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ICSE • Class 8 • Mathematics • Ch 9
Estimated Time: 45 Mins
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Profit, Loss and Discount

In ICSE Class 8 Mathematics, "Profit, Loss and Discount" provides an authoritative, commercial arithmetic master study guide investigating commercial transactions, cost accounting, markup percentages, successive trade discounts, and Value Added Tax (VAT) / Goods and Services Tax (GST). This comprehensive chapter explores Core Commercial Concepts (Cost Price [CP], Selling Price [SP], Overhead Expenses added to basic cost price to determine Effective/Total CP; Profit / Gain when $SP > CP$: $\text{Profit} = SP - CP$; Loss when $CP > SP$: $\text{Loss} = CP - SP$), Profit Percentage and Loss Percentage (Calculated strictly on the Effective Cost Price: $\text{Profit}\% = \frac{\text{Profit}}{CP} \times 100\%$; $\text{Loss}\% = \frac{\text{Loss}}{CP} \times 100\%$; Master formulas for calculating SP from CP and CP from SP), Marked Price (MP) and Trade Discount (Marked Price / List Price / Printed Price; Discount given on Marked Price: $\text{Discount} = MP - SP$; $\text{Discount}\% = \frac{\text{Discount}}{MP} \times 100\%$; Calculating SP from MP: $SP = MP\left(1 - \frac{D}{100}\right)$), Successive Discounts ($D_1\%$ and $D_2\%$: $\text{Effective Single Discount} = \left( D_1 + D_2 - \frac{D_1 D_2}{100} \right)\%$), Goods and Services Tax / GST (Calculated strictly on the final discounted selling price: $\text{GST Amount} = \frac{\text{GST}\%}{100} \times SP$; Total Price paid by consumer $= SP + \text{GST}$), Dishonest Dealer and False Weight Problems, and Complex Multi-Tier Trading Transactions aligned with the 2026–27 CISCE ICSE curriculum.

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How Did a Devious 19th-Century London Spice Merchant Make a 25% Profit While Swearing on His Mother's Bible That He Was Selling at Cost Price?

In Victorian London, an unscrupulous merchant hung a giant sign outside his spice shop: "CHARITY TO ALL! Pure ground cinnamon sold at exact Cost Price with ZERO profit!" Crowds gathered. The merchant weighed a kilogram of cinnamon, matched the wholesale bill, and smiled benevolently. Yet every evening, he counted bags of solid gold sovereigns in his back room! How could he amass fortune while selling at cost price? The merchant used a Counterfeit Faulty Scale: his iron "1 Kilogram" counter-weight actually weighed only $800\text{ grams}$! For every 800 grams of cinnamon he handed across the counter, he charged the full 1,000-gram cost price! His true profit was: $\frac{1000 - 800}{800} \times 100\% = \mathbf{25\%\text{ PROFIT}}$! Mathematics saw right through the fraud! Why is Profit or Loss ALWAYS calculated on Cost Price, while Discount is ALWAYS calculated on Marked Price? How do you calculate the single equivalent discount for successive cuts of $20\%, 10\%,$ and $5\%$? Let's master profit, loss, and discount.

Why This Chapter Matters

Commercial arithmetic is the essential practical skill of modern commerce: managing retail businesses, calculating corporate operating margins, deciphering GST sales invoices, evaluating festive e-commerce sales discounts, and budgeting household expenses. Scoring top marks in ICSE commercial arithmetic requires mastering these formula linkages.

Before You Begin (Prerequisites)

  • Percentages and fraction inter-conversions from Chapter 8.
  • Basic algebraic equations and solving for an unknown variable $x$.
  • Basic arithmetic operations.

What You Will Learn (Core Objectives)

  • Calculate Profit, Loss, Profit Percentage, and Loss Percentage on effective Cost Price.
  • Compute Selling Price (SP) given CP and Profit/Loss%, and reverse-compute CP given SP.
  • Calculate Marked Price (MP), Discount, and Discount Percentage.
  • Determine the single equivalent discount for two or more successive trade discounts.
  • Calculate Goods and Services Tax (GST) on discounted selling prices and compute final customer billing.
  • Solve complex multi-step retail word problems involving markups, discounts, and net profits.

Chapter Roadmap & Progression

1 1. Cost Price, Selling Price & Over...
2 2. Master Formulae for SP and CP
3 3. Marked Price & Trade Discount
4 4. Successive Discounts & GST Taxat...

