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Profit, Loss, and Discount
The Price Triangle
Every transaction revolves around two fundamental values: the cost price (what the seller pays) and the selling price (what the buyer pays). The relationship between them determines whether money was gained or lost.
$$\text{Profit} = \text{SP} - \text{CP} \quad | \quad \text{Loss} = \text{CP} - \text{SP}$$
Profit occurs when the selling price exceeds the cost price; loss occurs when the cost price exceeds the selling price.
$\text{CP}$=Cost price — the expense to acquire or produce the item(currency)
$\text{SP}$=Selling price — the amount received from the buyer(currency)
$\text{SP} = \text{CP}$
→No profit, no loss — the transaction is neutral.
Profit Direction: When SP > CP, the seller gains. When CP > SP, the seller loses. The signs tell the whole story.
Baseline Is Always CP: Every percentage — profit, loss, markup, discount — is calculated on the cost price (or a closely related base) unless stated otherwise.
Real-World Overheads: The true cost price may include transport, storage, and repair expenses, not just the purchase price.
Before a sale reaches the customer, stores often display a marked price (also called list price or tag price). This is the price printed on the label — the "before discount" number you see on shelves.
$$\text{MP} = \text{CP} + \text{Markup}$$
The marked price is the cost price plus the seller's desired markup — what the seller hopes to charge before any discounts.
$\text{MP}$=Marked price (tag price)(currency)
$\text{CP}$=Cost price(currency)
$\text{Markup}$=The amount added above cost to set the tag price(currency)
Markup vs Profit: Markup is calculated on CP and sets the MP. Profit is calculated on CP but uses the actual SP (after any discount). They are different.
Why Use MP?: Businesses set an inflated tag price so they can offer discounts without losing money — a psychological sales tactic.
Profit and Loss Percentages
Expressing profit as a percentage of the cost price lets you compare the profitability of different items regardless of their absolute prices. A Rs 50 profit on a Rs 200 item is far more impressive than the same Rs 50 on a Rs 5,000 item.
$$\text{Profit \%} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100 = \frac{\text{Profit}}{\text{CP}} \times 100$$
Profit percentage measures how much gain was made relative to the amount invested.
$\text{Profit \%}$=Profit expressed as a percentage of cost price(%)
$\text{SP}$=Selling price(currency)
$\text{CP}$=Cost price(currency)
$\text{SP} = \text{CP}$
→Profit % = 0% (no gain, no loss).
$\text{SP} = 2 \times \text{CP}$
→Profit % = 100% (selling for double the cost).
Base Is Always CP: Profit and loss percentages are always calculated on the cost price, never on the selling price.
Quick CP from SP: If profit % is known, $\text{CP} = \frac{\text{SP} \times 100}{100 + \text{Profit \%}}$. For a loss, $\text{CP} = \frac{\text{SP} \times 100}{100 - \text{Loss \%}}$.
Equal Gain and Loss: If two items are sold at the same SP, one at a% profit and the other at a% loss, there is always an overall loss of $\left(\frac{a}{10}\right)^2$%.
When a seller is forced to sell below cost — due to damage, competition, or urgency — the loss percentage quantifies the shortfall relative to the original investment.
$$\text{Loss \%} = \frac{\text{CP} - \text{SP}}{\text{CP}} \times 100 = \frac{\text{Loss}}{\text{CP}} \times 100$$
Loss percentage measures how much of the invested money was not recovered.
$\text{Loss \%}$=Loss expressed as a percentage of cost price(%)
$\text{CP}$=Cost price(currency)
$\text{SP}$=Selling price (which is less than CP)(currency)
$\text{SP} = 0$
→Loss % = 100% (total loss).
Negative Profit = Loss: A profit of -30% is the same as a loss of 30%. The sign convention keeps the formulas unified.
Loss Recovery: To recover from a 20% loss and still make a 20% overall profit, you need to earn $\frac{120}{80} \times 100 = 50$% profit on the reduced selling price — not 20%.
Markup and Its Percentage
Markup is the amount a business adds to its cost to set the marked price. Unlike profit (which depends on the actual selling price), markup is based on the cost price and determines the tag price before any discount is applied.
$$\text{Markup \%} = \frac{\text{MP} - \text{CP}}{\text{CP}} \times 100$$
Markup percentage tells you how much the tag price is inflated above the cost price.
$\text{Markup \%}$=Markup expressed as a percentage of cost price(%)
$\text{MP}$=Marked price(currency)
$\text{CP}$=Cost price(currency)
$\text{Markup \%} = 100$
→$\text{MP} = 2 \times \text{CP}$ (the tag price is double the cost).
Markup on CP: Like profit and loss, markup percentage is always calculated on the cost price.
Markup vs Profit: If MP = Rs 600 and CP = Rs 400, markup is 50%. But if a 10% discount is offered, SP = Rs 540 and actual profit = $\frac{540 - 400}{400} \times 100 = 35\%$ — different from markup.
