1. The three prices
A single item moves through three different prices on its way from the supplier to the bag you carry out of the store. Keeping them straight is the entire game.
What the seller paid to acquire (or produce) the item. The seller's investment. The denominator for every profit and loss percentage.
The price displayed on the shelf before any discount is applied. The number on the sticker. The denominator for every discount percentage.
What the customer actually pays — MP minus whatever discount was offered. The number that hits the register before tax.
The order, in a typical retail transaction, is CP → MP → SP. The shop buys at CP, marks it up to MP, then offers a discount that drops it to SP. Whether the shop makes money depends on whether SP is still above CP after all that.
The picture
Two percentages live in this topic, and each has its own base. Profit% and loss% are computed on CP. Discount% is computed on MP. Mixing them up — taking profit% on SP, or discount% on CP — is the most common single mistake on this material.
2. Profit, loss, and their percentages
Profit and loss are the same idea seen from two sides of zero. They compare what came in (SP) to what went out (CP).
$$ \text{profit} = \text{SP} - \text{CP} \qquad (\text{when } \text{SP} > \text{CP}) $$ $$ \text{loss} = \text{CP} - \text{SP} \qquad (\text{when } \text{SP} < \text{CP}) $$The percentage versions divide by CP — the seller's investment — to turn a dollar amount into a rate that's comparable across items of different sizes.
$$ \text{profit\%} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100 \qquad \text{loss\%} = \frac{\text{CP} - \text{SP}}{\text{CP}} \times 100 $$A $30 profit on a $150 item is 20%; the same $30 profit on a $600 item is only 5%. Same dollar; very different efficiencies of capital. That's what the percentage is telling you.
Rearranged for the unknown you need
$\text{SP} = \text{CP}\,(1 + p/100)$
$\text{SP} = \text{CP}\,(1 - \ell/100)$
$\text{CP} = \dfrac{\text{SP}}{1 + p/100}$
$\text{CP} = \dfrac{\text{SP}}{1 - \ell/100}$
If an item sold for $480 at a 20% profit, the cost was not "$480 minus 20% of $480." That would be $384, which is wrong. The 20% is of CP, not SP. The correct move is to divide: $\text{CP} = 480 / 1.20 = \$400$, and you can verify with $400 \times 1.20 = 480$. Whenever you reverse a percentage change, divide by $(1 + r)$ — never multiply by $(1 - r)$.
3. Discount: a percentage off the tag
A discount is the markdown from MP to SP. Same arithmetic as profit and loss, different reference point.
$$ \text{discount} = \text{MP} - \text{SP} \qquad \text{discount\%} = \frac{\text{MP} - \text{SP}}{\text{MP}} \times 100 $$And the direct multiplier form, which is the one you'll actually use:
$$ \text{SP} = \text{MP}\,(1 - d/100) $$A $500 jacket at 15% off sells for $500 \times 0.85 = \$425$. To reverse — to recover MP from the final price and the discount — divide:
$$ \text{MP} = \frac{\text{SP}}{1 - d/100} $$"Buy m, get n free"
Promotions like "buy 3, get 2 free" are discounts in disguise. The customer takes home $m + n$ items but pays for only $m$. The effective discount is the fraction they didn't pay for:
$$ \text{effective discount} = \frac{n}{m + n} \times 100\% $$Buy 3, get 2 free → $2/5 = 40\%$ off. Buy 1, get 1 free → $1/2 = 50\%$ off. It's a cleaner way to read those promotions than trying to translate them into a discount on a single item.
4. Successive discounts (and why 20+20 ≠ 40)
Here is the single most counterintuitive fact in this topic. If a store offers 20% off, then an additional 20% off the reduced price, the total discount is not 40%. It's 36%.
Why? Because the second 20% is taken from a smaller base. On a $100 item:
- First discount: $100 \times 0.80 = \$80$.
- Second discount: $80 \times 0.80 = \$64$.
- Total paid: $64. Total off: $36. Equivalent single discount: 36%.
Algebraically, two discounts of $a\%$ and $b\%$ combine multiplicatively:
$$ \text{SP} = \text{MP}\,(1 - a/100)(1 - b/100) $$Expanding gives a tidy net-discount formula:
$$ \boxed{\;\text{net discount} = a + b - \frac{ab}{100}\;\%\;} $$The third term $-ab/100$ is the "interaction" — what's saved because the second discount isn't applied to the full original. For $a = b = 20$: net $= 20 + 20 - 400/100 = 36\%$. For $a = 25, b = 20$: net $= 45 - 5 = 40\%$. (That one happens to round to a clean number; usually it won't.)
