Double Discount Calculator

Double discounts (also called stacked or successive discounts) are calculated by applying each discount to the price AFTER the previous discount, not to the original price. THE FORMULA: Final Price = Original Price × (1 - d₁) × (1 - d₂). Where d₁ and d₂ are the discounts as decimals. STEP-BY-STEP EXAMPLE: Original price: $100. First discount: 20%. Second discount: 10%. Step 1: Apply first discount. $100 × (1 - 0.20) = $100 × 0.80 = $80. Step 2: Apply second discount to $80 (not $100!). $80 × (1 - 0.10) = $80 × 0.90 = $72. Final price: $72. EQUIVALENT SINGLE DISCOUNT: You can calculate the equivalent single discount: Equivalent = 1 - (1 - d₁)(1 - d₂). = 1 - (0.80)(0.90). = 1 - 0.72. = 0.28 or 28%. So 20% + 10% stacked = 28% total (NOT 30%). KEY INSIGHT: The second discount always saves less than you might expect because it's applied to an already-reduced price. This is why 20% + 10% ≠ 30%.

This is one of the most common misconceptions in discount math. The reason 20% + 10% ≠ 30% is because the second discount is applied to the already-reduced price, not the original price. THE MATH EXPLAINED: First discount (20%): Applied to the full $100 → Saves $20. Second discount (10%): Applied to $80 (not $100) → Saves $8 (not $10). Total savings: $20 + $8 = $28 (not $30). That's why the total is 28%, not 30%. ANOTHER WAY TO THINK ABOUT IT: When you take 20% off, you pay 80% of the original. When you take 10% off that, you pay 90% of that 80%. 80% × 90% = 72% of original = 28% off. THE "LOST" DISCOUNT: Where did that extra 2% go? The 2% "loss" comes from: 10% of the $20 you already saved = $2. You can't save on money you've already saved! GENERAL FORMULA: For any two discounts d₁ and d₂: Actual total = d₁ + d₂ - (d₁ × d₂). = 20% + 10% - (20% × 10%). = 30% - 2%. = 28%.

For exactly two discounts, the order does NOT matter mathematically. However, there are important caveats to understand. WHY ORDER DOESN'T MATTER (TWO DISCOUNTS): Multiplication is commutative: a × b = b × a. Example with 20% and 10% discounts: 20% first, then 10%: $100 × 0.80 × 0.90 = $72. 10% first, then 20%: $100 × 0.90 × 0.80 = $72. Same result! MATHEMATICALLY: (1 - d₁)(1 - d₂) = (1 - d₂)(1 - d₁). Both equal 1 - d₁ - d₂ + d₁d₂. WHEN ORDER MIGHT MATTER: 1. Rounding: If prices round to nearest cent after each discount. Different order → Different rounding → Slightly different final price. Usually just pennies difference. 2. Three or More Discounts: Still mathematically equivalent. But psychological perception may differ. 3. Conditional Discounts: "Take 10% off if you spend over $50 after other discounts". Order could affect eligibility for conditional discount. 4. Caps or Limits: "Maximum discount $50" on first discount. Order would matter with caps. PRACTICAL TIP: In real shopping, always check if there are conditions or caps that make order matter. For pure math, order is irrelevant for multiple percentage discounts.

The equivalent single discount is the one discount percentage that would give you the same final price as applying multiple successive discounts. FORMULA FOR TWO DISCOUNTS: Equivalent = 1 - (1 - d₁)(1 - d₂). Or: Equivalent = d₁ + d₂ - (d₁ × d₂). FORMULA FOR THREE DISCOUNTS: Equivalent = 1 - (1 - d₁)(1 - d₂)(1 - d₃). EXAMPLE: 20% and 10% discounts: Method 1: 1 - (0.80)(0.90) = 1 - 0.72 = 0.28 = 28%. Method 2: 0.20 + 0.10 - (0.20 × 0.10) = 0.30 - 0.02 = 0.28 = 28%. THREE DISCOUNTS EXAMPLE: 30%, 20%, and 10% discounts: Equivalent = 1 - (0.70)(0.80)(0.90). = 1 - 0.504. = 0.496 or 49.6%. Not 60% (simple sum)! QUICK REFERENCE TABLE: | First | Second | Simple Sum | Actual Equivalent |. | 10% | 10% | 20% | 19% |. | 20% | 10% | 30% | 28% |. | 20% | 20% | 40% | 36% |. | 30% | 20% | 50% | 44% |. | 50% | 25% | 75% | 62.5% |. | 50% | 50% | 100% | 75% |. IMPORTANT INSIGHT: Two 50% discounts don't give you 100% off (free). They give you 75% off. You pay 50% of 50% = 25% of original price.

