Why percentages go wrong

Percentages sound simple but hide traps: margin vs. markup, percent change vs. percent difference, stacking discounts, removing tax, and weighted rates. This guide clarifies the concepts with formulas and worked examples, and shows how to compute everything quickly using our Advanced Percentage Calculators.

Key formulas at a glance

• Percent change from old → new: (new − old) / |old| × 100%
• Percent difference between A and B (symmetric): |A − B| / ((A + B)/2) × 100%
• Markup on cost C to price P: markup% = (P − C) / C × 100%
• Margin on price P with cost C: margin% = (P − C) / P × 100%
• Discounted price with rate d: P_final = P × (1 − d)
• Stacked discounts d1 then d2: P × (1 − d1) × (1 − d2) (not d1 + d2)
• Add tax at rate t: P_with_tax = P × (1 + t)
• Remove tax from gross G: P = G / (1 + t)

Worked examples

Percent change vs percent difference

Old price 80 → New price 100. Percent change = (100−80)/80 = 25%. Percent difference (symmetric) = |100−80|/((100+80)/2) = 20/90 ≈ 22.22%. Use percent change for time-based changes; percent difference for comparing two peers without implying direction.

Markup vs margin

Cost C = 60; Price P = 100. Markup% = (100−60)/60 = 66.67%. Margin% = (100−60)/100 = 40%. Same numbers, different denominators. Don’t interchange them.

Stacked discounts

Price 100; 20% off, then extra 10% off. Final = 100 × 0.8 × 0.9 = 72 (not 70). Combined discount = 28%.

Removing tax (reverse-tax)

Gross bill 118 with 18% tax. Pre-tax = 118 / 1.18 = 100. Removing tax is division, not subtraction.

Weighted success rate

Team A: 40/50 = 80%; Team B: 20/50 = 40%. Combined: (40+20)/(50+50) = 60/100 = 60%. You can’t average the percentages directly; weight by counts.

Advanced scenarios you’ll encounter

Comparing conversion lifts

Variant A converts 5%; Variant B 6%. Absolute lift: +1pp. Relative lift: (6−5)/5 = 20%. If traffic differs, compare with weighted or statistical techniques; our calculator can structure the math and export your table.

Margin after stacked fees

Price 100; platform fee 10%; processing 2.5% + fixed 3. Net ≈ 100 − 10 − 2.5 − 3 = 84.5 → Margin% = 15.5%. Accurate estimates avoid overpricing/underpricing.

Back-solving needed price

Target margin 30% with cost 70. Solve P from margin = (P−70)/P = 0.3 → P = 100. Use the calculator’s “solve for price” to avoid algebra slips.

Presenting percentages clearly

• Use absolute and relative values when it helps: “+1.2pp (from 6.3% to 7.5%), +19% relative”.
• Prefer one decimal place for percents in dashboards unless high precision matters.
• Show denominators: “75/120 (62.5%)” beats “62.5%” alone.

Quality checklist

• Pick the right formula (change vs difference; margin vs markup).
• Show math for non-obvious steps (stacking, reverse-tax).
• Provide context (counts, time window, baseline).

Frequently asked questions

What’s the difference between percentage points and percent?

Percentage points (pp) measure absolute change between rates (e.g., 10% → 12% is +2pp). Percent change is relative: (12−10)/10 = +20%.

Is a 50% decrease then a 50% increase back to original?

No. 100 → 50 (−50%), then +50% = 75. You need +100% to return to 100.

How do I handle negative numbers?

For percent change, divide by the old value’s magnitude. For variance-like comparisons around zero, percent difference is safer than percent change.

Use the calculators

Open Advanced Percentage Calculators for guided inputs and instant results. Export PDF via jsPDF/AutoTable and CSV for spreadsheets to share with your team.