"Take an extra 10% off our already-reduced prices!" sounds bigger than it is. A 30% original markdown followed by 10% more comes to $30 + 10 - 3 = 37\%$, not 40%. The marketing leverages the human instinct to add percentages, when in reality they always compound.
Trade discount, then cash discount
Wholesale and B2B transactions often stack discounts in a fixed order: trade discount on MP first (for the privilege of being a reseller), then cash discount on the result (for paying immediately rather than on credit). The arithmetic is the same as any successive discount — multiply the factors.
A $100 invoice with 15% trade + 5% cash discount: $100 \times 0.85 \times 0.95 = \$80.75$. Net discount: 19.25%.
The gain-then-loss asymmetry
The same multiplicative logic explains a famous "feels wrong" fact: a 10% gain followed by a 10% loss does not return you to where you started.
$$ (1 + 0.10)(1 - 0.10) = 1.10 \times 0.90 = 0.99 $$You end at 99% — a 1% net loss. In general, $(1+p)(1-p) = 1 - p^2$. The bigger $p$ gets, the more painful the asymmetry: a 50% gain followed by a 50% loss leaves you at 75%, not 100%.
5. Markup vs margin
Both words describe the same dollar profit, but they divide it by different bases — and the resulting numbers look strikingly different. Confusing one for the other is how retailers accidentally underprice products and manufacturers misjudge their distributors.
| Term | Formula | Whose perspective |
|---|---|---|
| Markup | $\dfrac{\text{SP} - \text{CP}}{\text{CP}}$ | "How much I added on top of my cost." Manufacturer / wholesaler view. |
| Margin | $\dfrac{\text{SP} - \text{CP}}{\text{SP}}$ | "What fraction of each sale is profit." Retailer / finance view. |
Consider an item with CP $= \$60$ and SP $= \$100$. The dollar profit is $40 in both cases. But:
- Markup $= 40 / 60 \approx 66.7\%$ — "I marked the price up by two-thirds of my cost."
- Margin $= 40 / 100 = 40\%$ — "Forty cents of every dollar that comes in is profit."
For any profitable sale, markup is always larger than margin. The conversion between them:
$$ \text{margin} = \frac{\text{markup}}{1 + \text{markup}}, \qquad \text{markup} = \frac{\text{margin}}{1 - \text{margin}} $$A store owner who wants a "40% profit margin" and naively sets prices at cost × 1.40 has actually only set a 40% markup — which is a 28.6% margin. To get a true 40% margin, the multiplier on cost should be $1 / (1 - 0.40) \approx 1.667$. The two phrases differ by enough to flip a profitable business into one quietly losing money.
6. Sales tax, VAT, and GST
Sales tax (and its cousins VAT, GST) sits on top of the selling price. It's paid by the customer, collected by the seller, and forwarded to the government. From the math's perspective, it's just one more multiplicative factor:
$$ \text{total paid} = \text{SP}\,(1 + t/100) $$A $150 item with 18% GST becomes $150 \times 1.18 = \$177$ at the register. The seller's profit calculation is unaffected — tax isn't part of revenue.
Reversing tax from a total
This trips people up. If a receipt shows $108 including 8% tax, the pre-tax price is not "$108 minus 8% of $108 = \$99.36$." It's the same reverse-percent trap from earlier:
$$ P = \frac{\text{total}}{1 + t/100} = \frac{108}{1.08} = \$100 $$The 8% is of $100, not of $108. The difference here is $0.64 — a small leak per receipt, but a real one if you ever need to back out a pre-tax base from a checkout total for bookkeeping or expense reporting.
Discount first, tax after
Retail convention applies discount before tax. A $200 item at 15% off plus 8% tax:
- Discount: $200 \times 0.85 = \$170$.
- Tax on discounted price: $170 \times 1.08 = \$183.60$.
The two factors commute — $(1 - 0.15)(1 + 0.08) = 0.918$ either way — but the discounted price is what shoppers and accountants think of as "the price." Tax always sits last.
7. Common pitfalls
20% + 10% successive is never 30%. The product is $0.80 \times 0.90 = 0.72$, so the equivalent single discount is 28%. Always multiply the factors, then subtract from 1 if you need the net rate.
Profit% goes on CP. Discount% goes on MP. Margin goes on SP. If you write a percentage without being sure of its base, you'll get an answer that's the right shape and the wrong number.
If $x$ is the pre-discount price and the item sold for $y$ after a $d\%$ discount, then $x = y / (1 - d/100)$. Do not compute $y \times (1 + d/100)$ — that takes a percentage of the wrong base.
A shopkeeper sells two pens for the same price; on one she made 20% profit, on the other a 20% loss. People feel this should break even. It doesn't. The two CPs are different (the profitable pen had a smaller CP than the loss-making one), and the net is always a loss of $x^2/100$% — here, $20^2/100 = 4\%$. This is a favorite exam question because the trap is so seductive.