Mathematically, a double discount and its equivalent single discount give the EXACT same result. However, there are practical considerations. MATHEMATICAL EQUIVALENCE: 20% + 10% (stacked) = 28% (single). Both result in paying 72% of original price. Final prices are identical. PSYCHOLOGICAL PERCEPTION: Double discounts often FEEL better to consumers. "20% + extra 10%" sounds like 30%. Retailers know this - it's a marketing strategy. The "extra" discount triggers additional excitement. PRACTICAL DIFFERENCES: 1. Coupon Stacking: Sometimes you can stack coupons that don't work together. Getting 20% + 10% vs. a 30% coupon that doesn't exist. 2. Timing: Sales at different times. 20% now + 10% clearance later. 3. Qualification: Might qualify for 20% but not 30%. Employee + sale discount combination. 4. Maximum Limits: Two 25% coupons (capped at $50 each). vs. one 50% coupon (capped at $50 total). Two = up to $100 savings; One = up to $50 savings. RETAILER PERSPECTIVE: Double discounts can: Move more inventory (psychological effect). Allow price discrimination. Clear old stock progressively. Make sales events seem more generous. CONSUMER TIP: Always calculate the actual equivalent discount. Compare stacked deals to single-discount alternatives. Don't be fooled by "extra" marketing language.

Sometimes discounts apply to different bases (e.g., item discount + order discount). This requires careful calculation. SAME-BASE DISCOUNTS (Simple): Both discounts apply to product price. Use standard stacking formula. Example: 20% off item + 10% off item = 28% off item. DIFFERENT-BASE DISCOUNTS: Item Discount + Order Discount: Calculate item discount first. Then apply order discount to the item total. Example: $100 item at 20% off = $80. $80 + $50 second item = $130 order. 10% off order: $130 × 0.90 = $117. Shipping Discounts: Some discounts exclude shipping. Calculate product total, then add shipping. Example: $100 item at 20% off = $80. $10 shipping (no discount applies). Total: $90. Tax Considerations: Discounts typically apply before tax. $100 at 20% off = $80. 8% tax on $80 = $6.40. Total: $86.40. MULTIPLE ITEM SCENARIOS: Buy One Get One (BOGO): "50% off second item" is NOT same as 25% off total. $100 + $50 (half of $100) = $150. vs. $100 + $100 = $200 at 25% off = $150. Same in this case, but check the terms! QUANTITY DISCOUNTS: "10% off when you buy 3+". Applied to all items or just extras?. Read the fine print! LOYALTY + SALE COMBINATIONS: Some stack, some don't. Check if "cannot be combined with other offers".

Theoretically, you can never reach 100% off with percentage discounts alone, but you can get very close. MATHEMATICAL LIMIT: Each discount multiplies by a fraction less than 1. No matter how many you stack, you never reach zero. Examples: 50% + 50% = 75% off (pay 25%). 50% + 50% + 50% = 87.5% off (pay 12.5%). 50% × 10 times = 99.9% off (pay 0.1%). THE FORMULA: After n discounts of d each: Remaining = (1 - d)ⁿ. Discount = 1 - (1 - d)ⁿ. EXAMPLES OF STACKED 50% DISCOUNTS: | # of 50% | Pay | Total Discount |. | 1 | 50% | 50% |. | 2 | 25% | 75% |. | 3 | 12.5% | 87.5% |. | 4 | 6.25% | 93.75% |. | 5 | 3.125% | 96.875% |. | 10 | 0.1% | 99.9% |. APPROACHING BUT NEVER REACHING 100%: This is mathematically similar to Zeno's Paradox. Each discount takes you half the remaining distance. You get closer and closer but never arrive at zero. PRACTICAL LIMITS: Real stores have: Maximum discount policies. Minimum price rules. Anti-fraud measures. Rounding (might round to $0.00 eventually). REAL-WORLD SCENARIO: Most aggressive realistic stack: Employee discount (20%). + Sale (40%). + Coupon (20%). + Loyalty (10%). = 1 - (0.80)(0.60)(0.80)(0.90). = 1 - 0.3456. = 65.44% total discount. Still leaves you paying 34.56% of original.

Retailers strategically use double discounts as a psychological marketing tool. Understanding this helps you shop smarter. COMMON MARKETING TACTICS: 1. "Take an EXTRA X% Off". "40% off + take an extra 20% off!". Sounds like 60%, actually 52%. The word "extra" is key psychological trigger. 2. "Clearance + Additional Discount". Already marked down 50%. Take an additional 30% off. Sounds massive (80%?), actually 65%. 3. "Sale on Sale". Double discount weekends. Creates urgency and excitement. Often no better than equivalent single discount. 4. "Stack Your Savings". Credit card discount + sale + coupon. Multiple small discounts feel bigger. 10% + 10% + 10% sounds like 30%, is 27.1%. WHY RETAILERS DO THIS: Psychological impact: Multiple discounts feel like more savings. Perceived value: "Three ways to save!" sounds generous. Price anchoring: Original price seems even higher. Inventory movement: Creates shopping excitement. Customer engagement: Requires participation (applying codes). HOW TO SHOP SMART: Always calculate the equivalent single discount. Compare to competitors' single discounts. Check if "extra" discount has conditions. Calculate final price, not just percentages. Watch for "up to" language (maximum, not guaranteed). EXAMPLE ANALYSIS: "50% off + extra 25% off + 10% loyalty discount". Consumer might think: 85% off!. Reality: 1 - (0.50)(0.75)(0.90) = 66.25% off. Still good, but not 85%. HONEST MARKETING: Some retailers now show: "60% off" (equivalent single discount). More transparent. Easier for consumers. Building trust. THE BOTTOM LINE: Double discounts are real savings. But always calculate the actual percentage. Don't let marketing language inflate your expectations.