Official Syllabus Item: 5 Commercial Mathematics

Profit, Loss, Discount & Commercial Transactions

Master Cost Price, Effective CP with overheads, Selling Price, Marked Price, Trade Discount, successive discounts, equal SP transactions, and dishonest dealer problems for RIMC Entrance Exam.

Overview & Fundamental Concept

Profit and Loss is one of the most reliable and heavily weighted 10-mark questions in the RIMC Mathematics Paper (Paper I, 200 Marks, 90 Minutes). Prescribed under Item 5 of Chapter 8 in the official RIMC brochure, questions in this module appear across both Part A (5 Marks) and Part B (10 Marks). The syllabus tests mastery over core financial mechanics: calculating effective Cost Price by integrating purchase price with overhead expenses (transportation, repairs, excise), computing Profit or Loss percentages strictly against the Cost Price baseline, navigating Marked Price and trade discounts (which are always anchored to Marked Price), computing single equivalent discounts for successive discount tiers, solving dual-article transactions with equal selling prices, and analyzing deceptive metric trades involving faulty weights. RIMC examiners demand formal declarations of given financial terms, explicit state inequality tests ($SP > CP$ or $SP < CP$), clean algebraic substitutions, and boxed final answers with appropriate rupee amounts and percentage symbols.

Official Syllabus Coverage Breakdown

Syllabus Item	Prescribed Scope	Exam Relevance & Pedagogical Objective
Item 5	Profit and Loss: Basic operations, overhead expenses, discount, marked price, and applied trade problems	Core commercial arithmetic syllabus item. Assesses financial literacy, base-anchor discipline (CP vs. MP), multi-tier retail supply chains, and multi-article transactions.

1. Foundational Vocabulary & The Governing Base-Anchor Rule

In commercial arithmetic, every transaction involves specific price anchors that must never be confused:

• Cost Price (CP): The actual purchase price of an article.
• Overhead Expenses: Additional expenses incurred after purchase before the article is ready for sale (e.g., transportation, freight, labour, cartage, repair, painting).
• Effective Cost Price:

$$\text{Effective CP} = \text{Purchase Price} + \text{Overhead Expenses}$$

• Selling Price (SP): The price at which the article is sold to the consumer.
• Marked Price (MP) / List Price: The printed catalogue price or tag price displayed on the article.

The Two Sacred Base-Anchor Rules of Commercial Mathematics:

1. Profit % and Loss % are ALWAYS calculated on COST PRICE (CP).
2. Discount % is ALWAYS calculated on MARKED PRICE (MP).

Mathematical Inequality Conditions

Case I (Gain / Profit): If $SP > CP$, the seller gains:
$$\text{Profit} = SP - CP \implies \text{Profit}\% = \left( \frac{\text{Profit}}{CP} \times 100 \right)\%$$
Case II (Loss): If $SP < CP$, the seller incurs a loss:
$$\text{Loss} = CP - SP \implies \text{Loss}\% = \left( \frac{\text{Loss}}{CP} \times 100 \right)\%$$
Case III (Break-Even): If $SP = CP$, there is neither profit nor loss.

2. Direct Formulas for Selling Price & Cost Price

In RIMC descriptive papers, candidates should quote standard algebraic conversion formulas rather than improvising unstandardized unitary sentences:

$$\text{When Gain\% is given:} \quad SP = \left( \frac{100 + \text{Gain}\%}{100} \right) \times CP \quad \iff \quad CP = \left( \frac{100}{100 + \text{Gain}\%} \right) \times SP$$

$$\text{When Loss\% is given:} \quad SP = \left( \frac{100 - \text{Loss}\%}{100} \right) \times CP \quad \iff \quad CP = \left( \frac{100}{100 - \text{Loss}\%} \right) \times SP$$

3. Marked Price, Trade Discount & The MP-CP Bridge Formula

To increase sales or attract customers, shopkeepers offer a deduction on the marked price called a Discount (or Trade Rebate):

$$\text{Discount} = MP - SP \implies \text{Discount}\% = \left( \frac{\text{Discount}}{MP} \times 100 \right)\%$$

$$SP = MP \times \left( 1 - \frac{\text{Discount}\%}{100} \right) = \left( \frac{100 - D\%}{100} \right) \times MP$$