Complete Concept Guide (100% Curriculum Coverage)

1. Cost Price, Selling Price & Overheads

Understand
A. Fundamental Terms:
  • Cost Price (CP): The purchase price of an article.
  • Overhead Charges: Additional expenditures incurred on transportation, freight, repairs, or labor. $$\mathbf{\text{Effective (Total) CP} = \text{Basic Cost Price} + \text{Overhead Expenses}}$$
  • Selling Price (SP): The price at which the article is sold to a customer.
B. Profit vs Loss:
  • If $SP > CP$: $\mathbf{\text{Profit (Gain)} = SP - CP}$ $$\mathbf{\text{Profit \%} = \frac{\text{Profit}}{\text{CP}} \times 100\%}$$
  • If $CP > SP$: $\mathbf{\text{Loss} = CP - SP}$ $$\mathbf{\text{Loss \%} = \frac{\text{Loss}}{\text{CP}} \times 100\%}$$
  • *Golden Law of Commerce:* Profit% and Loss% are ALWAYS CALCULATED ON COST PRICE (CP) unless explicitly stated otherwise!

2. Master Formulae for SP and CP

Master Formulas
A. Finding Selling Price (SP) when CP is Given:
$$\mathbf{SP = \frac{100 + \text{Profit \%}}{100} \times CP} \quad \Big| \quad \mathbf{SP = \frac{100 - \text{Loss \%}}{100} \times CP}$$
B. Finding Cost Price (CP) when SP is Given:
$$\mathbf{CP = \frac{100}{100 + \text{Profit \%}} \times SP} \quad \Big| \quad \mathbf{CP = \frac{100}{100 - \text{Loss \%}} \times SP}$$

3. Marked Price & Trade Discount

Discount & Markup
A. Marked Price (MP) & Discount:
  • Marked Price (List Price / Printed Price): The price printed on the label of an article.
  • Discount ($D$): Reduction offered by the seller on the Marked Price: $$\mathbf{\text{Discount} = MP - SP} \quad \Big| \quad \mathbf{SP = MP - \text{Discount}}$$
  • *Crucial Rule:* Discount is ALWAYS CALCULATED ON THE MARKED PRICE (MP)! $$\mathbf{\text{Discount \%} = \frac{\text{Discount}}{\text{MP}} \times 100\%}$$ $$\mathbf{SP = \frac{100 - \text{Discount \%}}{100} \times MP}$$

4. Successive Discounts & GST Taxation

Advanced Commerce
A. Single Equivalent Discount for Successive Discounts:

If two successive discounts of $D_1\%$ and $D_2\%$ are given on Marked Price:

$$\mathbf{\text{Single Equivalent Discount} = \left( D_1 + D_2 - \frac{D_1 D_2}{100} \right)\%}$$

Example: Two successive discounts of $20\%$ and $10\%$: $$20 + 10 - \frac{20 \times 10}{100} = 30 - 2 = \mathbf{28\% \text{ (NOT 30%!)}}.$$

B. Goods and Services Tax (GST):

GST is a consumption tax charged by the government strictly on the actual final Selling Price ($SP$) after deducting all discounts:

$$\mathbf{\text{GST Amount} = \frac{\text{GST \%}}{100} \times SP}$$ $$\mathbf{\text{Total Bill Paid by Consumer} = SP + \text{GST Amount}}$$

Key Formulas, Identities & Theorems

Selling Price with Profit Formula
$$SP = \frac{100 + \text{Profit \%}}{100} \times CP$$
Calculates selling price directly.
Equivalent Single Discount Formula
$$D_{\text{equiv}} = \left( D_1 + D_2 - \frac{D_1 D_2}{100} \right)\%$$
Single discount equivalent to two successive discounts.

Commercial Math: The Pricing Journey (CP to MP to SP to Bill)

Commercial Math: The Pricing Journey from Cost Price to Final Bill COST PRICE (CP) Purchase + Overheads Base of Profit/Loss % +Markup MARKED PRICE (MP) Printed List Price Base of Discount % -Discount SELLING PRICE (SP) Actual Sale Price SP - CP = Profit! +GST FINAL BILL Consumer Price SP + GST Amount GOLDEN RULES OF COMMERCIAL ARITHMETIC • Rule 1: Profit % or Loss % is ALWAYS calculated on COST PRICE (CP)! • Rule 2: Discount % is ALWAYS calculated on MARKED PRICE (MP)! • Rule 3: GST is ALWAYS calculated on the final discounted SELLING PRICE (SP)! • Rule 4: Successive Discounts D1 & D2 ⇒ Single Equivalent D = D1 + D2 - (D1×D2)/100 PROFIT% ON CP • DISCOUNT% ON MP • GST ON SP • EQUIVALENT DISCOUNT = D1 + D2 - D1*D2/100

