The percent off formula, and stacked discounts

Taking a percentage off a price is one multiplication. Almost every mistake with discounts comes from a second discount, from sales tax, or from confusing the amount saved with the amount payable.

The formula

saving = price × percent ÷ 100  ·  final price = price × (1 − percent ÷ 100)

The second form is the one to memorise, because it gets you to the answer in one step. A 25% discount means you pay 75%, so multiply by 0.75 and stop.

Stacked discounts never add up

"50% off, then an extra 20% off" is not 70% off. The second discount applies to the already-reduced price, so you multiply rather than add:

final = price × (1 − d₁) × (1 − d₂)

The gap widens as the discounts grow. Stacking is always worth less than the sum suggests, and retailers know this — "extra 20% off sale items" reads far better than "an additional 10 percentage points".

Order does not matter. 50% then 20% gives the same answer as 20% then 50%, because multiplication commutes. What matters is whether the second discount applies to the original or the reduced price, and it is almost always the reduced one.

Working backwards to the original price

If you know the sale price and the discount, divide rather than multiply:

original = sale price ÷ (1 − discount)

An item on sale at $48 after 40% off was originally 48 ÷ 0.60 = $80. Adding 40% back to $48 gives $67.20, which is wrong — a classic error, because the 40% was a percentage of the larger number.

To find the discount percentage from two prices, divide the saving by the original:

discount % = (original − sale) ÷ original × 100

From $80 to $60 you saved $20, and 20 ÷ 80 = 25% off.

BOGO offers in percentage terms

The condition column is the whole story. A BOGO is only 50% off if the second item was something you would have bought anyway. If it was not, the discount on what you actually wanted is zero, and you have spent more than you planned.

Discount and sales tax together

Discount first, tax second — tax applies to what is actually charged:

total = price × (1 − discount) × (1 + tax rate)

A $40 item at 25% off with 8% sales tax comes to $30 × 1.08 = $32.40. Because multiplication is commutative the order does not change the arithmetic, but it does change the reasoning, and it matters in one real case: manufacturer coupons. Where the retailer is reimbursed by the manufacturer, several states require tax on the pre-coupon price, because the retailer received full value. Store coupons reduce the taxable base; manufacturer coupons often do not.

In the UK and EU, shelf prices already include VAT, so a "20% off" sign reduces a gross price and nothing is added at the till.

In a spreadsheet

Enter 20% either as 0.2 in a plain cell or as 20% in a percentage-formatted one — never both, or the discount silently becomes 0.2%.

Doing it in your head

Discounts round nicely because they decompose into 10% and 5%. For 30% off $70: 10% is $7, so 30% is $21, and you pay $49. For 15% off: take 10% ($7), halve it ($3.50), add them ($10.50), and pay $59.50.

For 25% off, halve it twice — $70 → $35 → $17.50 saved, $52.50 to pay. For anything ending in 5%, work out the neighbouring 10% and adjust by half of it.