The Universal MP-CP Bridge Identity: When a merchant marks an item up so that he can offer a discount of $D\%$ and still realize a net profit of $G\%$, the ratio between Marked Price and Cost Price is given directly by:

$$\frac{MP}{CP} = \frac{100 + G\%}{100 - D\%}$$

Successive Discounts (Series Discounts)

When two or more discounts are offered in sequence, the second discount is calculated on the reduced price remaining after the first discount:

If successive discounts are $d_1\%$ and $d_2\%$, the Single Equivalent Discount is:
$$D_{\text{equiv}} = \left( d_1 + d_2 - \frac{d_1 \times d_2}{100} \right)\%$$
Never simply add them! A series discount of $20\%$ and $10\%$ is NOT $30\%$, but $20 + 10 - \frac{200}{100} = \mathbf{28\%}$.

4. High-Yield Special RIMC Exam Paradigms

Examiners frequently test two famous non-intuitive commercial problems:

Exam Paradigm	Mathematical Condition	Universal Identity / Shortcut
Equal SP with Equal Gain & Loss	Two articles sold at same SP: one at $x\%$ gain, other at $x\%$ loss.	Always a Net Loss: $$\text{Loss}\% = \left(\frac{x}{10}\right)^2\% = \frac{x^2}{100}\%$$
CP of m Articles = SP of n Articles	The cost price of $m$ items equals the selling price of $n$ items.	$$\text{Gain/Loss}\% = \left( \frac{m - n}{n} \times 100 \right)\%$$ (Gain if $m > n$, Loss if $m < n$)
Dishonest Dealer (False Weights)	Dealer claims to sell at CP but uses a faulty weight measuring $W_{\text{false}}$ instead of $W_{\text{true}}$.	$$\text{Gain}\% = \left( \frac{\text{Error}}{\text{True Weight} - \text{Error}} \times 100 \right)\% = \left( \frac{\text{Error}}{\text{False Weight}} \times 100 \right)\%$$

Illustrated Conceptual Diagrams & Visual Schematics

Commercial Math Value Stream: Cost Price to Selling Price

Pricing Sequence: Markup is strictly applied on Cost Price ($CP$); Discount is strictly applied on Marked Price ($MP$); Profit or Loss is determined exclusively between Selling Price ($SP$) and Cost Price ($CP$). $SP = MP imes (1 - rac{D}{100}) = CP imes (1 + rac{P}{100})$.

Topper Strategy & Descriptive Step Breakdown

Mastering RIMC Mathematics Standard

In RIMC, clear step-by-step layout is essential. Write down all given values, declare formulas before substitution, and conclude with full sentences with proper units.

Official Descriptive Marking Scheme & Step Allocation

In RIMC Stage-I, answers without step justification receive only 20-30% of total marks even if the numeric answer is correct. Follow this strict 4-tier layout on every question:

Step 1: Financial State Declaration & Effective CP 1 - 2 Marks

List the given purchase price, explicit overheads, and compute the Effective Cost Price.

Booklet Layout:
Given:\nPurchase Price of Almirah $= ₹ 6,250$\nOverhead Expenses (Repairs + Freight) $= ₹ 375$\n$$\text{Total Effective CP} = 6250 + 375 = ₹ 6,625$$

Step 2: Comparison Test (SP vs. CP) & Profit/Loss Formulation 1 - 2 Marks

State whether SP > CP (Gain) or SP < CP (Loss), and write out the algebraic difference formula.

Booklet Layout:
Selling Price $SP = ₹ 6,890$.\nSince $SP > CP$ ($₹ 6,890 > ₹ 6,625$), the seller makes a \textbf{Profit (Gain)}.\n$$\text{Profit} = SP - CP = 6890 - 6625 = ₹ 265$$

Step 3: Percentage Formula Citation & Calculation 2 - 4 Marks

State the percentage formula with Effective CP in the denominator, substitute values, and show arithmetic reduction.

Booklet Layout:
$$\text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{Effective CP}} \times 100 \right)\% = \left( \frac{265}{6625} \times 100 \right)\% = \left( \frac{1}{25} \times 100 \right)\% = 4\%$$

Step 4: Boxed Final Value with Currency / Percentage Units 1 Mark

State the final gain or loss clearly in a concluding sentence inside an answer box.

Booklet Layout:
$$\mathbf{\text{Answer: The shopkeeper gained } 4\%.}$$

Examiner Traps & Common Blunders (Marks Penalties)

Omitting Overhead Expenses from Cost Price -2 to -3 Marks

Common Mistake: Computing profit as $SP - \text{Purchase Price} = 6890 - 6250 = ₹ 640$, ignoring repair costs.