Chapter Summary & 10 Key Takeaways

Takeaway 1
Total Cost Price (CP) includes the purchase price plus all transportation and repair overheads.
Takeaway 2
Profit = SP - CP (when SP > CP); Loss = CP - SP (when CP > SP).
Takeaway 3
Profit% and Loss% are strictly calculated on the Cost Price: (Profit/CP) * 100%.
Takeaway 4
Formula for SP: SP = [(100 + Profit%) / 100] * CP or [(100 - Loss%) / 100] * CP.
Takeaway 5
Formula for CP: CP = [100 / (100 + Profit%)] * SP or [100 / (100 - Loss%)] * SP.
Takeaway 6
Marked Price (MP) is the list price; Discount = MP - SP.
Takeaway 7
Discount percentage is strictly calculated on Marked Price: (Discount / MP) * 100%.
Takeaway 8
Selling price after discount: SP = [(100 - Discount%) / 100] * MP.
Takeaway 9
Successive discounts D1% and D2% equal single discount D = D1 + D2 - (D1*D2)/100 %.
Takeaway 10
Goods and Services Tax (GST) is calculated strictly on the discounted Selling Price (SP).

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
A merchant buys an article for $\text{Rs. } 1,200$ and spends $\text{Rs. } 300$ on its transportation. He sells it for $\text{Rs. } 1,800$. Find his profit percentage.
Reveal Answer & Explanation
Answer: Step 1: Calculate Effective Total Cost Price:
$$\text{Effective CP} = \text{Basic CP} + \text{Overhead Expenses} = 1200 + 300 = \mathbf{\text{Rs. } 1,500}$$
Step 2: Selling Price $SP = \text{Rs. } 1,800$. Since $SP > CP$, there is a Profit:
$$\text{Profit} = SP - CP = 1800 - 1500 = \text{Rs. } 300$$
Step 3: Calculate Profit Percentage (on Effective CP):
$$\text{Profit \%} = \frac{\text{Profit}}{\text{Effective CP}} \times 100\% = \frac{300}{1500} \times 100\% = \frac{1}{5} \times 100\% = \mathbf{20\%}$$.
Effective CP $= 1200 + 300 = 1500$. Profit $= 1800 - 1500 = 300$. Profit% $= (300 / 1500) \times 100 = 20\%$.
2
By selling a watch for $\text{Rs. } 1,440$, a shopkeeper loses $10\%$. At what price should he sell it to gain $15\%$?
Reveal Answer & Explanation
Answer: Step 1: Find the Cost Price (CP) from the first condition:
Given $SP = \text{Rs. } 1,440$ and $\text{Loss} = 10\%$:
$$CP = \frac{100}{100 - \text{Loss \%}} \times SP = \frac{100}{100 - 10} \times 1440 = \frac{100}{90} \times 1440$$
$$CP = 100 \times 16 = \mathbf{\text{Rs. } 1,600}$$
Step 2: Calculate the new SP to achieve a Profit of $15\%$ on $CP = 1600$:
$$\text{New } SP = \frac{100 + \text{Profit \%}}{100} \times CP = \frac{100 + 15}{100} \times 1600 = \frac{115}{100} \times 1600$$
$$\text{New } SP = 115 \times 16 = \mathbf{\text{Rs. } 1,840}$$.
First find $CP = (1440 \times 100) / 90 = 1600$. Then new $SP = 1600 \times 1.15 = \text{Rs. } 1,840$.
3
Find the single discount equivalent to two successive discounts of $20\%$ and $10\%$.
Reveal Answer & Explanation
Answer: Apply the Successive Discount formula:
$$\text{Single Equivalent Discount} = \left( D_1 + D_2 - \frac{D_1 D_2}{100} \right)\%$$
Here $D_1 = 20\%$ and $D_2 = 10\%$:
$$= \left( 20 + 10 - \frac{20 \times 10}{100} \right)\% = \left( 30 - \frac{200}{100} \right)\% = 30 - 2 = \mathbf{28\%}$$.
*(Notice that it is $28\%$, NOT $30\%!$)*.
$20 + 10 - (20 \times 10)/100 = 30 - 2 = 28\%$.
4
A trader marks his goods $40\%$ above the cost price and allows a discount of $25\%$ on the marked price. Find his net profit or loss percentage.
Reveal Answer & Explanation
Answer: Let the Cost Price (CP) be $\text{Rs. } 100$.
