#9 Compulsory Commercial Mathematics Math 9

Percentage, Profit & Loss, Discountप्रतिशतता, लाभ एवं हानि, बट्टा (छूट)

Mastery of Module 2 (Commercial Mathematics: Percentage, Profit & Loss, Discount) is crucial for securing top ranks in the Super TET Junior Mathematics & Science paper. Candidates must master fractional equivalents, base conversions, successive percentage changes, and relationship formulas between Cost Price, Selling Price, Marked Price, and Discount to solve complex multi-layered word problems within seconds.

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Syllabus Focus & High-Yield Strategy: Mastery of Module 2 (Commercial Mathematics: Percentage, Profit & Loss, Discount) is crucial for securing top ranks in the Super TET Junior Mathematics & Science paper. Candidates must master fractional equivalents, base conversions, successive percentage changes, and relationship formulas between Cost Price, Selling Price, Marked Price, and Discount to solve complex multi-layered word problems within seconds.

1. Fundamental Axioms, Definitions & Foundational Properties

Definition of Percentage ($\%$): A percentage is a dimensionless ratio expressed as a fraction of $100$. Symbolized by $%$, $\text{Rate } \% = \frac{\text{Part}}{\text{Whole}} \times 100$. For any real number $x$, $x\% = \frac{x}{100}$.

Cost Price ($ ext{CP}$) & Selling Price ($ ext{SP}$): $\text{CP}$ is the total acquisition cost inclusive of overhead expenses ($O_h$) such as transportation and repairs: $\text{True CP} = \text{Nominal CP} + O_h$. $\text{SP}$ is the net revenue realized upon sale.

Profit & Loss Axioms: Profit ($ ext{Profit} > 0$) occurs strictly when $\text{SP} > \text{CP}$, defined as $\text{Profit} = \text{SP} - \text{CP}$. Loss ($ ext{Loss} > 0$) occurs strictly when $\text{CP} > \text{SP}$, defined as $\text{Loss} = \text{CP} - \text{SP}$.

Marked Price ($ ext{MP}$) & Discount ($D$): $\text{MP}$ is the catalog price printed on the article. Discount is the monetary concession offered on $\text{MP}$, given by $D = \text{MP} - \text{SP}$. The selling price is always computed on $\text{MP}$ after discount.

Exam Pro-Tip: Always compute Profit $\%$ or Loss $\%$ strictly with respect to $\text{CP}$ unless explicitly stated otherwise in the problem statement. Discount $\%$ is always calculated with respect to $\text{MP}$.

Memory Mnemonic: C-S-M-D — Cost gives Profit/Loss; Marked gives Discount!

2. Formulas, Standard Theorems & Algorithmic Procedures

Category / Case	Formula / Mathematical Rule	Topper Calculation Shortcut
Percentage Profit / Loss	$\text{Profit}\% = \frac{\text{Profit}}{\text{CP}} \times 100$; $\text{Loss}\% = \frac{\text{Loss}}{\text{CP}} \times 100$	$\text{SP} = \text{CP} \times \left(1 + \frac{P\%}{100}\right)$
Selling Price from CP	$\text{SP} = \text{CP} \pm \text{Gain/Loss Amount}$	$\text{CP} = \text{SP} \times \left(\frac{100}{100 \pm \text{Profit/Loss}\%}\right)$
Discount Calculations	$D\% = \frac{\text{MP} - \text{SP}}{\text{MP}} \times 100$	$\text{SP} = \text{MP} \times \left(1 - \frac{D\%}{100}\right)$

Successive Percentage Changes: If a quantity undergoes successive percentage changes of $a\%$ and $b\%$, the net percentage change $N$ is governed by the exact formula: $N = \left(a + b + \frac{a \times b}{100}\right)\%$.

Equivalence Theorem of CP and MP: The fundamental bridge connecting Cost Price and Marked Price under discount and profit is: $\frac{\text{CP}}{\text{MP}} = \frac{100 - \text{Discount}\%}{100 + \text{Profit}\%}$.

False Weight / Dishonest Dealer Theorem: If a dishonest merchant sells goods at cost price but uses a false weight of $W_{\text{false}}$ instead of $W_{\text{true}}$, the effective profit percentage is given by: $\text{Profit}\% = \frac{\text{True Weight} - \text{False Weight}}{\text{False Weight}} \times 100$.