Cadet Antidote: Overheads MUST be added to the purchase price before computing any profit or loss.

Calculating Discount on Cost Price -3 Marks

Common Mistake: Computing discount as $D\% \times CP$ instead of $D\% \times MP$.

Cadet Antidote: Discount is exclusively offered on the MARKED PRICE (the price tag), never on the seller's secret cost price.

Adding Successive Discounts Directly -2 Marks

Common Mistake: Claiming successive discounts of $20\%$ and $10\%$ equal a total discount of $30\%$.

Cadet Antidote: Apply the successive formula: $d_1 + d_2 - \frac{d_1 d_2}{100}$. Two discounts of $20\%$ and $10\%$ yield exactly $28\%$.

Claiming Equal Gain and Loss on Same SP Produces No Profit/Loss -4 Marks

Common Mistake: Stating in dual-article problems that a $20\%$ gain on one article and a $20\%$ loss on the other cancel out.

Cadet Antidote: Since the selling prices are identical, the cost prices are completely different! The transaction ALWAYS results in a net loss: $\frac{x^2}{100}\%$.

Cadet Topper Mental Verification Shortcuts

The MP-CP Bridge Formula Shortcut

Whenever a question links discount percentage $D$ with desired gain percentage $G$, bypass intermediate SP calculations using: $\frac{MP}{CP} = \frac{100 + G}{100 - D}$.

If a trader wants a $20\%$ profit after giving a $10\%$ discount: $\frac{MP}{CP} = \frac{100 + 20}{100 - 10} = \frac{120}{90} = \frac{4}{3}$. Thus he must mark goods $33\frac{1}{3}\%$ above CP.

Instant Single Equivalent Discount

For two successive discounts $d_1$ and $d_2$: $D_{\text{equiv}} = d_1 + d_2 - \frac{d_1 d_2}{100}$.

For $30\%$ and $20\%$: $30 + 20 - \frac{600}{100} = 50 - 6 = 44\%$ in 2 seconds.

False Weight Error Ratio

For dishonest dealer problems where a 1 kg weight is replaced by $W_{\text{false}}$: $\text{Gain}\% = \frac{1000 - W_{\text{false}}}{W_{\text{false}}} \times 100\%$.

Using $900\text{ g}$ for $1\text{ kg}$: $\text{Gain}\% = \frac{100}{900} \times 100\% = 11\frac{1}{9}\%$ instantly.

Key Formulas & Conversion Cheat Sheet

Profit (Gain)

$$\text{Profit} = SP - CP \quad (SP > CP)$$

Selling price exceeds cost price.

Loss

$$\text{Loss} = CP - SP \quad (CP > SP)$$

Cost price exceeds selling price.

Profit Percentage

$$\text{Profit}\% = \left( \frac{\text{Profit}}{\text{Effective CP}} \times 100 \right)\%$$

Gain expressed as a percentage of Cost Price.

Loss Percentage

$$\text{Loss}\% = \left( \frac{\text{Loss}}{\text{Effective CP}} \times 100 \right)\%$$

Loss expressed as a percentage of Cost Price.

Selling Price from Cost Price

$$SP = \left( \frac{100 \pm \text{Gain/Loss}\%}{100} \right) \times CP$$

Direct multiplier from CP to SP.

Cost Price from Selling Price

$$CP = \left( \frac{100}{100 \pm \text{Gain/Loss}\%} \right) \times SP$$

Inverse conversion from SP to CP.

Trade Discount Formula

$$\text{Discount} = MP - SP; \quad \text{Discount}\% = \left( \frac{\text{Discount}}{MP} \times 100 \right)\%$$

Deduction calculated exclusively on Marked Price.

Single Equivalent Discount

$$D_{\text{equiv}} = \left( d_1 + d_2 - \frac{d_1 \times d_2}{100} \right)\%$$

Single discount replacing two successive discounts.

Solved Model Examples (Descriptive Format with Step Mark Distribution)

Model 1: Overheads & Gain Percentage 5 Marks

A shopkeeper bought an almirah for ₹ $6,250$ and spent ₹ $375$ on its repair and transportation. He then sold it for ₹ $6,890$. Find his gain or loss percent.

