How do I correctly combine percentages without just adding them?

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1. Percent changes stack multiplicatively-think multipliers, not sprinkles: up 25% then down 20% is 1.25×0.80=1.00, so 0% overall; in general multiply all (1±r_i), order doesn’t matter, then subtract 1 for the net percent.
   For combined groups, use a weighted average by sizes: overall = (n_A·p_A + n_B·p_B)/(n_A + n_B), so 40% and 70% only average to 55% if the groups are the same size.

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2. Treat each percentage change as a scale factor: +r% = ×(1+r), −r% = ×(1−r); multiply the factors to stack them and convert back at the end (e.g., 25% up then 20% down: 1.25×0.80 = 1.00 ⇒ 0% net), and the order only matters if the percentages are taken on different bases (like coupon before vs after tax). For combining groups, use a weighted average by sizes: overall % = (pA·nA + pB·nB)/(nA+nB), not a plain average unless sizes match-percent changes are like gears that multiply, while group percents are like mixing paint by volume, not just tossing sprinkles!

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3. Short answer first: percentages don’t stack by simple addition; they stack by multiplying “factors.” Think of each percent as a stretchy-squishy lens you put over a price. Every new lens scales whatever came before. So you don’t sprinkle; you scale.

   Let me walk through the main situations and then give you a quick test-day checklist.

   1) Successive increases/decreases on one number (like a price)
   – Rule: Turn each percent change into a multiplier, multiply them all, then convert back to a percent at the end.
   – Increase p% → multiplier 1 + p
   – Decrease q% → multiplier 1 − q
   – Example: Up 25%, then down 20%
   – Multiply: 1.25 × 0.80 = 1.00
   – Final multiplier 1.00 means no net change: overall 0%.
   – Fast formula for two steps: overall percent = p + q + pq (with p and q as decimals, signed for up/down).
   – Here: 0.25 + (−0.20) + (0.25)(−0.20) = 0.25 − 0.20 − 0.05 = 0.00 → 0%
   – Gotchas:
   – Up 20%, then down 20% is not 0%. It’s 1.20 × 0.80 = 0.96 → −4%.
   – To “undo” a +p% increase, you need a decrease of p/(1 + p), not p itself. Example: after +25%, you need −20% to come back.

   2) Stacking discount, then tax, then coupon
   – If each step is a percent of the current price, you still multiply the factors. Multiplication order doesn’t matter mathematically, but the base each percent applies to does matter in real policies.
   – Discount c%, then tax t%: multiplier (1 − c)(1 + t).
   – Example: 30% off then 10% tax → 0.70 × 1.10 = 0.77 → net −23%.
   – Two discounts c1% and c2%: (1 − c1)(1 − c2). This is not c1 + c2; it’s less because of the cross-term.
   – Example: 20% then 10% off → 0.80 × 0.90 = 0.72 → 28% total off, not 30%.
   – Where order/policy matters:
   – Fixed-dollar coupon (say $k off) mixes addition/subtraction with multiplication. Then the path matters because tax might be applied before or after the coupon depending on the rules.
   – If coupon before tax: final = (P − k)(1 + t).
   – If coupon after tax: final = P(1 + t) − k.
   – These usually give different totals.
   – If a percent is taken off the original price rather than the current price (less common in clean math problems but possible), that changes the base and breaks the neat multiplier chain.

   3) Combining percentages from different groups (weighted average)
   – You can’t just average the percentages unless the group sizes are equal.
   – Rule: overall percent = (pA × nA + pB × nB) / (nA + nB).
   – pA, pB are decimals; nA, nB are group sizes.
   – Example: 40% of Group A, 70% of Group B
   – If A has 100 and B has 50: overall = (0.40×100 + 0.70×50) / 150 = (40 + 35)/150 = 75/150 = 50%.
   – If sizes are equal (say 100 and 100): overall = (40 + 70)/200 = 55%.
   – Moral: you need the sizes to combine group percentages correctly.

   A tiny analogy (because I can’t help myself)
   – Imagine your price is a balloon. A +25% breath inflates it by a factor of 1.25. Then a −20% pinch shrinks whatever balloon you have now by a factor of 0.80. You don’t add breaths and pinches; you stretch, then squish. The final size is the product of those scale effects, not their sum.

   Quick test-day checklist
   – For chained percent changes on one number:
   1) Convert each step to a multiplier: increase p% → 1 + p; decrease q% → 1 − q (use decimals).
   2) Multiply all the multipliers.
   3) Net percent change = final multiplier − 1, expressed as a percent.
   4) Shortcut for two steps: p + q + pq (with signs).
   – For discount-tax-coupon:
   – If all are percents of the current price: multiply the factors; order doesn’t matter.
   – If there’s a fixed-dollar amount or a special base (tax before coupon, etc.): compute in the stated order.
   – For combining group percentages:
   – Use a weighted average with group sizes: overall = (sum of successes) / (sum of totals).
   – If sizes aren’t given, you can’t find a unique overall percent (unless it’s stated or can be inferred that sizes are equal).
   – Mental tip: For small percentages, p and q, the exact two-step result is p + q + pq. If you’re sprinting, p + q is a quick approximation and pq is the “tiny correction.”

   One last wrap-up example, all-in-one
   – Original price $200. 15% off coupon, then 8% tax. What’s the final price and overall percent change?
   – Factors: 0.85 (discount), then 1.08 (tax). Final multiplier = 0.85 × 1.08 = 0.918.
   – Final price = 200 × 0.918 = $183.60.
   – Overall percent change = 0.918 − 1 = −0.082 → −8.2%.
   – If the coupon were $30 off (fixed) before tax:
   – Final = (200 − 30) × 1.08 = 170 × 1.08 = $183.60 (coincidentally the same number here), but the “effective percent” depends on the original price: 183.60/200 = 0.918 → −8.2%. If the original had been different, that effective percent would change.

   If you remember nothing else: Convert to multipliers, multiply, then convert back. For groups, weight by sizes. Your future self under test pressure will thank you.

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