Common percentage mistakes and how to avoid them

Most percentage mistakes are one mistake in different clothes: the number you divide by, the base, quietly changes. A 30% loss is 30% of the starting value, but the gain that follows is measured on the smaller amount left. A second discount comes off a price the first one already cut. Sales tax is a percentage of the pre-tax price, not of the receipt total. Keep track of the base and most of these traps go away.

What’s the difference between percent and percentage points?

Subtract one percentage from the other and the answer is in percentage points. Divide that difference by the percentage you started from and the answer is a percent change. When a pass rate rises from 72% to 81%, it rises 9 percentage points. Measured against the starting 72%, it rises 12.5%, because 9 ÷ 72 = 0.125.

$$\text{change in points} = B - A$$ $$\text{relative change} = \frac{B - A}{A} \times 100\%$$

The two measures drift far apart when the starting percentage is small: one point on top of 2% is a 50% rise, so a “small” and a “huge” headline about the same change can both be accurate. Differences between rates are also given in basis points: one basis point is one-hundredth of a percentage point, so the 0.75-point rise in the table is 75 basis points.

The practical risk is ambiguity. If a 5% rate “fell 1%”, it may now be 4% (down one point) or 4.95% (down 1% of 5%). When you write, say “percentage points” for a difference between percentages. When you read, work out which one the writer meant.

Why doesn’t a 50% gain undo a 50% loss?

Because the gain is calculated on a smaller base. Lose 50% of $100 and you have $50. A 50% gain on $50 adds only $25, so you finish at $75. Getting back to $100 means doubling the $50.

Successive percentage changes multiply; they don’t add. Two changes $a$ and $b$, written as decimals, combine to

$$\text{overall change} = (1 + a)(1 + b) - 1$$

Take an example investment worth $12,000 that falls 30% and then rises 30%:

1. After the fall: $12,000 × 0.70 = $8,400.
2. After the rise: $8,400 × 1.30 = $10,920.
3. Overall: 0.70 × 1.30 − 1 = −0.09, a 9% loss, although −30% and +30% “average” to zero.
4. Getting from $8,400 back to $12,000 takes a gain of $12,000 ÷ $8,400 − 1 = 42.86%.

Order makes no difference: +30% followed by −30% also ends at $10,920. The same happens in a shop, where a $50.00 item raised 30% to $65.00 and then offered at 30% off sells for $45.50.

The gain needed to recover from a loss $L$ is $L \div (1 - L)$, and it climbs fast:

Because changes multiply, a 12-month change in the CPI is not the sum of the monthly changes (the Bureau of Labor Statistics makes this point), and a simple average of yearly returns overstates growth when returns vary. The −30% and +30% years average 0%, but $12,000 becoming $10,920 over two years is a compound annual growth rate of −4.61% a year, which the CAGR calculator reproduces.

Do two discounts add up?

No. The second discount comes off the price the first one already reduced, so 25% off followed by an extra 15% off is 36.25% off, not 40%. Each percentage discount multiplies the price by (1 − discount):

$$\text{combined} = 1 - (1 - d_1)(1 - d_2)$$

For an example $64.00 jacket:

1. 25% off: $64.00 × 0.75 = $48.00.
2. Extra 15% off: $48.00 × 0.85 = $40.80.
3. Combined: 1 − 0.75 × 0.85 = 1 − 0.6375 = 0.3625, so 36.25% off. You save $23.20.

A true 40% off would have been $38.40. Swapping the order of two percentage discounts changes nothing ($64.00 × 0.85 × 0.75 is also $40.80), but mixing a dollar coupon with a percentage does. With the 25% sale and a $10 coupon, taking the coupon first gives ($64.00 − $10.00) × 0.75 = $40.50, while taking the percentage first gives $48.00 − $10.00 = $38.00. The store’s terms decide which order applies.

Does it matter whether tax is added before or after the discount?

Not for a percentage discount that also lowers the amount that is taxed, because multiplication gives the same result in any order. Adding 8.25% tax to $40.80 gives $44.17. Taxing the full $64.00 first ($69.28) and then taking 25% and 15% off also gives $44.17.

A dollar coupon is different. Take a $10 coupon off the $64.00 jacket before tax and the tax is $54.00 × 0.0825 = $4.455, which rounds to $4.46; tax the full $64.00 and it is $5.28.

Which amount is taxed is set by tax law, not arithmetic, for percentage and dollar discounts alike. In California, for example, a store’s own coupon lowers the amount that is taxed, but a manufacturer’s coupon that the manufacturer reimburses does not, according to the state’s Publication 113. If the extra 15% off came from a manufacturer’s coupon, the tax would still be figured on the $48.00 sale price: $48.00 × 0.0825 = $3.96 instead of $3.37, for a $44.76 total. Other states set their own rules.