Step-by-Step Model Solution:

Step 1: Compute Total Effective Cost Price 2 Marks

Purchase price of almirah $= ₹ 6,250$.
Overhead expenses (repairs + cartage) $= ₹ 375$.
$$\text{Total Effective CP} = 6250 + 375 = ₹ 6,625$$

Step 2: Determine Profit or Loss 1 Mark

Selling price $SP = ₹ 6,890$.
Since $SP > CP$ ($₹ 6,890 > ₹ 6,625$), the shopkeeper made a Profit.
$$\text{Profit} = SP - CP = 6890 - 6625 = ₹ 265$$

Step 3: Calculate Gain Percentage 2 Marks

$$\text{Gain}\% = \left( \frac{\text{Profit}}{\text{Effective CP}} \times 100 \right)\% = \left( \frac{265}{6625} \times 100 \right)\%$$
Divide numerator and denominator by $265$: $6625 \div 265 = 25$.
$$= \frac{1}{25} \times 100\% = 4\%$$
$$\mathbf{\text{Answer: Gain percent = } 4\%}$$

Examiner Scoring Key: Adding the ₹ 375 overheads to CP is mandatory. Award 2 marks for Effective CP = 6625.

Model 2: Re-selling to Convert Loss into Profit 5 Marks

By selling a wristwatch for ₹ $1,440$, a trader incurs a loss of $10\%$. At what price should he sell it in order to make a gain of $10\%$?

Step-by-Step Model Solution:

Step 1: Compute Original Cost Price (CP) 2.5 Marks

Given $SP_1 = ₹ 1,440$ and $\text{Loss} = 10\%$.
Formula: $CP = \frac{100}{100 - \text{Loss}\%} \times SP_1$
$$CP = \frac{100}{100 - 10} \times 1440 = \frac{100}{90} \times 1440$$
$$CP = 100 \times 16 = ₹ 1,600$$

Step 2: Calculate Required Selling Price (SP₂) 2.5 Marks

Now, desired $\text{Gain} = 10\%$ on $CP = ₹ 1,600$.
$$SP_2 = \left( \frac{100 + \text{Gain}\%}{100} \right) \times CP = \left( \frac{110}{100} \right) \times 1600 = 110 \times 16 = ₹ 1,760$$
$$\mathbf{\text{Answer: He must sell the wristwatch for } ₹\;1,760.}$$

Examiner Scoring Key: Calculating CP = ₹ 1,600 carries 2.5 marks. Computing new SP = ₹ 1,760 carries 2.5 marks.

Model 3: Successive Discounts & Finding Marked Price 7 Marks

A dealer offers two successive discounts of $20\%$ and $5\%$ on the marked price of a leather jacket. If a customer purchases the jacket for ₹ $3,420$, find:
(a) The single equivalent discount percentage.
(b) The marked price of the leather jacket.

Step-by-Step Model Solution:

Step 1: Compute Single Equivalent Discount (Part a) 3 Marks

Let successive discounts be $d_1 = 20\%$ and $d_2 = 5\%$.
$$\text{Equivalent Discount } D_{\text{equiv}} = \left( d_1 + d_2 - \frac{d_1 \times d_2}{100} \right)\%$$
$$= 20 + 5 - \frac{20 \times 5}{100} = 25 - \frac{100}{100} = 25 - 1 = \mathbf{24\%}$$

Step 2: Relate Selling Price to Marked Price 2 Marks

Since the single equivalent discount is $24\%$:
$$SP = (100\% - 24\%) \times MP = 76\% \times MP = \frac{76}{100} MP$$
Given $SP = ₹ 3,420$.

Step 3: Solve for Marked Price (Part b) 2 Marks

$$\frac{76}{100} MP = 3420 \implies MP = 3420 \times \frac{100}{76}$$
Notice $3420 \div 76$: $3420 = 76 \times 45$.
$$MP = 45 \times 100 = ₹ 4,500$$
$$\mathbf{\text{Answer: (a) Single Equivalent Discount = } 24\%; \quad \text{(b) Marked Price = } ₹\;4,500}$$

Examiner Scoring Key: Award 3 marks for D_equiv = 24%, and 4 marks for finding MP = ₹ 4,500.

Model 4: Dual-Article Equal Selling Price Transaction 7 Marks

A merchant sold two mobile phones for ₹ $9,900$ each. On one he gained $10\%$ and on the other he lost $10\%$. Find his overall gain or loss percentage on the whole transaction.