15-Second Speed-Math Trick: Memorize fractional equivalents: $12\frac{1}{2}\% = \frac{1}{8}$, $16\frac{2}{3}\% = \frac{1}{6}$, $33\frac{1}{3}\% = \frac{1}{3}$, $11\frac{1}{9}\% = \frac{1}{9}$. If $\text{CP}$ is increased by $16\frac{2}{3}\%$, multiply $\text{CP}$ directly by $\frac{7}{6}$ instead of computing decimal percentages!

3. Word Problem Modeling & Practical Applications

Problem Type 1 (Population & Depreciation Growth): Modeled using compound exponential formulas. If initial population is $P_0$ and annual growth rate is $r\%$, population after $n$ years is $P_n = P_0 \left(1 \pm \frac{r}{100}\right)^n$, where minus sign denotes depreciation.

Problem Type 2 (Two Articles Sold at Same SP): When two articles are sold at the same selling price $\text{SP}$, one at a profit of $x\%$ and the other at a loss of $x\%$, there is always a net loss. The net loss percentage is given by $\frac{x^2}{100}\%$.

Problem Type 3 (Mixture & Alligation Pricing): When two grades of commodities priced at $C_1$ and $C_2$ per kg are mixed in ratio $Q_1 : Q_2$ to form a mixture of mean price $C_m$, the ratio is given by $\frac{Q_1}{Q_2} = \frac{C_2 - C_m}{C_m - C_1}$.

Problem Type 4 (Equivalent Successive Discounts): Two successive discounts of $a\%$ and $b\%$ are mathematically equivalent to a single single discount (equivalent discount) of $\left(a + b - \frac{a \times b}{100}\right)\%$.

UP Exam High-Yield Tip: SCERT UP Class 7 and 8 textbooks frequently feature word problems on municipal taxes, GST (Goods and Services Tax), and commercial rebates. Always add GST to the discounted price to find the final billing amount.

4. Mathematical Edge Cases, Sign Conventions & Common Traps

Critical Distinction (Base Shift Trap): Profit percentage is calculated on $\text{CP}$, whereas profit margin (often used in business accounting) is calculated on $\text{SP}$. Unless 'margin' is explicitly stated, assume profit percentage is on $\text{CP}$.

Edge Case (Zero Cost Price): If $\text{CP} = 0$ (e.g., free promotional items or inherited goods), standard percentage profit formulas are undefined ($\frac{\text{Profit}}{0}$ causes division by zero).

Common Calculation Trap (Discount Accumulation): Successive discounts of $20\%$ and $10\%$ do not equal a $30\%$ discount. The net discount is $20 + 10 - \frac{20 \times 10}{100} = 28\%$, not $30\%$.

Trap Prevention Trick: Always convert percentage rates to vulgar fractions before performing multiplication chains in word problems to avoid rounding errors and extraneous decimal digits.

Memory Mnemonic: NET — Never Add Successive Discounts Directly! Use the product formula.

Comprehensive Mathematical Matrix & Method Comparison

Parameter / Dimension	Method / Concept A	Method / Concept B	Exam Speed & Accuracy Takeaway
Base Reference	Percentage Profit (Base = CP)	Discount Percentage (Base = MP)	Never confuse denominators when calculating profit vs discount.
Successive Change	Additive Formula: $a + b + \frac{ab}{100}$	Multiplier Chain: $(1 + \frac{a}{100})(1 + \frac{b}{100})$	Use Multiplier Chain for 3 or more successive percentage changes.
Equal Selling Price	Same SP with $\pm x\%$ always results in Net Loss of $\frac{x^2}{100}\%$.	Same CP with $\pm x\%$ results in Zero Net Profit/Loss.	Quickly identify whether SP or CP is constant in transaction problems.
Dishonest Dealing	Error calculated against False Weight.	Profit $\% = \frac{\text{Error}}{\text{True} - \text{Error}} \times 100$.	Always use the denominator as the weight actually delivered to the customer.

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Question 1 of 10 Super TET Junior Mathematics

1. If the cost price of $15$ articles is equal to the selling price of $12$ articles, what is the profit percentage?

Correct Answer: Option B

Let the cost price of 1 article be $1$. Then, $\text{CP of } 15 ext{ articles} = 15$. According to the problem, $\text{SP of } 12 ext{ articles} = 15$. Therefore, $\text{SP of } 1 ext{ article} = \frac{15}{12} = \frac{5}{4} = 1.25$. Profit per article $= \text{SP} - \text{CP} = 1.25 - 1 = 0.25$. Profit $\% = \frac{0.25}{1} \times 100 = 25\%$. Option 1 is correct.

