How Percentage Discounts Are Calculated — and Why Shops Make It Confusing

The Basic Formula

A percentage discount calculation has three components: the original price, the discount rate, and the final price. The formula is:

Discount amount = Original price × (Discount % ÷ 100)

Final price = Original price − Discount amount

Or combined into a single step: Final price = Original price × (1 − Discount % ÷ 100)

Example: A $80 item at 25% off. Final price = $80 × (1 − 0.25) = $80 × 0.75 = $60. The discount amount is $20, and the final price is $60.

The single-step version is faster and less prone to rounding errors. Multiplying by 0.75 (for a 25% discount), 0.70 (for 30%), or 0.85 (for 15%) is the most reliable mental approach.

Why Sequential Discounts Don't Add Up

One of the most common and consequential misunderstandings in discount arithmetic is the assumption that sequential discounts add together. They do not — and the difference can be significant.

If an item is discounted by 10% and then discounted again by 10%, the total is not a 20% discount. Here is why:

Start with $100. First discount: $100 × 0.90 = $90. Second discount: $90 × 0.90 = $81. Total discount: $19, which is a 19% reduction from the original $100 — not 20%.

The reason is that the second discount is applied to the already-reduced price, not the original. Each subsequent discount operates on a smaller base, so its absolute effect is smaller than it would be if applied to the original price.

The correct formula for combined sequential discounts is: Combined rate = 1 − (1 − r₁) × (1 − r₂), where r₁ and r₂ are the decimal equivalents of each discount rate.

10% + 10% = 1 − (0.90 × 0.90) = 1 − 0.81 = 0.19, confirming the 19% combined discount.

For 20% + 20%: 1 − (0.80 × 0.80) = 1 − 0.64 = 0.36 — a 36% combined discount, not 40%.

Percentage Off vs. Percentage of Original

A subtle but important distinction: "20% off" and "20% of the original price" mean very different things.

"20% off $100" means you pay $80 — the price is reduced by 20% of the original.

"20% of the original price" means you pay $20 — you are paying only 20% of what the item originally cost, which is an 80% discount.

This distinction becomes practically relevant in contexts like end-of-season clearances where "only 10% of original price" means a 90% discount, or in manufacturing and wholesale contexts where goods are sold "at 40% of list price." The framing matters: "percentage off" and "percentage of" are arithmetic inverses, and conflating them leads to significant errors.

The Asymmetry of Discounts and Increases

Perhaps the most counterintuitive fact about percentage discounts: a percentage decrease cannot be reversed by the same percentage increase. This is because the base changes.

A product priced at $100 that drops 30% falls to $70. To return to $100 from $70, you need an increase of ($100 − $70) / $70 × 100 = 42.86%.

This asymmetry becomes important in investment contexts (a 50% portfolio loss requires a 100% gain to recover), in salary negotiations (a 10% pay cut followed by a 10% raise leaves you earning less than before), and in retail pricing strategies.

The general relationship: if a price falls by X%, the percentage increase needed to recover is X / (100 − X) × 100. For a 30% drop: 30/70 × 100 ≈ 42.86%. For a 50% drop: 50/50 × 100 = 100%.

Buy One Get One: How BOGOF Compares to a Straight Discount

"Buy one get one 50% off" sounds like a generous deal. Is it better or worse than "25% off both items"? The answer is that they are mathematically equivalent — but the framing makes BOGOF feel more generous.

Two items at $40 each = $80 total. BOGOF 50% off: you pay $40 + $20 = $60. Saving: $20, which is 25% of $80. "25% off both items": you pay $30 + $30 = $60. Saving: $20, which is 25% of $80. Identical.

What changes is the psychology. BOGOF framing triggers a sense of getting something for less (or free), which is more emotionally compelling than a blanket percentage reduction applied to both items. Retailers choose BOGOF when they want to move higher volumes of a product; they choose straight percentage discounts when they want to appear broadly competitive on price.

VAT and Tax-Inclusive Pricing Confusion

In countries where prices are displayed inclusive of sales tax or VAT, percentage discounts on the listed price produce a different effective saving than the percentage suggests — because you are discounting an amount that includes tax.

In the UK, where 20% VAT applies to most goods: a product listed at £120 (which includes £20 VAT) discounted by 25% costs £90. The VAT on £90 is £15, so the pre-tax price has gone from £100 to £75 — a genuine 25% reduction on the pre-tax value. In this case the arithmetic is consistent because the discount is applied to the full inclusive price and the tax adjusts proportionally.

The confusion arises when discounts are advertised on ex-VAT prices and the final checkout price is presented inclusive of VAT. A "20% off" promotion applied to a £100 ex-VAT price reduces it to £80 ex-VAT, which becomes £96 including VAT — but if the shopper saw £120 on the shelf, they may expect to pay £96 and feel cheated if the promotional mechanics are unclear. Always check whether advertised prices and discounts refer to tax-inclusive or tax-exclusive figures.

Why Retailers Prefer "Save £X" vs. Percentages

Retailers choose between "save £X" and "X% off" based on which sounds bigger for any given product. Research confirms that consumers are more responsive to the presentation that produces the larger-feeling number.

On a £10 item, "save £2" sounds modest; "20% off" sounds significant. On a £500 appliance, "15% off" sounds minor; "save £75" sounds substantial. The rule is that absolute amounts feel larger on expensive items, and percentages feel larger on inexpensive ones.

When evaluating any discount, the most useful habit is to calculate the other representation — if you're shown a percentage, work out the absolute saving; if you're shown an absolute saving, calculate the percentage. The number the retailer didn't show you is usually the one that makes the deal look less impressive. A discount calculator makes this conversion instant and removes the ambiguity entirely.