TL;DR: Sequential percents multiply factors, they do not add. Two +10% moves are ×1.10 ×1.10 = ×1.21, a 21% increase, not 20%. A +10% then −10% is ×1.10 ×0.90 = ×0.99, a 1% decrease, not 0. Addition is the shortcut that fails as soon as the second percent gets its own base.
Write a factor per step
+10% → 1.10. −10% → 0.90. −100% → 0. +0% → 1.00. +50% → 1.50. Factors less than 1 shrink. Factors greater than 1 grow. A chain is a product. Start with 100 if you like nice numbers: 100 → 110 → 121 for two +10%s.
If a step is “plus $10,” that is not a factor. Convert only percents to factors. Mixed chains of dollars and percents must apply in the stated order; they do not commute. $10 then +10% is not +10% then $10. Try it on $100: 110 then 121 vs 110 then 120.
Why the extras appear
The second +10% is 10% of a bigger number. That extra 1% on a 100 start is the 21 vs 20. The extra is not a bug. It is the definition of percent of the current value. If you wanted +20% from the start, say one +20% (×1.20), not two +10%s.
Inflation talk: “2% a year for 10 years” is 1.02^10 ≈ 21.9%, not 20%. The 1.9 points are compounding. Whether that is “what people feel” is sociology. The product is the math.
Discounts in a chain
20% off then 10% off: 0.80 × 0.90 = 0.72, pay 72% of original, 28% off total, not 30%. Stores that print “up to 30% off” with stacked coupons are counting the additive dream. Your receipt counts the product. Believe the receipt.
A third 5% off: 0.72 × 0.95 = 0.684, 31.6% off original. Diminishing dollars off: each percent hits a smaller ticket. That is why stacking many small percents is not a shopping superpower equal to one large percent of original—unless the large percent is of original and the stack is of remainders.
Growth then shrink, prices then wages
Price +8%, wage +8% does not preserve purchasing if they are percents of different baskets, but as pure numbers on one index, ×1.08 ×1.08 on the ratio depends whether both apply to the same personal index. Keep one variable per chain. A price chain and a wage chain are two products. Dividing the two net factors is a real comparison. Adding 8 and 8 to 16 is not.
Use percent of a number to apply each factor as “10% of current,” then add, if you refuse decimals. 10% of 100 = 10, 110; 10% of 110 = 11, 121. Same product. The “of current” phrase is the whole lesson.
Order when percents differ
+20% then −10%: 1.20 × 0.90 = 1.08. −10% then +20%: 0.90 × 1.20 = 1.08. Pure percent factors commute. Adding a +$5 fee in the middle breaks commuting. Percents of different bases (20% of food, 10% of food+tax) also do not commute with each other in a receipt sense because the second whole changed. Commuting is for factors on one quantity.
If a worksheet asks for “net percent,” convert the net factor: 1.08 means +8%. 0.99 means −1%. 1.00 means 0%. Net percent = (net factor − 1) × 100.
Spreadsheets
A column of +10%, +10%, +10% should be a running product, not a running sum of 10s. SUM of percents is the trap in costume. PRODUCT of (1+p) is the grown-up. Then subtract 1. If you chart the running sum, you will present 30% after three months of 10% and be wrong by the compounding extras.
Named cells for each factor beat a 20-long formula of 1.1*1.1*… typos. Typos in chains hide because the result still “looks like a percent.” Check with a 100 start every time.
Investment posters versus one-year homework
A poster that says “average 10% a year” may mean an arithmetic average of yearly percents, which is not the compounded factor product. Arithmetic 10+12+8 / 3 = 10 does not equal 1.10×1.12×1.08. Homework that gives three yearly percents and asks for ending value wants the product. Homework that asks for the average of the three percents wants the mean of 10, 12, and 8. Read “ending” vs “average.” Those words pick the machine.
If you only need a sanity number, 1.10^3 = 1.331 for three 10% years. If your product is 1.05, you used 5% or added wrong. If it is 1.30, you added 10+10+10 as 30% once. 1.30 is the additive dream. 1.331 is the chain.
Population and repeated percent growth
A town of 10,000 that grows 3% a year is not +300 people forever. Year two’s 3% is of 10,300. After 10 years, 1.03^10 ≈ 1.344, about 13,440, not 13,000. Linear +300×10 is the add-percents cousin. Biology homework that says “3% per year compound” wants the product. If it says “300 more each year,” it wants linear. The word percent in “3% per year” is not linear unless they also say of the original every year, which is a weird, explicit freeze of the base. Frozen bases should be printed in bold because they are unusual.
Decay at 3% per year is ×0.97 each year, not −3 percentage points off a 100 scale forever down to negative people. Factors stay positive. Counts approach 0, not −500.
Raises stacked with bonuses
A 3% raise then a 3% bonus of the new salary is not 6% of the old if the bonus is of the new: 1.03 × 1.03. If the bonus is of the old salary, it is 1.03 + 0.03 = 1.06 of old, which equals a different story. HR language “3% raise and 3% bonus” needs “of which base.” Stacked percents of mixed bases are how offer letters disagree with first payslips. Copy the bases from the letter, not from a 3+3 habit.
Inflation talk is not a raise formula
A headline 3% CPI is not your raise. Your raise is the factor on your letter. If rent rose 8% and pay rose 3%, you felt a cut in rent-units, which is a useful household story and a different percent with a different whole (rent, not salary). Do not subtract 8 from 3 and call it a −5% raise. Subtraction of unrelated percents is collage. Keep two lines: pay factor, rent factor. Then decide housing, not arithmetic theater.
