Back to Course
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 × $\frac{3}{2}$.
Application Strategy
Worked Examples
Problem Statement
A shopkeeper buys a crate of mangoes for Rs 1,200 and sells the entire crate for Rs 1,560. What is the profit percentage?
profitcost priceselling price
Strategic Toolkit
$$\text{Profit \%} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100$$
Step 1
Identify CP and SP
CP = Rs 1,200, SP = Rs 1,560
Step 2
Calculate profit
Profit = SP − CP = 1,560 − 1,200 = Rs 360
$$\text{Profit} = 1560 - 1200 = 360$$
Step 3
Calculate profit percentage
Profit % = (360 / 1200) × 100 = 30%
$$\text{Profit \%} = \frac{360}{1200} \times 100 = 30\%$$
Problem Statement
A bookstore sells a textbook for Rs 640 and incurs a 20% loss on it. What was the cost price of the textbook?
losscost priceselling price
Strategic Toolkit
$$\text{CP} = \frac{\text{SP} \times 100}{100 - \text{Loss \%}}$$
Step 1
Identify given values
SP = Rs 640, Loss = 20%
Step 2
Apply the reverse formula
CP = (640 × 100) / (100 − 20) = 64000 / 80 = Rs 800
$$\text{CP} = \frac{640 \times 100}{100 - 20} = \frac{64000}{80} = 800$$
Step 3
Verify
Loss = 800 − 640 = Rs 160. Loss % = (160 / 800) × 100 = 20%. Confirmed.
Problem Statement
An electronics store marks a laptop at Rs 80,000 and offers two successive discounts of 20% and 10%. What is the final selling price, and what is the effective single discount percentage?
successive discountmarked priceselling price
Strategic Toolkit
$$\text{SP} = \text{MP} \times \left(1 - \frac{20}{100}\right)\left(1 - \frac{10}{100}\right)$$
Step 1
Apply the first discount
After 20% off: 80,000 × 0.80 = Rs 64,000
$$80{,}000 \times 0.80 = 64{,}000$$
Step 2
Apply the second discount
After 10% off the reduced price: 64,000 × 0.90 = Rs 57,600
$$64{,}000 \times 0.90 = 57{,}600$$
Step 3
Calculate effective discount
Total discount = 80,000 − 57,600 = Rs 22,400. Effective % = (22,400 / 80,000) × 100 = 28%
$$\text{Effective discount} = 20 + 10 - \frac{20 \times 10}{100} = 28\%$$
Problem Statement
A wholesaler marks goods at 50% above cost and then offers a 25% discount during a sale. If the cost price of a batch is Rs 4,000, find the marked price, selling price, and profit percentage.
markupdiscountprofitcost priceselling pricemarked price
Strategic Toolkit
$$\text{MP} = \text{CP}\left(1 + \frac{m}{100}\right), \quad \text{SP} = \text{MP}\left(1 - \frac{d}{100}\right)$$
Step 1
Find the marked price
MP = 4,000 × (1 + 50/100) = 4,000 × 1.50 = Rs 6,000
$$\text{MP} = 4000 \times 1.50 = 6000$$
Step 2
Find the selling price after discount
SP = 6,000 × (1 − 25/100) = 6,000 × 0.75 = Rs 4,500
$$\text{SP} = 6000 \times 0.75 = 4500$$
Step 3
Calculate profit percentage
Profit = 4,500 − 4,000 = Rs 500. Profit % = (500 / 4,000) × 100 = 12.5%
$$\text{Profit \%} = \frac{500}{4000} \times 100 = 12.5\%$$
Problem Statement
A trader sells two identical watches. One is sold at a 20% profit and the other at a 20% loss. If the selling price of each watch is Rs 600, find the overall profit or loss percentage.