Step-by-Step Model Solution:

Step 1: Calculate Cost Price of First Phone (10% Gain) 2.5 Marks

$SP_1 = ₹ 9,900$, $\text{Gain} = 10\%$.
$$CP_1 = \frac{100}{100 + 10} \times 9900 = \frac{100}{110} \times 9900 = 100 \times 90 = ₹ 9,000$$

Step 2: Calculate Cost Price of Second Phone (10% Loss) 2.5 Marks

$SP_2 = ₹ 9,900$, $\text{Loss} = 10\%$.
$$CP_2 = \frac{100}{100 - 10} \times 9900 = \frac{100}{90} \times 9900 = 100 \times 110 = ₹ 11,000$$

Step 3: Compare Total Cost Price and Total Selling Price 2 Marks

$$\text{Total CP} = CP_1 + CP_2 = 9000 + 11000 = ₹ 20,000$$
$$\text{Total SP} = SP_1 + SP_2 = 9900 + 9900 = ₹ 19,800$$
Since $\text{Total SP} < \text{Total CP}$, the merchant incurred a Loss:
$$\text{Net Loss} = 20000 - 19800 = ₹ 200$$
$$\text{Net Loss Percentage} = \left( \frac{200}{20000} \times 100 \right)\% = \mathbf{1\%}$$
(Alternatively, shortcut: $\text{Loss}\% = \frac{x^2}{100}\% = \frac{10^2}{100}\% = 1\%$).

Examiner Scoring Key: In RIMC, the full descriptive evaluation (computing CP1 = 9000 and CP2 = 11000) is mandatory. Merely quoting the 1% shortcut receives only 2 out of 7 marks.

Model 5: Articles Cost-to-Selling Equivalence 10 Marks

If the cost price of $16\text{ pens}$ is equal to the selling price of $12\text{ pens}$, determine the gain or loss percentage on the transaction. Show complete algebraic working.

Step-by-Step Model Solution:

Step 1: Declare Cost Price of One Article 2 Marks

Let the Cost Price of $1\text{ pen}$ be ₹ $x$.
Then, the Cost Price of $16\text{ pens} = ₹ 16x$.

Step 2: Relate to Selling Price 3 Marks

According to the question:
$$\text{Selling Price of } 12\text{ pens} = \text{Cost Price of } 16\text{ pens} = ₹ 16x$$
Therefore, the Selling Price of $1\text{ pen}$:
$$SP \text{ of } 1\text{ pen} = \frac{16x}{12} = ₹ \frac{4}{3}x$$

Step 3: Compare SP and CP of One Pen 3 Marks

For $1\text{ pen}$:
$CP = x = \frac{3}{3}x$, and $SP = \frac{4}{3}x$.
Since $SP > CP$ ($\frac{4}{3}x > x$), there is a Gain (Profit):
$$\text{Gain on } 1\text{ pen} = SP - CP = \frac{4}{3}x - x = \frac{1}{3}x$$

Step 4: Compute Gain Percentage & Conclude 2 Marks

$$\text{Gain Percentage} = \left( \frac{\text{Gain}}{CP} \times 100 \right)\% = \left( \frac{\frac{1}{3}x}{x} \times 100 \right)\% = \frac{100}{3}\% = 33\frac{1}{3}\%$$
$$\mathbf{\text{Answer: The gain percentage is } 33\frac{1}{3}\% \text{ (or } 33.33\%).}$$

Examiner Scoring Key: Setting CP of 1 pen as x (or ₹ 1) and finding SP of 1 pen as ₹ 4/3 carries 5 marks. Arriving at 33 1/3% earns full 10 marks.

Model 6: Multi-Stage Commercial Supply Chain 10 Marks

A manufacturer produces a military field radio at a cost of ₹ $4,000$. He sells it to a wholesale distributor at a profit of $20\%$. The wholesaler sells it to a retailer at a profit of $12.5\%$, and the retailer marks up the price by $25\%$ above his purchase price and allows a trade discount of $10\%$ to the final customer. Calculate:
(a) The price paid by the retailer.
(b) The marked price set by the retailer.
(c) The final price paid by the customer, and the retailer's net profit in rupees.

