Discount Rates — Single vs Multiple (Stacked) Discounts

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Single Discount

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Single discount: \(\displaystyle P_f = P_0 (1-d)\).   With tax \(t\) after discount: \(\displaystyle P_{f,\text{tax}} = P_0 (1-d)(1+t)\).

Multiple (Stacked) Discounts

Example: 10, then 15, then 5. Each discount applies to the reduced price.

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Final price with stacked \(d_1,\dots,d_n\): \(\displaystyle P_f = P_0 \prod_{i=1}^n (1-d_i)\).
Equivalent single discount: \(\displaystyle D = 1 - \prod_{i=1}^n (1-d_i)\).

Compare Single vs Stacked

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Stacked equivalent \(D = 1 - \prod(1-d_i)\). We report pre-tax and after-tax prices.

Reverse (Target Price)

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Needed single discount (pre-tax target): \(\displaystyle D = 1 - \frac{P_f}{P_0}\). If tax \(t\) applies after discount, back out \(P_f = \tfrac{P_{\text{target}}}{1+t}\). If split into \(n\) equal stacked discounts: \(\displaystyle d = 1 - (1-D)^{1/n}\).

How Discount Rates Work: Single vs. Multiple (Stacked) Discounts

Discounts reduce a list price by a percentage or fixed amount. With a single discount of rate d, the final price is Pfinal = P0(1 − d). With multiple or stacked discounts (e.g., 10% then 15%), each discount applies to the price produced by the previous step: Pfinal = P0 × ∏(1 − di). The equivalent single discount that produces the same outcome is D = 1 − ∏(1 − di). This is why 10% + 15% is not 25% off; it’s actually 23.5%.

Order, Coupons, and “Up To” Offers

If the retailer specifies the order (e.g., “member 5% then promo 10%”), use it—order only matters when rounding rules or minimum price floors apply. For coupon stacks, check terms: some coupons apply to pre-tax price, others exclude shipping or certain categories. “Up to X%” offers usually mean different items or tiers receive different rates; treat them as separate discounts in the tool.

Taxes, Fees, and Regional Practices (GEO)

Most jurisdictions apply sales tax, VAT, or GST after discounts are taken: Pfinal, tax = Pfinal(1 + t). In some places, environmental fees, recycling charges, or service fees are added regardless of discounts. If your region includes tax in shelf prices (common with VAT/GST), the “discount” effectively reduces a tax-inclusive price. Our calculator models tax added after discounts by default. For international users, note decimal separators (12,5% vs 12.5%) and currency rounding conventions; always enter percentages as whole numbers (e.g., 12.5 for 12.5%).

Business Use: Promotions, Wholesale, and MAP

Merchants often combine a base promotion with a loyalty or payment-method discount. Wholesalers may publish trade discounts (e.g., 30/10) that are explicitly stacked. If you must respect a minimum advertised price (MAP) or margin target, use the Reverse (Target Price) tab: pick a target final (with or without tax), and compute the single or equal stacked discount(s) that hit the goal without violating constraints.

Common Pitfalls

  • Adding percentages directly. Stacks compound; do not sum percentages.
  • Rounding at the wrong step. Some policies round after each discount; others round only once at checkout. Results can differ.
  • Minimum spend / exclusions. A discount might not apply to shipping, subscription items, or already marked-down goods.
  • Returns and exchanges. Refunds usually reflect the discounted price; exchanges may reapply active promos differently.

Worked Example

Original price = 250. Discounts: 10%, then 15%, then 5%. Product factor: (1 − 0.10) × (1 − 0.15) × (1 − 0.05) = 0.90 × 0.85 × 0.95 = 0.72675. Final pre-tax price = 250 × 0.72675 = 181.6875. Equivalent single discount: D = 1 − 0.72675 = 27.325%. With 7.5% tax after discounts, final = 181.6875 × 1.075 ≈ 195.82.

When to Use Single vs. Stacked Discounts

If you control the promotion, an equivalent single discount is cleaner for messaging and customer perception. If you’re modeling real-world checkout flows (multiple codes, loyalty tiers, bulk breaks), stacked discounts mirror reality and make it easier to audit edge cases.

Tip for Power Users

To split a required overall discount D into n equal stacked discounts, use d = 1 − (1 − D)1/n. This is handy when negotiating staged concessions or designing tiered promos.

5 Fun Facts about Discounts

Stacking is multiplicative, not additive

10% then 10% is 19% off, not 20%. Stacks multiply their “keep” factors, so (1−0.10)×(1−0.10)=0.81 → 19% equivalent.

Compound cuts

Order only matters with rounding rules

Pure math ignores order, but if a checkout rounds after each discount, swapping “15% then 5%” vs “5% then 15%” can change the price by a few cents.

Sequence quirks

Tax timing flips headline savings

Most places apply tax after discounts, but fixed fees (shipping, environmental) often sit outside the discount. “25% off everything” rarely means 25% off the whole bill.

After the cut

“Up to” is usually far less

A banner shouting “up to 50% off” might average nearer 15–25% across a cart. The true equivalent discount depends on your mix.

Real rate

Margin recovery is asymmetric

A 20% price drop needs more than a 20% hike to recover; you need a 25% markup to get back to a 20% margin. Percentages don’t mirror perfectly.

Percent trap

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