profitlosscost priceselling price
Strategic Toolkit
$$\text{Overall loss \%} = \left(\frac{a}{10}\right)^2 \%$$
Step 1
Find CP of first watch (sold at profit)
CP₁ = (600 × 100) / (100 + 20) = 60,000 / 120 = Rs 500
$$\text{CP}_1 = \frac{600 \times 100}{120} = 500$$
Step 2
Find CP of second watch (sold at loss)
CP₂ = (600 × 100) / (100 − 20) = 60,000 / 80 = Rs 750
$$\text{CP}_2 = \frac{600 \times 100}{80} = 750$$
Step 3
Calculate overall result
Total CP = 500 + 750 = Rs 1,250. Total SP = 600 + 600 = Rs 1,200. Loss = Rs 50.
$$\text{Loss \%} = \frac{50}{1250} \times 100 = 4\% = \left(\frac{20}{10}\right)^2 \%$$
Memorize formulas
Profit %
Profit as a fraction of cost price
Formula
$\text{Profit \%} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100$
Loss %
Loss as a fraction of cost price
Formula
$\text{Loss \%} = \frac{\text{CP} - \text{SP}}{\text{CP}} \times 100$
Markup %
Markup as a fraction of cost price
Formula
$\text{Markup \%} = \frac{\text{MP} - \text{CP}}{\text{CP}} \times 100$
Discount %
Discount as a fraction of marked price (not CP)
Formula
$\text{Discount \%} = \frac{\text{MP} - \text{SP}}{\text{MP}} \times 100$
Successive Discount (2)
Effective single discount from two successive discounts
Formula
$a + b - \frac{ab}{100}$
Review
1
A fruit vendor buys a crate of mangoes for Rs. 1,200 and sells them for Rs. 1,560. What is the profit percentage?
Ans
30%
Review
Tap to reveal
Calculate the profit
Subtract the cost price from the selling price.
$$\text{Profit} = 1560 - 1200 = 360$$
Express as a percentage of cost price
Divide the profit by the cost price and multiply by 100.
$$\text{Profit \%} = \frac{360}{1200} \times 100 = 30\%$$
2
A retailer sells a blender for Rs. 4,200, incurring a loss of 16\frac{2}{3}%. What was the cost price of the blender?
Ans
Rs. 5,040
Review
Tap to reveal
Convert the loss percentage to a fraction
16\frac{2}{3}\% = \frac{1}{6}, so the selling price is \frac{5}{6} of the cost price.
$$\text{SP} = \text{CP} \times \frac{5}{6}$$
Solve for cost price
Rearrange to find CP and substitute the selling price.
$$\text{CP} = 4200 \times \frac{6}{5} = 5040$$
3
A wholesaler marks an item 50% above cost price and sells it at a discount of 16% on the marked price. What is the profit percentage?
Ans
26%
Review
Tap to reveal
Assume a base cost price
Set CP = 100 for easy percentage calculation.
$$\text{CP} = 100, \quad \text{MP} = 100 \times 1.50 = 150$$
Apply the discount and find profit
Apply 16% discount on the marked price to get the selling price, then calculate profit.
$$\text{SP} = 150 \times 0.84 = 126 \implies \text{Profit \%} = 26\%$$
4
A furniture store offers two successive discounts of 15% and 10% on a sofa with a marked price of Rs. 40,000. What is the final selling price?
Ans
Rs. 30,600
Review
Tap to reveal
Apply the first discount
Reduce the marked price by 15%.
$$40000 \times 0.85 = 34000$$
Apply the second discount on the reduced price
Take 10% off the price after the first discount.
$$34000 \times 0.90 = 30600$$
5
The marked price of a watch is Rs. 3,200 and it is sold for Rs. 2,560. What is the discount percentage?
Ans
20%
Review
Tap to reveal
Calculate the discount amount
Subtract the selling price from the marked price.
$$\text{Discount} = 3200 - 2560 = 640$$
Express discount as percentage of marked price
Divide the discount by the marked price and multiply by 100.
$$\text{Discount \%} = \frac{640}{3200} \times 100 = 20\%$$
6
Two successive discounts of 25% and x% are equivalent to a single discount of 40%. What is the value of x?