How do you take sales tax out of a total?

Divide the total by 1 plus the tax rate. Don’t take the tax rate off the total: tax was charged on the pre-tax price, which is smaller than the total, so 8.25% of the total is more than the tax that was charged.

$$\text{pre-tax price} = \frac{\text{total}}{1 + r}$$ $$\text{tax} = \text{total} - \text{pre-tax price}$$

Suppose the jacket’s receipt shows $44.17 and the combined tax rate is 8.25%. That rate is only an example; rates depend on where you buy, so use your own.

1. Pre-tax price: $44.17 ÷ 1.0825 = $40.8037…, which rounds to $40.80.
2. Tax: $44.17 − $40.80 = $3.37.
3. Check: $40.80 × 0.0825 = $3.366, which rounds to $3.37, and $40.80 + $3.37 = $44.17.

The shortcut gives $44.17 × 0.0825 = $3.64 of “tax” and a pre-tax price of $40.53, which is 27 cents too low. Adding 8.25% back to $40.53 gives only $43.87, so the shortcut fails its own check. Put another way, tax at 8.25% makes up 0.0825 ÷ 1.0825 = 7.62% of a tax-inclusive total, not 8.25%.

The same division undoes any percentage change. A salary of $57,200 after a 4% raise was $57,200 ÷ 1.04 = $55,000 before it, not $57,200 × 0.96 = $54,912. The $40.80 sale price, 36.25% off, came from $40.80 ÷ 0.6375 = $64.00.

Which number is the base?

The base is the value after “of”, “than” or “from” in a carefully worded sentence. A name-brand item at $4.50 and a store brand at $3.60 differ by $0.90. Measured against the store brand, the name brand costs $0.90 ÷ $3.60 = 25% more. Measured against the name brand, the store brand costs $0.90 ÷ $4.50 = 20% less. Both statements are true; swapping the percentages is the mistake.

In pricing, markup divides the profit by the cost; margin divides it by the price. A seller who wants a 30% margin but adds a 30% markup ends up with a margin of 0.30 ÷ 1.30 = 23.08%. The guide to margin vs. markup covers pricing to a target margin.

How do you calculate percent change from a negative number or zero?

Divide the change by the size of the starting value, ignoring its minus sign; from zero, there is no percent change. Often it’s clearer to skip the percentage and give the amount. The usual formula divides by the starting value, and when that value is negative the sign of the answer flips. Suppose, as an example, a small business goes from a $4,000 loss to a $6,000 profit:

• Change: $6,000 − (−$4,000) = $10,000, an improvement.
• Dividing by the starting value: $10,000 ÷ (−$4,000) = −250%, which reads as a decline.
• Dividing by the size of the starting value, $4,000: +250%.

Dividing by the absolute value $|A|$ is a common workaround, and it’s the convention the percentage calculator uses:

$$\text{percent change} = \frac{B - A}{|A|} \times 100\%$$

Even then, the result is hard to read. A loss that narrows from $4,000 to $1,000 comes out as +75%, and one that widens to $6,000 as −50%, but neither means what the same percentage means for positive values. When the starting value is zero there is no percent change at all, because you can’t divide by zero: sales going from $0 to $6,000 have no percentage increase.

In these cases, give the amount and say what happened. “Swung from a $4,000 loss to a $6,000 profit” is clearer than any percentage.

What the arithmetic doesn’t settle

• Rounding. Rounding to the cent at every step and rounding once at the end can differ by a cent. When you check a figure, keep full precision until the last step.
• Store and tax rules. Whether promotions stack, which applies first and whether sale items are excluded are set by the store’s terms. Which discounts reduce the taxable amount is set by state tax rules.
• Missing bases. A percentage with no stated base (points or percent? of what? since when?) can’t be checked. Find the underlying numbers before you compare.

Try it

You can reproduce the worked numbers with these calculators:

• Discount calculator: enter an original price of 64 and 25 as the first discount’s percent off, then choose Add discount and enter 15 for the second. The sale price is $40.80, you save $23.20 and the percent off is 36.25%.
• Sales tax calculator: choose Remove tax, then enter a total with tax of 44.17 and a sales tax rate of 8.25. The price before tax is $40.80 and the tax is $3.37. With Add tax, a price before tax of 40.80 at 8.25 gives $3.37 of sales tax and a $44.17 total.
• Percentage calculator: under Calculation, choose Percent change from A to B. A starting value of 8400 and a new value of 12000 give +42.86%, the recovery from the 30% loss, and −4000 to 6000 gives +250%, with a note that the starting value is negative. Then choose Percentage points vs percent change: a starting percentage of 72 and a new percentage of 81 give +9 percentage points and a relative change of +12.5%.
• CAGR calculator: a starting value of 12000, an ending value of 10920 and a period of 2 years give a CAGR of −4.61%.