Question 2 of 10 Super TET Junior Mathematics

2. A shopkeeper marks his goods $40\%$ above the cost price and allows a discount of $25\%$ on the marked price. What is his net profit or loss percentage?

Correct Answer: Option D

Let $\text{CP} = 100$. Marked Price $\text{MP} = 100 + 40\% \text{ of } 100 = 140$. Discount $= 25\% \text{ of } 140 = \frac{25}{100} \times 140 = 35$. Selling Price $\text{SP} = \text{MP} - \text{Discount} = 140 - 35 = 105$. Profit $= \text{SP} - \text{CP} = 105 - 100 = 5$, which is a $5\%$ profit. Alternatively, using the net percentage formula: $a = 40, b = -25$. Net Change $= 40 - 25 - \frac{40 \times 25}{100} = 15 - 10 = +5\%$. Option 3 is correct.

Question 3 of 10 Super TET Junior Mathematics

3. What single discount percentage is equivalent to successive discounts of $20\%$, $10\%$, and $5\%$?

Correct Answer: Option B

Using multiplier fractions: Remaining price factor $= \left(1 - \frac{20}{100}\right) \times \left(1 - \frac{10}{100}\right) \times \left(1 - \frac{5}{100}\right) = \frac{4}{5} \times \frac{9}{10} \times \frac{19}{20} = \frac{684}{1000} = 0.684$. Total Single Discount $\% = (1 - 0.684) \times 100 = 0.316 \times 100 = 31.6\%$. Option 1 is correct.

Question 4 of 10 Super TET Junior Mathematics

4. A man sells two horses for ₹ $17,100$ each. On one he gains $10\%$ and on the other he loses $10\%$. What is his overall gain or loss in the entire transaction?

Correct Answer: Option B

For the first horse, cost price is 17100/1.10, approximately 15545.45. For the second, it is 17100/0.90=19000. Total cost is about 34545.45 and total selling price is 34200, giving a loss of about 345.45, or 346 to the nearest rupee. The answer is Option B.

Question 5 of 10 Super TET Junior Mathematics

5. A dishonest dealer professes to sell his goods at cost price, but uses a weight of $900$ grams for a kilogram. What is his actual profit percentage?

Correct Answer: Option B

True weight $= 1000$ g, False weight $= 900$ g. Profit $\% = \frac{\text{True Weight} - \text{False Weight}}{\text{False Weight}} \times 100 = \frac{1000 - 900}{900} \times 100 = \frac{100}{9} = 11\frac{1}{9}\%$. Option 1 is correct.

Question 6 of 10 Super TET Junior Mathematics

6. If the price of sugar increases by $25\%$, by what percentage must a household reduce its consumption so that expenditure remains constant?

Correct Answer: Option B

If price increases by $r\%$, reduction in consumption to keep expenditure constant is given by $\frac{r}{100 + r} \times 100\%$. Substituting $r = 25$: $\frac{25}{125} \times 100 = \frac{1}{5} \times 100 = 20\%$. Option 1 is correct.

Question 7 of 10 Super TET Junior Mathematics

7. A book was sold for ₹ $540$ at a loss of $10\%$. At what price should it be sold to earn a profit of $20\%?

Correct Answer: Option C

Given $\text{SP}_1 = 540$ at $10\%$ loss. $\text{CP} = \frac{540 imes 100}{90} = 600$. To earn a profit of $20\%$, the new selling price $\text{SP}_2 = \text{CP} \times \left(1 + \frac{20}{100}\right) = 600 \times 1.20 = 720$. Option 2 is correct.

Question 8 of 10 Super TET Junior Mathematics

8. The population of a town increases by $5\%$ in the first year and by $10\%$ in the second year. If the present population is $23,100$, what was the population $2$ years ago?

Correct Answer: Option A

Let population 2 years ago be $P_0$. Then $P_0 \times \left(1 + \frac{5}{100}\right) \times \left(1 + \frac{10}{100}\right) = 23100 \implies P_0 \times \frac{21}{20} \times \frac{11}{10} = 23100 \implies P_0 \times \frac{231}{200} = 23100 \implies P_0 = 23100 \times \frac{200}{231} = 20000$. Option 0 is correct.