Step-by-Step Model Solution:

Step 1: Compute Manufacturer's Selling Price (Wholesaler's Cost) 2.5 Marks

Manufacturer's Cost $= ₹ 4,000$, Profit $= 20\%$.
$$\text{Wholesaler's CP} = 4000 \times \left(1 + \frac{20}{100}\right) = 4000 \times 1.20 = ₹ 4,800$$

Step 2: Compute Wholesaler's Selling Price (Retailer's Purchase Price - Part a) 2.5 Marks

Wholesaler gains $12.5\% = \frac{1}{8}$.
$$\text{Retailer's CP} = 4800 \times \left(1 + \frac{1}{8}\right) = 4800 \times \frac{9}{8} = 600 \times 9 = \mathbf{₹ 5,400}$$

Step 3: Compute Retailer's Marked Price (Part b) 2.5 Marks

Retailer marks up price by $25\% = \frac{1}{4}$ above his CP of ₹ $5,400$:
$$\text{Marked Price (MP)} = 5400 + \left(\frac{1}{4} \times 5400\right) = 5400 + 1350 = \mathbf{₹ 6,750}$$

Step 4: Compute Customer's Final Price & Retailer's Profit (Part c) 2.5 Marks

Retailer allows $10\%$ discount on MP of ₹ $6,750$:
$$\text{Discount} = 10\% \text{ of } 6750 = ₹ 675$$
$$\text{Customer's SP} = 6750 - 675 = \mathbf{₹ 6,075}$$
Retailer's Profit $= SP - CP = 6075 - 5400 = \mathbf{₹ 675}$.
Conclusion:
(a) Retailer paid = $\mathbf{₹ 5,400}$
(b) Marked Price = $\mathbf{₹ 6,750}$
(c) Customer paid = $\mathbf{₹ 6,075}$, and Retailer's profit = $\mathbf{₹ 675}$.

Examiner Scoring Key: Sequential calculation CP1 -> SP1 -> SP2 -> MP -> SP3 evaluated in 4 distinct blocks of 2.5 marks each.

RIMC Practice & Drill Exercises

Attempt these questions on paper following proper step-wise conventions, then click Reveal Solution to verify your answers.

Question 1 5 Marks

A cycle was purchased for ₹ $2,400$ and sold for ₹ $2,700$. Find the profit and profit percentage.

Model Descriptive Solution:

Step 1: Compute Profit 2.5 Marks

$CP = ₹ 2,400$, $SP = ₹ 2,700$.
$$\text{Profit} = SP - CP = 2700 - 2400 = ₹ 300$$

Step 2: Compute Profit Percentage 2.5 Marks

$$\text{Profit}\% = \left(\frac{300}{2400} \times 100\right)\% = \frac{3}{24} \times 100\% = \frac{1}{8} \times 100\% = 12.5\%$$

Final Answer: Profit: ₹ $300$; Profit \%: $12.5\%$

Question 2 5 Marks

An article marked at ₹ $840$ is sold for ₹ $714$. Find the discount and the discount percentage.

Model Descriptive Solution:

Step 1: Compute Discount 2.5 Marks

$$MP = ₹ 840, \quad SP = ₹ 714$$
$$\text{Discount} = MP - SP = 840 - 714 = ₹ 126$$

Step 2: Compute Discount Percentage 2.5 Marks

$$\text{Discount}\% = \left(\frac{126}{840} \times 100\right)\% = \frac{1260}{84}\% = 15\%$$

Final Answer: Discount: ₹ $126$; Discount \%: $15\%$

Question 3 5 Marks

A dealer uses a false weight of $900\text{ grams}$ for a kilogram ($1000\text{ grams}$) weight. What is his real gain percent if he claims to sell goods at cost price?

Model Descriptive Solution:

Step 1: Identify Error and Effective Cost 2.5 Marks

True weight $= 1000\text{ g}$. False weight sold $= 900\text{ g}$.
$$\text{Error (Goods retained)} = 1000 - 900 = 100\text{ g}$$

Step 2: Compute Gain Percentage on Goods Sold 2.5 Marks

$$\text{Gain}\% = \left(\frac{\text{Error}}{\text{False Weight}} \times 100\right)\% = \left(\frac{100}{900} \times 100\right)\% = \frac{100}{9}\% = 11\frac{1}{9}\%$$

Final Answer: $11\frac{1}{9}\%$ (or $11.11\%$)

Question 4 7 Marks

Find the single discount equivalent to three successive discounts of $20\%$, $10\%$, and $5\%$.

