Fifty per cent off and then twenty is not seventy

The options around successive discounts are set out side by side below, with the conditions that genuinely favour one over the other.

The arithmetic people expect

Two reductions announced together read as though they add, because addition is what percentages feel like when they are stacked in a sentence. In practice the second reduction applies to the already reduced figure, so the combined effect is always less than the sum.

The gap widens as the individual percentages grow, which is why the largest advertised combinations are the most misleading. Nothing about the presentation is false, and the arithmetic is standard, but the intuition it exploits is nearly universal. Working in cash instead of percentages removes the problem entirely.

Mixed offers and order of operations

When a percentage reduction and a fixed cash reduction are combined, the order in which they are applied changes the final figure. Applying the cash amount first and then the percentage produces a different result from the reverse, and the system decides which happens.

Terms sometimes specify the order and often do not, in which case the only way to know is to compare totals. The same applies to whether delivery, taxes and fees are calculated before or after a discount. This is a further reason to compare final payable amounts rather than reasoning about the offer.

Where the confusion is exploited

Extra discounts applied at checkout on already reduced items are the most common presentation of this arithmetic. Percentage-off-percentage promotions in clearance are another, since the base has already fallen and the second cut is smaller than it sounds.

At the till, loyalty discounts applied to promotional prices behave the same way, and the combined figure is rarely stated. None of these are improper, and all of them are read incorrectly by most people most of the time. The remedy is arithmetic rather than suspicion.

Percentages are unanchored

A percentage is meaningless without knowing what it is a percentage of, and that base is chosen by the seller. A large percentage off an inflated reference is worth less than a small percentage off a competitive price.

Comparing two offers by their percentages therefore compares two numbers with different denominators, which is not a comparison at all. Converting to cash puts every offer on the same footing immediately.

The cash figure is also the only one that tells you whether you can afford the thing.

Doing it in your head

For a rough check, take ten per cent of the price and scale it, which handles most common percentages quickly. For successive discounts, apply the first, then take the second from the result rather than adding the two.

Round in whichever direction is unfavourable to the offer, so any error works in your favour. For anything large, use a calculator rather than an estimate, since the amounts justify the extra seconds. Mental arithmetic is not the point; getting a cash figure is the point.

Consumer protection rules are national, and what is unlawful in one market is routine in another.

A habit worth keeping

Whenever an offer is expressed in percentages, write down the cash price before and after and compare those. Whenever two offers compete, compare their cash outcomes for the exact basket you intend to buy.

Whenever a percentage seems remarkable, check what it is a percentage of before being impressed. This one habit defuses most promotional arithmetic without any specialist knowledge. Cash is the common denominator, and marketing prefers percentages precisely because they are not.

Side by side

The takeaway

Convert every percentage to cash before you compare, because percentages hide both their base and their order.

Decide what you would pay before you look at what they are asking.