The Mathematics of Discounts, Markups, and Sales Tax
A store advertises "20% off, plus take an extra 10% off at checkout." Most shoppers do the mental math and think 30% off. The real discount is 28%.
Meanwhile, the store owner who marks up a product by 50% isn't earning a 50% margin. It's 33.3%. Retail pricing is full of these small percentage traps, and every one of them comes down to a single idea: percentages multiply.
Tax First or Discount First? It Doesn't Matter
People sometimes argue about whether a discount should be applied before or after sales tax. Mathematically, for a percentage discount, the order makes no difference.
A $100 item, 20% off, with 8% sales tax:
Discount first: 100 × 0.80 = 80 → 80 × 1.08 = $86.40
Tax first: 100 × 1.08 = 108 → 108 × 0.80 = $86.40
Both give $86.40, because multiplication is commutative: a × b = b × a. (Order does matter for a fixed-dollar coupon, such as $10 off, and many jurisdictions have specific rules about whether coupons reduce the taxable amount.)
Stacked Discounts Multiply
A 20% discount means you pay 80%. An extra 10% off means you pay 90% of that:
Price paid = 0.80 × 0.90 = 0.72 → 72% of the original
Total discount = 28%
The general formula for discounts d₁, d₂, …:
Total discount = 1 − (1 − d₁)(1 − d₂)…(1 − dₙ)
| Stacked discounts | Naive sum | Actual total |
|---|---|---|
| 20% + 10% | 30% | 28% |
| 30% + 20% | 50% | 44% |
| 50% + 50% | 100% ("free!") | 75% |
| 40% + 25% + 10% | 75% | 59.5% |
Stacking two 50% discounts never makes something free. You'd pay 25%.
Decoding Common Deals
"Buy One, Get One 50% Off"
You pay 100% + 50% = 150% for 200% of the goods:
Effective discount = 1 − 150/200 = 25% off each item
"Buy Two, Get One Free"
You pay for 2 and receive 3:
Effective discount = 1 − 2/3 ≈ 33.3% off
"Spend $100, Save $20"
That's 20% off if you spend exactly $100. Spend $140 and it's only 14.3%. The effective percentage falls as your total rises.
Markup vs. Margin
This is the most important distinction in retail math, and the one most often confused.
- Markup is profit as a percentage of cost
- Margin is profit as a percentage of selling price
An item costs a store $60 and sells for $90. Profit is $30:
Markup = 30 / 60 = 50%
Margin = 30 / 90 = 33.3%
Same product, same profit, two different percentages. The conversions are:
Margin = Markup / (1 + Markup)
Markup = Margin / (1 − Margin)
| Markup | Margin |
|---|---|
| 25% | 20% |
| 50% | 33.3% |
| 100% | 50% |
| 200% | 66.7% |
A 100% markup, called keystone pricing in retail, gives a 50% margin. No markup, however large, can reach a 100% margin, because that would mean the item cost nothing.
Why "70% Off" Can Still Be Profitable
Say a clothing retailer buys a shirt for $12 and prices it at $60, a 400% markup and an 80% margin. At 70% off, the sale price is $18. The store still makes $6 per shirt, a 50% markup on cost.
The deep discount is the markup being unwound, not a loss. That's why outlet stores and seasonal sales can advertise enormous percentages.
An Insider Reference: J.C. Penney's Pricing Experiment
In early 2012, J.C. Penney's new CEO Ron Johnson, who had previously led Apple's retail stores, tried to end what he saw as fake discounting. The company scrapped most sales and coupons in favor of "Fair and Square" everyday low prices. The math of the final price was often the same or better for shoppers.
Customers hated it. J.C. Penney's revenue fell by about 25% in 2012, and Johnson was replaced in April 2013. The lesson for pricing strategists: people don't just respond to the final number. They respond to the size of the percentage they save, even when that percentage is calculated from an inflated "original" price.
Research on left-digit bias tells a similar story. A 2005 study by Manoj Thomas and Vicki Morwitz found that people judge $2.99 as noticeably cheaper than $3.00, even though the gap is a single cent.
Two Concepts Worth Knowing
Multiplicative Inverses
To undo a percentage change, divide by the multiplier. To find a pre-tax price from a total with 8% tax, divide by 1.08. To find the original price before 25% off, divide by 0.75. Multiplication and division are inverse operations.
Proportional Reasoning
Discounts, taxes and markups are all ratios. Setting up a proportion such as cost / price = 60 / 90 is often the clearest way to avoid confusion.
Quick Answer: How Do You Calculate Stacked Discounts?
Convert each discount into the fraction you pay (20% off becomes 0.80), multiply them together, and subtract the result from 1. For 20% off plus an extra 10% off: 0.80 × 0.90 = 0.72, so the total discount is 28%, not 30%.
Try Them Yourself
- Basic Calculator: chain discounts and tax quickly
- Multiplication Tables: build fast mental math for percentages
- Division Tables: reverse a price back to its original
- Algebra Formulas: solve for unknown prices
- Financial Formulas: percentage and profit formulas
- Why Percentages Are So Easy to Get Wrong: more percentage traps
Next time you see "extra 15% off," multiply instead of adding. Your answer will be right, and you'll know exactly how much the deal is worth.