Step 1: Calculate Marked Price ($40\%$ markup on CP):
$$MP = 100 + 40 = \text{Rs. } 140$$
Step 2: Calculate Selling Price ($25\%$ discount on MP):
$$\text{Discount} = 25\% \text{ of } 140 = \frac{25}{100} \times 140 = \text{Rs. } 35$$
$$SP = MP - \text{Discount} = 140 - 35 = \text{Rs. } 105$$
Step 3: Compare SP and CP:
Since $SP (105) > CP (100)$, there is a Profit:
$$\text{Profit} = 105 - 100 = \text{Rs. } 5$$
$$\mathbf{\text{Profit \%} = 5\%}$$.
Let $CP = 100 \implies MP = 140$. $25\%$ discount gives $SP = 140 \times 0.75 = 105$. Profit is $5\%$.
5
The marked price of a refrigerator is $\text{Rs. } 28,000$. The shopkeeper offers a discount of $10\%$. If a GST of $18\%$ is levied on the sales price, calculate the total amount the consumer pays.
Reveal Answer & Explanation
Answer: Given: Marked Price $MP = \text{Rs. } 28,000$, Discount $= 10\%$, GST $= 18\%$.
Step 1: Calculate Selling Price after discount:
$$\text{Discount Amount} = 10\% \text{ of } 28,000 = \text{Rs. } 2,800$$
$$SP = 28,000 - 2,800 = \mathbf{\text{Rs. } 25,200}$$
Step 2: Calculate GST on Selling Price:
$$\text{GST Amount} = 18\% \text{ of } 25,200 = \frac{18}{100} \times 25,200 = 18 \times 252 = \mathbf{\text{Rs. } 4,536}$$
Step 3: Calculate Final Amount Paid by Consumer:
$$\text{Total Bill} = SP + \text{GST} = 25,200 + 4,536 = \mathbf{\text{Rs. } 29,736}$$.
Discounted $SP = 28000 \times 0.90 = 25200$. Add $18\%$ GST ($4536$) to get final bill: $\text{Rs. } 29,736$.
6
A dishonest dealer professes to sell his goods at cost price, but uses a false weight of $950\text{ grams}$ for a $1\text{ kilogram}$ weight. Find his gain percentage.
Reveal Answer & Explanation
Answer: • True value $= 1\text{ kg} = 1000\text{ grams}$.
• False weight used $= 950\text{ grams}$.
• Error / Cheating quantity $= 1000 - 950 = 50\text{ grams}$.
• Apply the False Weight Profit formula:
$$\text{Gain \%} = \frac{\text{Error}}{\text{True Value} - \text{Error}} \times 100\% = \frac{\text{Error}}{\text{False Weight}} \times 100\%$$
$$\text{Gain \%} = \frac{50}{950} \times 100\% = \frac{5}{95} \times 100\% = \frac{1}{19} \times 100\% = \mathbf{5 \frac{5}{19}\% \approx 5.26\%}$$.
Gain% $= (\text{Error} / \text{False Weight}) \times 100\% = (50 / 950) \times 100\% = 5\frac{5}{19}\%$.
7
An article is sold for $\text{Rs. } 680$ at a loss of $15\%$. Find its cost price.
Reveal Answer & Explanation
Answer: Given: $SP = \text{Rs. } 680$, $\text{Loss} = 15\%$.
Apply formula:
$$CP = \frac{100}{100 - \text{Loss \%}} \times SP = \frac{100}{100 - 15} \times 680 = \frac{100}{85} \times 680$$
Notice that $680 \div 85 = 8$:
$$CP = 100 \times 8 = \mathbf{\text{Rs. } 800}$$.
$CP = (680 \times 100) / 85 = \text{Rs. } 800$.
8
Why is Discount always calculated on the Marked Price and never on the Cost Price?
Reveal Answer & Explanation
Answer:

• The Cost Price (CP) is a private, confidential figure known only to the merchant that represents their manufacturing or procurement expense.
• The Marked Price (MP) is the publicly printed label price displayed to the prospective customer.
• Since a commercial discount is an advertised concession offered to entice the public to buy, it must mathematically be deducted from the publicly displayed Marked Price.


Marked price is the public price tag seen by the buyer; cost price is private to the seller.
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