Fundamental and Advanced Concepts of Profit and Loss: Loss and Article-Based Quantitative Analysis

Pedagogical Overview of Profit and Loss Methodology

The instructional session delivered by Abhishek Ojha Sir (SSC CGL AIR 112, CBI Officer) focuses on a comprehensive curriculum for Profit and Loss based on the Eduquity pattern for the SSC CGL 2026 examination. The course architecture is designed to transition from basic definitions to advanced problem-solving techniques. Key conceptual pillars include the Number Line Concept, the Cost Price (CP) Same Concept, and the Selling Price (SP) Same Concept. Advanced analytical tools taught include the Concept of Weighted Average, Article-based quantitative analysis, the Cross Product Method, the Allegation Method, and the Succession Method. Each method is tailored to specific problem types frequently encountered in competitive exams like SSC Selection Post, Delhi Police Constable, SSC CGL Pre/Mains, and RRB NTPC.

The Selling Price (SP) Same Concept: Type 1 Scenarios

A fundamental scenario in profit and loss involves a transaction where two items are sold at the identical Selling Price ($SP$), with one yielding an $x\%$ profit and the other an $x\%$ loss. In such instances, the mathematical constant remains that there is always a net loss calculated by the formula $\frac{x^2}{100}\% \text{ Loss}$. For example, if a man sells two items at ₹ $4000$ per item, making a profit of $25\%$ on one and a loss of $25\%$ on the other, the net result for the entire transaction is a loss of $\frac{25^2}{100} = 6.25\%$. Using the ratio method, a $25\%$ profit implies a $CP:SP$ ratio of $4:5$, while a $25\%$ loss implies a ratio of $4:3$. To equate the Selling Price (the condition of the problem), the ratios are multiplied to achieve a common $SP$ of $15$, resulting in cumulative $CP$ of $12 + 20 = 32$ and cumulative $SP$ of $15 + 15 = 30$, verifying the loss.

Supporting this principle, further examples include selling two houses at ₹ $6,75,958$ each with a profit of $16\%$ and a loss of $16\%$. The approximate loss remains $\frac{16^2}{100} = 2.56\% \text{ loss}$. In a transaction involving items sold for ₹ $5,000$ each with a $20\%$ gain and $20\%$ loss, the percentage loss is $4\%$. To find the absolute loss in currency, one calculates that $96\%$ of the total $CP$ equals the combined $SP$ of ₹ $10,000$. Therefore, a $4\%$ loss accounts for ₹ $416.67$, derived from the proportion $\frac{10000}{96} \times 4$.

Advanced SP Same Variants: Type 2 and Type 3 Analysis

When the profit and loss percentages are not identical but the $SP$ remains constant, a weighted approach using ratios is necessary. For a trader selling two items for ₹ $3450$ each (or any identical price), with a $15\%$ profit on the first and a $10\%$ loss on the second, we convert percentages to fractions: $15\% = \frac{3}{20}$ and $10\% = \frac{1}{10}$. The ratios become $20:23$ and $10:9$. Scaling these to match the $SP$ (using $207$ as the common $SP$), the total $CP$ becomes $180 + 230 = 410$ and total $SP$ becomes $207 + 207 = 414$. This results in an overall profit of $\frac{4}{410} \times 100 \approx 0.98\% \text{ profit}$.

In more complex variations, such as SSC CGL Pre 2019 problems, two articles might be sold for ₹ $4096$ each with a $32\%$ gain and $28\%$ loss. Converting these to the lowest terms ($\frac{32}{100} = \frac{8}{25}$ and $\frac{28}{100} = \frac{7}{25}$) yields $CP:SP$ ratios of $25:33$ and $25:18$. After equating the $SP$ to $198$ through multiplication, the combined $CP$ is $150 + 275 = 425$ and combined $SP$ is $198 + 198 = 396$. The resulting net loss is Calculated as $\frac{29}{425} \times 100 \approx 6.8\% \text{ loss}$.

For triple transactions (Type 3), such as selling three wristwatches for ₹ $2,800$ each at profits of $40\%$, $25\%$, and $12\%$, the ratios are calculated as $5:7$, $4:5$, and $25:28$. Setting a common $SP$ of $140$, the cost prices scale to $100$, $112$, and $125$. The total $CP$ is $337$ against a total $SP$ of $420$, leading to a total profit percentage of $\frac{83}{337} \times 100 \approx 24.63\% \text{ or } 0.2463$.