A "30% markup" and a "30% margin" are not the same number. Markup is on cost; margin is on selling price. Pricing software, spreadsheets, and conversations between buyers and sellers all routinely confuse them — and the difference can swing a quoted price by 10% or more.
8. Worked examples
Try each before opening the solution. The goal is to recognise the shape of the problem — which prices are known, which percentage applies, and which base it's on.
Example 1 · Profit% from CP and SP
An item has CP $= \$150$ and SP $= \$180$. Find the profit percentage.
Step 1. Profit in dollars: $180 - 150 = \$30$.
Step 2. Divide by CP (the base for profit%):
$$ \text{profit\%} = \frac{30}{150} \times 100 = 20\% $$Check. $150 \times 1.20 = 180$ ✓
Example 2 · Recover CP from SP and profit%
An item sold for $480 at a 20% profit. What was the cost price?
Use $\text{CP} = \text{SP} / (1 + p/100)$:
$$ \text{CP} = \frac{480}{1.20} = \$400 $$Check. $400 \times 1.20 = 480$ ✓. (Common wrong move: $480 \times 0.80 = 384$ — that takes 20% off SP, but the 20% is supposed to be of CP.)
Example 3 · Markup followed by discount
A shopkeeper buys at $500, marks it up 50%, then offers a 20% discount. Find the selling price and the resulting profit%.
Step 1. Markup gives MP: $500 \times 1.50 = \$750$.
Step 2. Discount gives SP: $750 \times 0.80 = \$600$.
Step 3. Profit on CP:
$$ \text{profit\%} = \frac{600 - 500}{500} \times 100 = 20\% $$Example 4 · Two successive discounts
A $1000 item is offered at 20% off, then a further 10% off the already-reduced price. Find the final SP and the equivalent single discount.
Step 1. $1000 \times 0.80 = \$800$.
Step 2. $800 \times 0.90 = \$720$.
Equivalent single discount: $1000 - 720 = \$280$ off, i.e. $28\%$.
Cross-check with the formula $a + b - ab/100$ for $a=20, b=10$: $20 + 10 - 2 = 28\%$ ✓.
Example 5 · Markup needed for break-even after discount
A shop plans a 20% discount on every item. What markup on cost will leave them at break-even (SP = CP) after the discount?
Let the markup be $m$. Then $\text{MP} = \text{CP}\,(1+m)$, and after 20% off:
$$ \text{SP} = \text{CP}\,(1+m)(0.8) $$Break-even means $\text{SP} = \text{CP}$, so $(1+m)(0.8) = 1$, giving $1+m = 1.25$ and $m = 25\%$.
The lesson: if your store always discounts by 20%, marking up by 20% loses money — you need 25% to break even.
Example 6 · Tax on a discounted price
An item marked at $200 is given 15% off; 8% sales tax is added at checkout. What's the final total?
Step 1. Discount: $200 \times 0.85 = \$170$.
Step 2. Tax: $170 \times 1.08 = \$183.60$.
Order matters for narrative — the receipt will show "$170 + $13.60 tax" — but the factors commute mathematically.
Example 7 · Pre-tax price from a total
A receipt shows $216 total, including 8% sales tax. What was the pre-tax price?
Let $P$ be the pre-tax price. Then $1.08 P = 216$, so
$$ P = \frac{216}{1.08} = \$200 $$Wrong move to avoid: $0.08 \times 216 = \$17.28$ tax, giving "pre-tax = $198.72." That takes 8% of the wrong base. The 8% is of the pre-tax price, not the total.
Example 8 · Markup vs margin on the same sale
An item has CP $= \$80$ and SP $= \$100$. Find the markup on cost and the margin on selling price.
Dollar profit: $100 - 80 = \$20$.
Markup $= 20/80 = 25\%$ — what was added on top of cost.
Margin $= 20/100 = 20\%$ — the fraction of the sale that is profit.
Same dollars, different bases, different percentages — and yes, this is the cause of countless real-world pricing arguments.
Example 9 · The equal-SP trap
A shopkeeper sells two pens at the same SP. On one she makes 20% profit; on the other, a 20% loss. What is her net gain or loss on the pair?
Let each SP = $100. Then the CPs are different:
- Profitable pen: $\text{CP} = 100/1.20 \approx \$83.33$.
- Loss pen: $\text{CP} = 100/0.80 = \$125.00$.
Total CP $\approx \$208.33$, total SP $= \$200$. Net loss $\approx \$8.33$, or
$$ \frac{8.33}{208.33} \approx 4\% $$The shortcut: equal-percent profit and loss on equal SPs always nets a loss of $x^2/100$%. Here, $20^2/100 = 4\%$.