The Discount Maths Shops Rely on You Not Doing

Retail pricing is designed around a small number of arithmetic mistakes that almost everyone makes. None of the calculations are hard. They are just slightly harder than doing them in your head while standing in a shop, which is the entire point.

1. Stacked discounts do not add up

"30% off, then an extra 20% off at the till" is not 50% off.

The second discount applies to the already-reduced price. On a £100 item:

• 30% off → £70
• 20% off £70 → £56

That is 44% off, not 50%. The gap widens with larger numbers: 50% then 50% is 75% off, not 100%.

The quick method: multiply the remaining fractions. 0.70 × 0.80 = 0.56, so you pay 56%.

Usefully, the order does not matter. 20% then 30% gives the same £56. If a shop offers to apply discounts in a particular order "to your advantage", that part is not true — though the order genuinely does matter when a fixed amount is involved. £20 off then 30% off gives £56; 30% off then £20 off gives £50. Take the percentage first when you can.

2. "50% extra free" is not 50% off

A 50% larger pack at the same price is a 33% discount, not 50%.

Say the normal product is 100 ml for £3, and the promotion gives 150 ml for £3. The normal unit price is 3p/ml; the promotional price is 2p/ml. You save a third.

The general rule: getting X% extra free is a discount of X ÷ (100 + X). 25% extra free is a 20% discount. 100% extra free — buy one get one free — is a 50% discount, which is the one case where the intuitive number is right.

3. The bigger box is not automatically cheaper

Bulk pricing is a strong convention, not a rule, and retailers are aware that shoppers assume it.

Studies of supermarket shelf pricing have repeatedly found a meaningful share of products where the larger size costs more per unit — often when the smaller size is on promotion, or when the large size is the "family" pack in a premium position.

The complication is that shelf labels make comparison deliberately awkward by using different units: one product priced per 100 g, the next per kg, a third per item. Converting in your head between "£1.20 per 100 g" and "£11.50 per kg" is exactly the friction that stops people checking.

The unit price comparator normalises different sizes and units to a common basis. The general habit — always compare per unit, never per pack — is worth more than any individual saving.

4. Percentage off versus percentage of

"20% off" and "20% of" are different quantities, and mixing them up is the most common percentage error there is.

• 20% of £80 = £16 — the saving.
• 20% off £80 = £64 — the price.

The same confusion appears in reverse when working out an original price. If something costs £64 after 20% off, the original was not £64 + 20% = £76.80. It was £64 ÷ 0.8 = £80.

Percentage increases and decreases are not symmetrical. A price that rises 20% and then falls 20% does not return to where it started: £100 → £120 → £96. To reverse a 20% increase you need a 16.7% decrease.

Two more worth knowing

"Was" prices are often not what you paid before. Many jurisdictions require the reference price to have been charged for some minimum period, and enforcement varies. A permanent "was £99, now £49" tells you what the retailer wants the anchor to be, not what the market price is.

Tax inclusion differs by country. In the UK and EU, displayed prices include VAT. In the US, sales tax is added at the till and varies by state, county and often city — so an advertised $50 can settle anywhere from $50 to about $55. Comparing a US price to a European one without adjusting for this understates the US price. The sales tax calculator handles the combined rates.

The short version

The discount calculator handles stacked offers and reverse calculations directly, and the percentage calculator covers increases, decreases and "what percent of" in one place — which is the three-way distinction most of these errors come down to.