — what the sticker takes off
← Finance & MoneyA discount is one multiplication, and the easy half of shopping is doing it. The hard half is the second sticker: thirty per cent off and then a further twenty per cent off the sale price is 44% off, not 50%, because the second one is a share of the smaller number. Put in the ticket price and up to two discounts, and this page shows the price, the saving at each step, and the single rate the two of them come to together — which is the figure worth comparing when two tickets are stuck to one rail.
You pay
The same price at the discounts that turn up most
| Discount | You pay | You save |
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Arithmetic, not shopping advice. One discount is the price times what is left of a hundred, and a second one is the same multiplication done again on the answer — which is why the two together are less than they look, and why their order makes no difference to the result. Prices are shown to the cent; a shop that rounds a discounted price to the nearest five or ten cents will differ from the figure here by a few cents. Nothing on this page knows about GST, and it does not need to: shop prices in Australia already include it, so a discount comes off the GST-inclusive price. To add GST to a figure that does not include it, the percentage calculator's "plus" line is the one to use.
Why use this calculator?
A single discount is one multiplication, and anyone can do it on a phone. What goes wrong is the second sticker. "A further 20% off" is a share of the price the first discount left, not of the ticket price, so the two never add up to the number they look like — and the shop is not lying, the arithmetic is just not additive. That gap is worth a page of its own for two reasons: it is the difference between thinking you are getting 50% off and working out that you are getting 44%, and it is the same gap that turns "half price on the second" into 25% off the pair.
Where the discount comes from
The word is older than the shop, and the psychology is newer than both:
- A discount began as a count-off. The word came into English from Old French desconter — des-, away, and conter, to count — and its first sense was an allowance for paying a bill early: a merchant counted something off the amount owed because the money arrived before it was due. The retail price cut is the later meaning, and it is the same idea seen from the other side of the counter.
- The "was" price has to be real. In Australia a crossed-out price is a claim, not decoration: the Australian Consumer Law forbids misleading pricing, and the ACCC's position is that a higher "was" price has to be one the item was genuinely offered at, for a reasonable period, close to the sale. A saving no shopper could ever have refused is the kind of claim that has been to court.
- Prices ending in 9 are not an accident. The left-digit effect is the name for why $9.95 reads as "nine" rather than "ten": the first figure is what the eye takes, and the rest is treated as change. It is the oldest trick in retail and it still works, in both directions — which is why a discount is usually quoted against a price that was already chosen for how it reads.
- A discount is also a reference. Fifty dollars means very little on a rail; fifty dollars beside a crossed-out hundred is a gain, and the brain measures the gain rather than the price. That is why the "was" figure is printed in the same size as the "now" one, and why the effective rate — the figure this page works out — is the one to compare between two shops.
- And the multibuy is a discount in disguise. "Half price on the second" sounds like 50% off, and it is 25% off the pair: one item at full price and one at half is 1½ items for the price of 2, which is 75% of the money for both. The same arithmetic runs the other way when a shop offers three for two — that is a third off, not a half.
How does this calculator work?
A discount leaves a share of the price behind, and the share left is what gets multiplied:
sale price = price × (1 − discount ÷ 100)
effective discount = 1 − (1 − first) × (1 − second)
- One discount, one multiplication. Thirty per cent off $120 is $120 × 0.7, which is $84 — and the $36 saved is the other 30%. Multiplying by what is left beats subtracting the discount, because it is one step rather than two and it cannot be got backwards.
- The second discount goes on the answer. "A further 20% off" is $84 × 0.8, which is $67.20. Taken off the ticket price it would have been $24; taken off the $84 it is $16.80, and that difference is the whole point of this page.
- The two together are one rate. Multiplying the factors — 0.7 × 0.8 = 0.56 — gives the share of the original price being paid, so 44% came off. Nothing is rounded on the way through: the figures on the card are the exact arithmetic, shown to the cent.
- The order does not matter, and the reason is worth a moment: 0.7 × 0.8 and 0.8 × 0.7 are the same number, so 30% then 20% is identical to 20% then 30%. Two discounts only add up when they are shares of the same amount, and neither of these is.
Frequently asked questions
Is 30% off and then 20% off the same as 50% off?
No — it is 44% off, on any price. The first discount takes 30% off the ticket price, and the second takes 20% off what is left, which is a smaller amount of money. On $120: $36 comes off first, then $16.80, so $52.80 in total rather than the $60 that half would be. The rate that describes both steps together is 44%, because 0.7 × 0.8 is 0.56.
Does it matter which discount is applied first?
Not to the price. Twenty per cent and then thirty is the same $67.20 as thirty and then twenty, because multiplication does not care about order. What does matter is what each one is a share of: the second discount is always taken off what the first one left, never off the ticket price.
How do I work out a discount in my head?
Build it from ten per cent, which is a decimal point moved one place. Ten per cent of $120 is $12, so 30% is $36 and 40% is $48. For 25%, divide by four. Then keep the rest rather than subtracting: 30% off means keeping 70%, and 70% of $120 is $84 — one multiplication instead of two.
What is "half price on the second" actually worth?
25% off the pair. You pay full price for one and half for the other, which is one and a half prices for two items — 75% of what both would cost. "Three for two" works the same way and comes out at 33% off the three. Both are worth having; neither is the discount the sticker suggests.
How do I work out what something cost before the discount?
Divide what you paid by what is left of a hundred. $84 paid after 30% off means $84 ÷ 0.7, which is $120 — and that is the whole-behind-a-part line on the percentage calculator, which is the page for that question.
Is the discount taken off before or after GST?
After it, in effect. Australian shops show GST-inclusive prices, and a discount comes off the displayed price — so 30% off takes 30% off the GST as well, and the tax on what you pay is smaller because you are paying less. For a business claiming GST back, that means the credit is an eleventh of the discounted price rather than of the ticket price.
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