Model Descriptive Solution:

Step 1: Combine First Two Discounts 3 Marks

$$D_{12} = 20 + 10 - \frac{20 \times 10}{100} = 30 - 2 = 28\%$$

Step 2: Combine with Third Discount 4 Marks

$$D_{\text{final}} = 28 + 5 - \frac{28 \times 5}{100} = 33 - \frac{140}{100} = 33 - 1.40 = 31.6\%$$

Final Answer: $31.6\%$

Question 5 7 Marks

A shopkeeper marks his goods $20\%$ above the cost price and allows a discount of $10\%$ on marked price. Find his net gain or loss percent.

Model Descriptive Solution:

Step 1: Set CP = ₹ 100 and Find MP 2.5 Marks

Let $CP = ₹ 100$. Markup $= 20\% \implies MP = ₹ 120$.

Step 2: Compute Selling Price after Discount 2.5 Marks

$$SP = 120 - (10\% \text{ of } 120) = 120 - 12 = ₹ 108$$

Step 3: State Net Gain Percentage 2 Marks

Since $SP = ₹ 108 > CP = ₹ 100$, Net Gain $= 108 - 100 = 8\%$.

Final Answer: Net Gain $= 8\%$

Question 6 7 Marks

If selling an umbrella for ₹ $336$ gives a profit of $12\%$, at what price should it be sold to make a profit of $20\%$?

Model Descriptive Solution:

Step 1: Calculate Cost Price 3.5 Marks

$$CP = \frac{100}{112} \times 336 = 100 \times 3 = ₹ 300$$

Step 2: Calculate New Selling Price for 20% Profit 3.5 Marks

$$SP_2 = 300 \times \left(1 + \frac{20}{100}\right) = 300 \times 1.20 = ₹ 360$$

Final Answer: ₹ $360$

Question 7 7 Marks

A fruit vendor buys oranges at $5$ for ₹ $10$ and sells them at $6$ for ₹ $15$. Find his profit or loss percent.

Model Descriptive Solution:

Step 1: Compute CP and SP of 1 Orange 3 Marks

$$CP \text{ of } 1\text{ orange} = \frac{10}{5} = ₹ 2.00$$
$$SP \text{ of } 1\text{ orange} = \frac{15}{6} = ₹ 2.50$$

Step 2: Compute Profit Percentage 4 Marks

$$\text{Profit on } 1\text{ orange} = 2.50 - 2.00 = ₹ 0.50$$
$$\text{Profit}\% = \left(\frac{0.50}{2.00} \times 100\right)\% = \frac{1}{4} \times 100\% = 25\%$$

Final Answer: $25\%$ Profit

Question 8 10 Marks

Two horses were sold for ₹ $19,200$ each. One was sold at a loss of $20\%$ and the other at a gain of $20\%$. Find the entire loss or gain in rupees and percent.

Model Descriptive Solution:

Step 1: Compute CP of First Horse (Loss 20%) 3 Marks

$$CP_1 = \frac{100}{80} \times 19200 = 100 \times 240 = ₹ 24,000$$

Step 2: Compute CP of Second Horse (Gain 20%) 3 Marks

$$CP_2 = \frac{100}{120} \times 19200 = 100 \times 160 = ₹ 16,000$$

Step 3: Compare Totals & Conclude 4 Marks

$$\text{Total CP} = 24000 + 16000 = ₹ 40,000$$
$$\text{Total SP} = 19200 + 19200 = ₹ 38,400$$
$$\text{Loss} = 40000 - 38400 = ₹ 1,600$$
$$\text{Loss}\% = \left(\frac{1600}{40000} \times 100\right)\% = 4\%$$

Final Answer: Loss in rupees: ₹ $1,600$; Loss percentage: $4\%$

Question 9 10 Marks

By selling $45\text{ lemons}$ for ₹ $40$, a man loses $20\%$. How many lemons should he sell for ₹ $24$ to gain $20\%$ in the transaction?