Question 9 of 10 Super TET Junior Mathematics

9. An article is marked at ₹ $800$ and is sold for ₹ $680$. What is the discount percentage offered on the article?

Correct Answer: Option C

Marked Price $\text{MP} = 800$, Selling Price $\text{SP} = 680$. Discount amount $D = 800 - 680 = 120$. Discount $\% = \frac{D}{\text{MP}} \times 100 = \frac{120}{800} \times 100 = \frac{120}{8} = 15\%$. Option 2 is correct.

Question 10 of 10 Super TET Junior Mathematics

10. If a $10\%$ discount is allowed on the marked price of an article, a shopkeeper still gains $20\%$. What is the ratio of the Cost Price to the Marked Price?

Correct Answer: Option B

Using the fundamental CP-MP ratio theorem: $\frac{\text{CP}}{\text{MP}} = \frac{100 - \text{Discount}\%}{100 + \text{Profit}\%} = \frac{100 - 10}{100 + 20} = \frac{90}{120} = \frac{3}{4}$. Thus, the ratio is $3 : 4$. Option 1 is correct.

5-Minute Exam Simulation Mode

Experience the real pressure of Super TET Junior (Part 2C Mathematics). You will face 10 randomized speed-math questions with negative marking (-1 for incorrect answers) and a strict 5-minute countdown.

Time Limit:

5 Minutes (300s)

Marking Scheme:

+3 Marks | -1 Negative

Passing Benchmark:

21+ Marks (70%)

High-Yield Rapid Recall Formulas & Laws (Must-Remember Pointers)

Percentage Rate Formula: $\text{Rate}\% = \frac{\text{Part}}{\text{Whole}} \times 100$.

Profit and Loss Base: Always calculated with respect to Cost Price ($\text{CP}$).

Discount Base: Always calculated with respect to Marked Price ($\text{MP}$).

Successive Percentage Change: Net Change $N = \left(a + b + \frac{ab}{100}\right)\%$.

CP-MP Relationship Theorem: $\frac{\text{CP}}{\text{MP}} = \frac{100 - D\%}{100 + P\%}$.

Equal SP Profit/Loss Rule: Selling two items at same SP with equal gain/loss $x$ always yields a net loss of $\frac{x^2}{100}\%$.

Dishonest Dealer Formula: $\text{Profit}\% = \frac{\text{True} - \text{False}}{\text{False}} \times 100$.

Consumption-Price Inverse Rule: If price increases by $r\%$, required consumption reduction is $\frac{r}{100+r} \times 100\%$. Deductions apply symmetrically.

Speed Math & Memory Mnemonics Vault (Exam Retention Anchors)

Commercial Math Formula Anchor

Memory Key: CP-SP-MP

C - Cost Price is base for Profit, S - Selling Price is revenue, M - Marked Price is base for Discount.

Successive Discount Acronym

Memory Key: NET

N - Never add directly, E - Equivalent discount formula, T - Total percentage product.

Negative Marking Traps & Common Blunders (-1 Penalty Avoidance)

Negative Marking Alert #1:

Negative Marking Trap 1: Calculating profit percentage on Selling Price instead of Cost Price.

Negative Marking Alert #2:

Negative Marking Trap 2: Directly adding successive percentage discounts (e.g., treating $20\%$ and $10\%$ as $30\%$).

Negative Marking Alert #3:

Negative Marking Trap 3: Confusing Marked Price with Cost Price when calculating shopkeeper rebates.

Negative Marking Alert #4:

Negative Marking Trap 4: Neglecting overhead expenses when calculating the true Cost Price of commodities.

Self-Assessment Checklist

Mastered fractional percentage equivalents for rapid mental calculation. Memorized all core formulas connecting CP, SP, MP, Profit, Loss, and Discount. Practiced successive percentage change and dishonest dealer word problems. Completed all 10 practice MCQs under strict negative marking simulation.

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Percentage, Profit & Loss, Discountप्रतिशतता, लाभ एवं हानि, बट्टा (छूट)

1. Fundamental Axioms, Definitions & Foundational Properties

2. Formulas, Standard Theorems & Algorithmic Procedures

3. Word Problem Modeling & Practical Applications

4. Mathematical Edge Cases, Sign Conventions & Common Traps

Comprehensive Mathematical Matrix & Method Comparison

5-Minute Exam Simulation Mode

Test Completed!

High-Yield Rapid Recall Formulas & Laws (Must-Remember Pointers)

Speed Math & Memory Mnemonics Vault (Exam Retention Anchors)

Commercial Math Formula Anchor

Successive Discount Acronym

Negative Marking Traps & Common Blunders (-1 Penalty Avoidance)

Self-Assessment Checklist