Percentages that trip people up: change, points, margin and markup

Percentages look simple until two of them interact. A rise and a fall of the same percentage don’t cancel out; a rate “up 1%” might mean up one point or up one hundredth; a 40% markup isn’t a 40% margin. Almost every mistake comes from the same root: a percentage only means something relative to its base, and the base keeps changing. This guide works through the cases that catch people out, with the formulas to get them right.

Percentage change: always divide by the old value

percentage change = (new − old) ÷ old × 100

The old value is the base. A subscription that goes from 64.00 to 72.00 has risen by 8.00, which is 12.5% of the old price. To check a claimed change, multiply back: 64.00 × 1.125 = 72.00.

Reversing a known change uses division, not subtraction. If a price is 72.00 after a 12.5% rise, the original is 72.00 ÷ 1.125 = 64.00. The same rule takes VAT or GST out of a gross price, explained in our VAT and sales tax guide.

Why a rise and a fall don’t match

Going from 64.00 to 72.00 is a rise of 12.5%. Going back from 72.00 to 64.00 is a fall of only -11.11%, because the base is now the larger number. Equal percentage moves up and down therefore leave you worse off: a 10% gain followed by a 10% loss multiplies by 1.10 × 0.90 = 0.99, a net loss of 1%.

This matters for investments and prices alike. After a 50% fall, a 100% rise is needed to get back to the start.

Percent vs percentage points

When the thing being measured is itself a percentage, “up 1%” is ambiguous. Central banks avoid the ambiguity by announcing changes in percentage points: the Bank of England, for instance, describes Bank Rate moves in percentage points.

Both statements are true, but they answer different questions. Headlines sometimes pick whichever sounds bigger. When you read “up 21%”, check whether the thing that rose was already a percentage.

Repeated changes and the right average

Successive percentage changes multiply rather than add. Over three years of +12%, −5% and +8%, the total growth factor is 1.12 × 0.95 × 1.08 = 1.14912, a cumulative change of 14.91%.

The simple average of the three rates is 5.00%, but compounding at that rate for three years would overstate the result. The steady yearly rate that produces the same total, the geometric mean or compound annual growth rate, is about 4.74%. Use the geometric figure for investment returns and price indices; the inflation guide shows the same idea applied to consumer prices, and the compound interest calculator projects a steady rate forward.

Discounts, tax and the order of operations

Percentage discounts and percentage taxes are both multiplications, so their order does not change the final price. A 15% discount and 20% VAT on 200 give 200 × 0.85 × 1.20 = 204.00 either way round.

Order does matter in two cases. A fixed-amount discount (10 off) gives a different result before and after tax. And for tax records, the discount should reduce the price the tax is calculated on; HMRC’s VAT guide, for example, treats a discount given at the time of sale as reducing the value on which VAT is due. Stacked percentage discounts multiply too: two successive 15% discounts are 1 − 0.85 × 0.85 = 27.75% off, not 30%.

Margin vs markup

Both describe the same profit, measured against different bases:

• markup = profit ÷ cost
• margin = profit ÷ selling price
• margin = markup ÷ (1 + markup) and markup = margin ÷ (1 − margin)

Margin is always smaller than markup for the same profit, and it can never reach 100%, while markup has no upper limit. Retailers often talk in margin and suppliers in markup, which is a classic source of misunderstanding in negotiations.

Pricing to a target margin

To hit a target margin, divide the cost by one minus the margin. Adding the margin percentage to the cost is the common mistake:

If the price must also include VAT or GST, work out the net price for your margin first and then add tax; the VAT calculator does the second step.

Shares that don’t add to 100, and fees that look small

Rounded percentages of a whole don’t always sum to 100. Split a budget three equal ways and each share is 33.3333%; shown to one decimal place that is 33.3% three times, or 99.9%. Rounded to whole numbers, other splits can total 101%. Neither is an error, but a table that must add up needs one share adjusted, conventionally the largest, and a note saying so.

The base matters for charges too. A fee of 1% a year on an investment of 20000.00 is 200.00. If the investment returns 5% that year, a gain of 1000.00, the same fee is 20% of the return. “One percent” of the money you hold and “one percent” of what it earns are very different amounts. Fees quoted as a share of assets are best compared with the return you expect, and you can see the long-run effect by running the compound interest calculator twice, once at the gross return and once at the return minus the fee.

Percentages of percentages work the same way. A sales commission of 30% on a 5% booking fee is 1.5% of the booking, not 35%. When two percentages are chained, multiply them; add them only when they apply to the same base, as stacked tax rates on one price do.

Percentage change vs percentage difference

Percentage change has a direction and a base: from A to B. When comparing two values with no natural “before”, such as two quotes, some fields use the percentage difference, which divides by the average of the two: |A − B| ÷ ((A + B) ÷ 2). For quotes of 64.00 and 72.00 that gives 11.76%, between the -11.11% and 12.5% you get by picking a base. Say which measure you are using; the two are not interchangeable.

The percentage calculator covers percent of, percent change, reverse percentages, stacked discounts and margin versus markup, with the formula shown for each. For tax bands, which are percentages of slices of income, see the income tax bracket calculator.

This guide explains how things work in general terms. It isn’t financial, tax or legal advice. Spotted something out of date? Email errors@moneyotter.com and we’ll check it against the source.