Model Descriptive Solution:

Step 1: Compute Cost Price of 45 Lemons 3 Marks

$SP_1 = ₹ 40$, $\text{Loss} = 20\%$.
$$CP \text{ of } 45\text{ lemons} = \frac{100}{80} \times 40 = ₹ 50$$

Step 2: Compute Required Selling Price for 20% Gain 3 Marks

To gain $20\%$, selling price of $45\text{ lemons}$ must be:
$$SP_2 = 50 \times 1.20 = ₹ 60$$

Step 3: Apply Unitary Method for ₹ 24 4 Marks

For ₹ $60$, lemons sold $= 45$.
For ₹ $1$, lemons sold $= \frac{45}{60} = \frac{3}{4}$.
For ₹ $24$, lemons sold $= \frac{3}{4} \times 24 = 3 \times 6 = 18\text{ lemons}$.

Final Answer: $18\text{ lemons}$

Question 10 10 Marks

A trader marks his goods at such a price that after allowing a discount of $12.5\%$ on marked price, he still makes a profit of $10\%$. If the cost price of an article is ₹ $2,800$, find its marked price.

Model Descriptive Solution:

Step 1: State MP-CP Bridge Formula 3 Marks

$$\frac{MP}{CP} = \frac{100 + G\%}{100 - D\%}$$
Here $G = 10\%$ and $D = 12.5\% = \frac{25}{2}\%$.

Step 2: Calculate Ratio MP : CP 4 Marks

$$\frac{MP}{CP} = \frac{100 + 10}{100 - 12.5} = \frac{110}{87.5} = \frac{1100}{875} = \frac{44}{35}$$

Step 3: Solve for Marked Price 3 Marks

$$MP = \frac{44}{35} \times CP = \frac{44}{35} \times 2800$$
Notice $2800 \div 35 = 80$:
$$MP = 44 \times 80 = ₹ 3,520$$

Final Answer: Marked Price $= ₹ 3,520$

High-Yield FAQs & Revision Notes

What is the difference between Cost Price and Effective Cost Price?

Cost price is the invoice purchase price. Effective Cost Price includes all mandatory overhead expenses such as transport, repairs, and labour needed to make the item ready for sale.

Why is discount always calculated on Marked Price?

Discount is a price deduction offered to the consumer from the advertised or printed list price (the tag price).

Why is profit or loss always calculated on Cost Price?

Because profit or loss measures the return on the financial capital invested by the merchant.

Why do successive discounts of 20% and 10% not equal 30%?

Because the second 10% discount is applied to the already reduced 80% price, not the original 100% value. The effective discount is $28\%$.

When two items are sold at the same price with equal profit and loss %, why is it always a loss?

Because the item sold at a loss had a much larger cost price than the item sold at a profit. The loss in rupees exceeds the profit in rupees.

How do you calculate gain from false weights?

The gain is based on the goods actually surrendered: $\text{Gain}\% = \frac{\text{Error}}{\text{False Weight}} \times 100\%$.

What is the MP-CP Bridge formula?

$\frac{MP}{CP} = \frac{100 + \text{Gain}\%}{100 - \text{Discount}\%}$. It links Marked Price and Cost Price directly without calculating intermediate selling prices.

What is break-even point in RIMC?

The transaction state where Selling Price equals Cost Price ($SP = CP$), meaning zero profit and zero loss.

Can discount percentage ever exceed 100%?

No, a 100% discount means the item is given away for free ($SP = 0$).

How does this module map to the official RIMC syllabus?

Directly covers Serial No. 5 ('Profit and Loss') of Chapter 8 broader syllabus in the official RIMC brochure.

How does this topic relate to the official RIMC syllabus?

This module directly covers Serial No. 5 of the broader syllabus published in Chapter 8 of the official RIMC admission brochure, ensuring 100% syllabus alignment.

Last-Minute Cadet High-Yield Revision Matrix

Topic / Concept	High-Yield Exam Rules & Memory Anchor
Effective CP	$\text{Effective CP} = \text{Purchase Price} + \text{Overhead Expenses}$. Always add repairs & transport.
Profit / Loss %	Always divide by CP: $\text{Gain/Loss}\% = (\text{Difference} / CP) \times 100\%$.
Discount %	Always divide by MP: $\text{Discount}\% = (\text{Discount} / MP) \times 100\%$.
MP-CP Ratio	$\frac{MP}{CP} = \frac{100 + G\%}{100 - D\%}$. Direct bridge between markup and discount.
Dual-Article Equal SP	Gain $x\%$ & Loss $x\% \implies$ Always Net Loss $= (x/10)^2\% = \frac{x^2}{100}\%$.
False Weight	$\text{Gain}\% = \frac{\text{Error}}{\text{False Weight}} \times 100\%$.