Chain Formula: $\text{SP} = \text{CP} \times \left(1 + \frac{\text{Markup \%}}{100}\right) \times \left(1 - \frac{\text{Discount \%}}{100}\right)$ — this connects CP, markup, discount, and SP in one expression.
Discount and Discount Percentage
A discount is a reduction from the marked price offered to the buyer. It is the difference between what the item was tagged at and what it actually sells for. Businesses use discounts to attract customers, clear inventory, or compete in the market.
$$\text{Discount} = \text{MP} - \text{SP} \quad | \quad \text{Discount \%} = \frac{\text{MP} - \text{SP}}{\text{MP}} \times 100$$
Discount is always calculated on the marked price (not the cost price). It is a reduction from the tag price.
$\text{MP}$=Marked price (tag price)(currency)
$\text{SP}$=Selling price (what the buyer actually pays)(currency)
$\text{Discount \%}$=Discount expressed as a percentage of the marked price(%)
Base Is MP, Not CP: Discount percentage is calculated on the marked price. This is the single most common source of errors in these problems.
Profit After Discount: Even after giving a discount, the seller can still make a profit if the markup was large enough: Profit = SP - CP.
Quick SP from MP: $\text{SP} = \text{MP} \times \left(1 - \frac{\text{Discount \%}}{100}\right)$. For example, 25% off Rs 800 gives SP = $800 \times 0.75 = 600$.
Successive Discounts
When multiple discounts are applied one after another (not added together), the actual reduction is less than the sum of the individual discounts. Each discount is calculated on the remaining price after the previous discount was applied.
$$\text{SP} = \text{MP} \times \left(1 - \frac{a}{100}\right)\left(1 - \frac{b}{100}\right)\left(1 - \frac{c}{100}\right)$$
Each successive discount shrinks a smaller base — the price remaining after the previous discount.
$a, b, c$=The discount percentages applied in sequence(%)
$\text{MP}$=Original marked price(currency)
$\text{SP}$=Final selling price after all discounts(currency)
$a = b$
→Two equal successive discounts of $a$% are equivalent to a single discount of $\left(2a - \frac{a^2}{100}\right)$%.
$a + b + c$ is compared to effective discount
→The effective discount is always less than $a + b + c$ because each percentage acts on a shrinking base.
Two-Discount Shortcut: For discounts of $a$% and $b$%, the effective single discount is $\left(a + b - \frac{ab}{100}\right)$%. Example: 20% + 10% effective = $20 + 10 - 2 = 28$% (not 30%).
Three-Discount Formula: For $a$%, $b$%, $c$%: effective discount $= a + b + c - \frac{ab + bc + ca}{100} + \frac{abc}{10000}$.
Order Does Not Matter: Successive discounts are commutative — 20% then 10% gives the same SP as 10% then 20%.
Connecting CP, SP, and MP
In most business problems, the cost price, marked price, and selling price are linked through markup and discount. Understanding how to move between any two of these three values is essential for solving combined problems.
$$\text{SP} = \text{MP}\left(1 - \frac{d}{100}\right) = \text{CP}\left(1 + \frac{p}{100}\right)$$
The selling price can be reached from the marked price (via discount) or from the cost price (via profit). Setting them equal connects all three.
$d$=Discount percentage(%)
$p$=Profit percentage(%)
Finding MP Given CP and Discount for No Loss: If CP = 500 and discount = 20% with zero profit, then $\text{MP} = \frac{500}{0.80} = 625$.
Finding Discount for Target Profit: If CP = 400, MP = 600, and the seller wants a 10% profit, then target SP = $400 \times 1.10 = 440$, so discount $= \frac{600 - 440}{600} \times 100 = \frac{160}{600} \times 100 \approx 26.67$%.
Profit or Loss After Discount: Profit $= \text{SP} - \text{CP} = \text{MP}(1 - d/100) - \text{CP}$. The sign of this expression determines profit or loss.
Many word problems give you the selling price and the profit or loss percentage, asking you to work backwards to find the original cost price. This is a frequent pattern in business aptitude questions.
$$\text{CP} = \frac{\text{SP} \times 100}{100 + p} \qquad \text{CP} = \frac{\text{SP} \times 100}{100 - l}$$
Given the selling price and profit/loss percentage, divide SP by the multiplier to recover the original cost price.
$p$=Profit percentage(%)
$l$=Loss percentage(%)
Profit Multiplier: When SP is known and profit = p%, think of CP as SP divided by $1 + p/100$. For profit = 25%, CP = SP / 1.25.
Loss Multiplier: When SP is known and loss = l%, CP = SP divided by $1 - l/100$. For loss = 20%, CP = SP / 0.80.
Fractions Trick: Using fractional equivalents speeds up calculations. 25% profit → CP = SP × $\frac{4}{5}$. 33\frac{1}{3}$% loss → CP = SP × $\frac{3}{2}$.