Zero-Net-Gain Scenarios and Article-Based Problems

A specific category of problems involves transactions with "neither loss nor profit" (Break-Even). If Shashi sells two items at ₹ $5000$ each and one yields a loss of $16\frac{2}{3}\%$ ($\text{ratio } 6:5$), the second item must yield a gain sufficient to cover the $1$ unit loss from the first. Since both items sell at the same $SP$ ($5$ units), the $CP$ of the second item must be $4$ (since total $CP$ must equal total $SP$, i.e., $6 + 4 = 5 + 5$). The profit on the second item is thus $\frac{1}{4} \times 100 = 25\%$. Similarly, for triple articles $X$, $Y$, and $Z$ sold at ₹ $2520$ each with zero overall profit/loss, and $X$ at $30\%$ loss and $Y$ at $36.36\%$ profit, the ratios determine the $CP$ of $Z$. At a common $SP$ of $105$, the combined gains/losses must cancel out, leading to a calculated $CP$ for $Z$ of ₹ $2112$.

Article-based questions require calculating price per unit. If items are bought at $5 \text{ for } ₹ 8$ ($\text{CP} = \frac{8}{5}$) and sold at $4 \text{ for } ₹ 9$ ($\text{SP} = \frac{9}{4}$), the ratio method ($CP:SP = 32:45$) reveals a profit of $\frac{13}{32} \times 100 = 40.625\%$ or approximately $325/8\%$. When dealing with quantity adjustments for a fixed budget (e.g., ₹ $1$), the relationship between price and quantity is inverse. Buying $7$ lemons for ₹ $1$ and seeking a $75\%$ profit ($\text{Ratio } 4:7$) requires selling $4$ lemons for ₹ $1$.

Cross Product and Allegation Methods in Profit and Loss

The Cross Product Method is utilized when both $CP$ and $SP$ are adjusted by specific numerical amounts. For instance, if an item sold at $20\%$ profit (ratio $5:6$) has its $CP$ increased by ₹ $50$ and $SP$ increased by ₹ $30$, resulting in a $10\%$ profit (ratio $10:11$), the cross product equation $|55 - 60| = |300 - 550|$ leads to $5x = 250$, identifying the original $CP$ as ₹ $250$. If $CP$ and $SP$ decrease by ₹ $75$, and profit moves from $25\%$ to $30\%$, the same logic leads to identifying the original selling price.

The Allegation Method is applied when a total quantity is split into segments with different profit/loss rates. If a vendor buys $240$ TVs and sells some at $20\%$ profit and the rest at $30\%$ profit for an overall $28\%$ profit, the allegation ratio ($2:8$ or $1:4$) shows that $1/5$ of the units ($48$ TVs) were sold at $20\%$. This method also resolves investment problems where a total sum (e.g., ₹ $5,000$) is split between a $10\%$ profit item and a $5\%$ loss item to achieve a net profit of ₹ $100$ (effectively a $2\%$ overall profit). The resulting ratio ($7:8$) allows for the calculation of the specific investment amounts.

The Succession Method for Chain Transactions

The Succession Method addresses the chain of sales from person A to B to C and so on. For a chain $A \rightarrow B \text{ (10\% loss)} \rightarrow C \text{ (15\% profit)} \rightarrow D \text{ (10\% profit)}$ where $D$ pays ₹ $4,554$, the $CP$ for $A$ is found by the equation $x \times \frac{90}{100} \times \frac{115}{100} \times \frac{110}{100} = 4554$, which solves to $x = ₹ 4,000$. This is applied similarly to manufacturer-wholesaler-retailer chains. In a case where a customer pays ₹ $330$ after three successive profit stages of $25\%$, $20\%$, and $10\%$, the initial manufacturer cost is calculated as $x \times \frac{5}{4} \times \frac{6}{5} \times \frac{11}{10} = 330$, resulting in $x = ₹ 200$. For high-precision requirements, such as a water geyser with successive profits of $5\%$, $10\%$, and $15\%$ on a retail price of ₹ $11,300$, the $CP$ is found to be approximately ₹ $8,507.43$.