Ans
20%
Review
Tap to reveal
Apply the successive discount formula
Set up $a + b - \frac{ab}{100}$ equal to the equivalent single discount.
$$25 + x - \frac{25x}{100} = 40$$
Solve for x
Simplify and solve the linear equation.
$$25 + 0.75x = 40 \implies 0.75x = 15 \implies x = 20$$
7
A trader sells a shirt at a profit of 20% and a pair of trousers at a loss of 20%. Both items were sold for the same price of Rs. 600. What is the overall profit or loss percentage?
Ans
4% loss
Review
Tap to reveal
Find the cost prices of both items
Use the selling price and profit/loss percentage to recover each cost price.
$$\text{Shirt CP} = \frac{600}{1.20} = 500, \quad \text{Trousers CP} = \frac{600}{0.80} = 750$$
Calculate total cost, total revenue, and net result
Add up the cost prices and selling prices to find the overall profit or loss.
$$\text{Total CP} = 1250, \quad \text{Total SP} = 1200, \quad \text{Loss \%} = \frac{50}{1250} \times 100 = 4\%$$
8
An electronics item is marked up by 60% on cost price. A customer avails a discount of 25% on the marked price and pays Rs. 960. What was the cost price?
Ans
Rs. 800
Review
Tap to reveal
Express SP in terms of CP
Apply the markup and then the discount to get the relationship between SP and CP.
$$\text{SP} = \text{CP} \times 1.60 \times 0.75 = \text{CP} \times 1.20$$
Solve for the cost price
Substitute the given selling price and divide.
$$\text{CP} = \frac{960}{1.20} = 800$$
9
A bakery produces a cake at a cost of Rs. 350. It applies a markup of 100% and then offers a festival discount of 35%. What is the selling price and the profit percentage?
Ans
Rs. 455, 30% profit
Review
Tap to reveal
Calculate the marked price and selling price
Apply the 100% markup to get MP, then apply the 35% discount to get SP.
$$\text{MP} = 350 \times 2 = 700, \quad \text{SP} = 700 \times 0.65 = 455$$
Find the profit percentage
Calculate profit relative to cost price.
$$\text{Profit} = 455 - 350 = 105 \implies \text{Profit \%} = \frac{105}{350} \times 100 = 30\%$$
10
A single discount of 28% is equivalent to two successive discounts of 20% and x%. What is the value of x?
Ans
10%
Review
Tap to reveal
Set up the successive discount equation
Use the formula $a + b - \frac{ab}{100}$ with $a = 20$ and the equivalent discount of 28%.
$$20 + x - \frac{20x}{100} = 28$$
Solve for x
Simplify the equation and isolate x.
$$20 + 0.8x = 28 \implies 0.8x = 8 \implies x = 10$$
11
A camera is sold after two successive discounts of 20% and 15%. If the selling price is Rs. 3,060, what was the marked price?
Ans
Rs. 4,500
Review
Tap to reveal
Find the effective price multiplier
Multiply the two discount factors: (1 − 0.20) and (1 − 0.15).
$$\text{Effective multiplier} = 0.80 \times 0.85 = 0.68$$
Recover the marked price
Divide the selling price by the effective multiplier to find MP.
$$\text{MP} = \frac{3060}{0.68} = 4500$$
12
A wholesaler buys 100 units of a product at Rs. 200 each. She marks each unit at 40% above cost. During a clearance sale, she offers a flat discount of Rs. 48 per unit and sells all 100 units. What is her total profit and profit percentage per unit?
Ans
Rs. 3,200 total, 16% per unit
Review
Tap to reveal
Find the marked price and selling price per unit
Apply the 40% markup to cost price, then subtract the flat discount.
$$\text{MP} = 200 \times 1.40 = 280, \quad \text{SP} = 280 - 48 = 232$$
Calculate profit per unit and total profit
Subtract CP from SP for per-unit profit, then multiply by 100 units.
$$\text{Profit/unit} = 32, \quad \text{Profit \%} = \frac{32}{200} \times 100 = 16\%, \quad \text{Total} = 32 \times 100 = 3200$$
Question 1 of 12
0:00
1
A fruit vendor buys a crate of mangoes for Rs. 1,200 and sells them for Rs. 1,560. What is the profit percentage?