Percentage math: increases, percentage points, and common pitfalls

4 min read
MathStatistics

Percentages are elementary math, but real-world use hits subtle gotchas — does “+20% then −20%” return to the start? This article walks through the common operations and traps.

The basic formula

“What percent of the whole”:

percent = part / whole × 100

Example: 20 women out of 80 → 20 / 80 × 100 = 25%.

Change rate: from A to B

change = (B - A) / A × 100
  • 100 → 120 — +20%
  • 100 → 80 — −20%
  • 50 → 70 — +40%

100 → 120 and 120 → 100 are not symmetric:

  • 100 → 120 — +20%
  • 120 → 100 — −16.67%

“20% above 100” is “16.67% below 120”. The base matters.

The “+20% then −20%” trap

Start at 100, go up 20% → 120. From 120, go down 20% → 96 (not 100).

100 × 1.20 × 0.80 = 96

Bites in serial discounts and serial price hikes.

Percent vs percentage point

Percent (%) and percentage point (pp) are different:

  • “Approval rose from 30% to 33%.”
    • Change in points — 3 pp.
    • Change in percent — 10% (3 / 30 = 10%).

When the news says “rate moved from 0.5% to 1.0%“, that’s 0.5 pp in points, but 100% in relative change. Often misreported.

Sales tax (10% example)

  • Pre-tax → tax-inclusive — × 1.10
  • Tax-inclusive → pre-tax — ÷ 1.10
Tax-inclusive 1100 → pre-tax = 1100 / 1.10 = 1000
Pre-tax 1000 → tax-inclusive = 1000 × 1.10 = 1100

“Subtract 10% from 1100” gives 990, not 1000. Use division to back out tax.

Discounts

20% off:

discounted = original × (1 - 0.20) = original × 0.80

Discounts that don’t stack:

  • 20% + 10% = 30% off → original × 0.70

Discounts that do stack (sequential):

  • First 20% off → original × 0.80.
  • Then 10% off → original × 0.80 × 0.90 = original × 0.72.

“30% off” and “20% off then 10% off” differ (72% vs 70% of original).

Compound interest

5% annual rate, 10 years:

principal × (1 + 0.05)^10 = principal × 1.6289

$10,000 → ~$16,289 (62.9% growth). Not “50% (5% × 10 years)“.

The Rule of 72

Approximation for “years to double”:

years ≒ 72 / rate(%)
  • 6% — about 12 years.
  • 8% — about 9 years.
  • 12% — about 6 years.

Lets you skip the compound formula for ballpark estimates.

Margin and markup

“Cost $1,000, sell at $1,500”:

  • Margin (sale-based) — (1500 − 1000) / 1500 = 33.3%.
  • Markup (cost-based) — (1500 − 1000) / 1000 = 50%.

Different industries default to different bases. Retail uses margin, manufacturing often uses markup.

When someone says “50% margin”, verify which base they mean.

Weighted average

Combining group averages — easy to get wrong:

  • Group A — 1,000 people, mean 60.
  • Group B — 100 people, mean 80.
  • Naive average — (60 + 80) / 2 = 70 → wrong.
  • Correct — (1000 × 60 + 100 × 80) / 1100 = 61.8.

Always weight by group size. Plain averaging overweights the small group.

Percentiles

“Top 10%”, “bottom 25%” — related but distinct from percentages:

  • 50th percentile = median.
  • 90th percentile = top 10% threshold.

Used in test scores, latency p99, etc.

Rounding choices

Percent math creates fractional values that need a rounding rule:

  • $3.33 × 1.10 = $3.663 → $3.66? $3.67?
  • Some jurisdictions mandate truncation (sales tax often allows merchant choice).

Pick a rule (truncate / ceil / round half-up) and apply it consistently in accounting.

Percentages above 100%

Going past 100% is fine when the base permits it:

  • “Year-over-year 250%” — 2.5×.
  • “Achievement 120%” — exceeded goal by 20%.

But “110% of the total” is nonsensical when “total” means the whole.

Summary

  • Change rates aren’t symmetric (+20% then −20% ≠ start).
  • Percent and percentage point are different — don’t conflate.
  • Stacked discounts multiply, not add.
  • Margin vs markup differ by base.
  • Weighted averages account for group size.

For any percentage calculation, the percentage tool on this site handles the common operations.