Stacked discounts: 30% off ≠ 20% off then 10% off
“30% off” vs “20% off, then another 10% off” — which is better? The first, but the math is counterintuitive. This article walks through discount arithmetic.
Basic discount
20% off means:
discounted = original × (1 - 0.20) = original × 0.80$10 with 20% off → $8.
Stacking: not additive
“20% off, then another 10% off”:
10 × 0.80 × 0.90 = 7.20“20% + 10% = 30%” would be $7. Stacking comes out 20¢ higher.
| Method | Multiplier | $10 → |
|---|---|---|
| 30% off | × 0.70 | $7.00 |
| 20% off then 10% off | × 0.80 × 0.90 | $7.20 |
| 10% off then 20% off | × 0.90 × 0.80 | $7.20 |
Order doesn’t matter (multiplication commutes). But it’s not addition.
Coupon plus sale
“30%-off sale item with a 10%-off coupon”:
10 × 0.70 × 0.90 = 6.30Feels like “40% off” but it’s only ~37% off.
Cashback (points) vs discount
“10% cashback” and “10% discount” are different:
- 10% off — pay $9 for a $10 item.
- 10% cashback — pay $10 and get $1 back as points.
- That $1 is worth $1 toward your next purchase, so effective discount is 10 / 110 ≒ 9.09%.
Effective discount rate from cashback is smaller than the nominal:
effective = cashback / (1 + cashback)- 10% cashback → 9.09% effective.
- 20% cashback → 16.67% effective.
- 50% cashback → 33.33% effective.
Why “5× points day” is less amazing than the marketing implies.
Order with sales tax
Sales tax 10% example:
Discount on pre-tax, then tax
10 × 0.80 × 1.10 = 8.80Discount on tax-inclusive
11 × 0.80 = 8.80Mathematically the same (× 0.80 × 1.10 = × 1.10 × 0.80).
But rounding can differ:
- 10 × 1.10 = 11, × 0.80 = 8.80
- 10 × 0.80 = 8, × 1.10 = 8.80
For integer prices, you can be off by one cent depending on order.
“Up to 50% off”
“Up to 50% off” usually means:
- A few items at 50% off.
- Most items at 10–20% off.
- The minimum discount isn’t disclosed.
Legal in many places, but consumer-protection laws in some jurisdictions (e.g., California) restrict it.
“Buy one, get one half off”
“First at full, second at 50% off”:
- First — $10
- Second — $5
- Total — $15
- Original total — $20
- Effective discount — (20 − 15) / 20 = 25%
Feels like “50% off” but it’s 25% off two items.
“Buy 2, get 1 free”:
- Original total — $30
- Paid — $20
- Effective discount — 33.33%
Volume discounts
Tiered pricing by quantity:
| Quantity | Unit price |
|---|---|
| 1–9 | $10 |
| 10–49 | $9 |
| 50–99 | $8.50 |
| 100+ | $8 |
Usually the new tier applies to the whole order (not just the units above the threshold). Some contracts apply tier-by-tier — read the fine print.
Seasonal sale stacking
Black Friday, year-end, etc.:
- Item already 30% off.
- “Today only, additional 20% off.”
- Members-only: another 10%.
10 × 0.70 × 0.80 × 0.90 = 5.04Feels like 60% off, actually 49.6%. Three discounts don’t add up to their sum.
Financing offers
“0% APR for 12 months”:
- Lower monthly cost, but the merchant absorbs the interest.
- Hence “no other discounts with financing” is common — would double-dip the merchant.
Rounding rules
Discount math creates fractions. Pick a rounding rule:
- Floor / truncate — consumer-friendly (typical retail).
- Ceiling — merchant-friendly.
- Round half-up — neutral.
$3.33 with 20% off = $2.664:
- Floor — $2.66.
- Ceiling — $2.67.
- Round — $2.66.
POS systems must standardize this.
Behavioral economics tidbits
How discounts are framed matters:
- “$1 off” vs “10% off” — at low prices, percent feels bigger.
- “10% off” vs “$10 off” — at high prices, dollars feel bigger.
- “Buy one, get one half off” vs “25% off two items” — same math, BOGO sounds better.
The perceived deal often diverges from the actual deal. Always do the math.
Summary
- Stacked discounts multiply (not add).
- Cashback’s effective rate is lower than the nominal.
- BOGO half-off is 25% off two items.
- Pre-tax vs tax-inclusive discount order is math-equivalent (rounding aside).
- “Total 60% off” sale framings are usually 50% or less in reality.
For discount calculations, the discount calculator on this site handles single and stacked rates.