3.2 Cost Price, Selling Price, Profit/Loss Percentage, Dishonest Dealer & Discount Schemes

Core Commercial Parameters & Accounting Foundations

Every commercial transaction revolves around three primary price thresholds:

1. Cost Price (CP): The total capital outlay required to procure, manufacture, or transport an item. Any auxiliary transportation, installation, or refurbishment overheads must be added to the initial purchase price to establish the effective CP:

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

1. Selling Price (SP): The actual monetary value realized upon the transfer or sale of the asset.
2. Marked Price (MP) / List Price: The catalogue price or base quoted tariff prior to concessions.

Foundational Mathematical Relationships

$$\text{Profit} = SP - CP \quad (\text{when } SP > CP)$$ $$\text{Loss} = CP - SP \quad (\text{when } CP > SP)$$ $$\mathbf{\text{Profit}\% = \left(\frac{SP - CP}{CP}\right) \times 100 = \left(\frac{SP}{CP} - 1\right) \times 100}$$ $$\mathbf{\text{Loss}\% = \left(\frac{CP - SP}{CP}\right) \times 100 = \left(1 - \frac{SP}{CP}\right) \times 100}$$ $$\text{Markup}\% = \left(\frac{MP - CP}{CP}\right) \times 100$$ $$\text{Discount}\% = \left(\frac{MP - SP}{MP}\right) \times 100$$

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The Master Unified Bridge Formula ($MP / CP$)

Selling price ($SP$) serves as the connecting link between Cost Price and Marked Price:

$$SP = CP \times \left(\frac{100 + P}{100}\right)$$ $$SP = MP \times \left(\frac{100 - D}{100}\right)$$

Equating both expressions eliminates the selling price entirely:

$$CP \times (100 + P) = MP \times (100 - D)$$ $$\mathbf{\frac{MP}{CP} = \frac{100 + P}{100 - D}}$$

If the transaction results in a net commercial loss, the formulation adapts directly:

$$\mathbf{\frac{MP}{CP} = \frac{100 - L}{100 - D}}$$

This single ratio eliminates the need to calculate intermediate selling prices, cutting exam calculation steps in half.

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Successive Discounts & Promotional Schemes

Single Equivalent Discount Formula

When successive commercial discounts of $d_1\%$ and $d_2\%$ are granted sequentially on Marked Price, each subsequent discount applies to the diminished balance:

$$\mathbf{D_{\text{eq}} = d_1 + d_2 - \frac{d_1 d_2}{100}}$$

For three successive discounts $d_1\%$, $d_2\%$, and $d_3\%$:

$$SP = MP \left(1 - \frac{d_1}{100}\right) \left(1 - \frac{d_2}{100}\right) \left(1 - \frac{d_3}{100}\right)$$ $$D_{\text{eq}} = 100 \times \left[1 - \left(1 - \frac{d_1}{100}\right) \left(1 - \frac{d_2}{100}\right) \left(1 - \frac{d_3}{100}\right)\right]$$

Buy $x$, Get $y$ Free Promotional Architecture

The effective-discount calculation assumes identical unit prices for the paid and free items. In commercial tenders and retail concessions, promotions often offer free items upon purchasing a specified batch. The true discount percentage represents the ratio of free goods to total goods delivered:

$$\mathbf{\text{Discount}\% = \frac{\text{Free Quantity}}{\text{Total Quantity Delivered}} \times 100 = \frac{y}{x + y} \times 100}$$

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Dishonest Dealers & Weighbridge Errors

In bulk freight loading yards, mineral sidings, and parcel hubs, weighbridge calibration directly determines billing accuracy. Dishonest dealer models analyze situations where a vendor claims to sell at cost price but dispenses a fraudulent weight or measure.

Core Formulation

When a seller claims to deliver a true measure $W_{\text{true}}$ but dispenses only $W_{\text{false}}$:

• Revenue collected corresponds to $W_{\text{true}}$.
• Actual cost outlay incurred corresponds only to $W_{\text{false}}$.

$$\mathbf{\text{Profit}\% = \frac{W_{\text{true}} - W_{\text{false}}}{W_{\text{false}}} \times 100 = \frac{\text{Error}}{\text{True Value} - \text{Error}} \times 100}$$

Compound Fraud Architecture (Markup Combined with False Weight)

When a dealer marks up the price by $m\%$, grants an optional discount of $d\%$, and uses a false measure $W_{\text{false}}$ for $W_{\text{true}}$:

$$\text{Net Multiplier} = \left(1 + \frac{m}{100}\right) \times \left(1 - \frac{d}{100}\right) \times \left(\frac{W_{\text{true}}}{W_{\text{false}}}\right)$$ $$\mathbf{\text{Net Profit}\% = (\text{Net Multiplier} - 1) \times 100}$$

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Twin Items at Equal Selling Price with Symmetrical Gain and Loss

A classic railway CBT question involves selling two distinct items at the exact same selling price ($SP_1 = SP_2 = SP>0$), where one item yields a profit of $x\%$ and the other incurs a loss of $x\%$, with $0<x<100$.

Equal-Selling-Price Loss Rule ($0<x<100$)

Because the base cost prices ($CP_1$ and $CP_2$) are unequal, the percentage loss operates on a larger base than the percentage profit. Therefore, the overall transaction always results in a net loss:

$$\mathbf{\text{Net Loss}\% = \left(\frac{x}{10}\right)^2\% = \frac{x^2}{100}\%}$$

Absolute Monetary Net Loss Formula

To compute the absolute rupee loss directly from the common selling price $SP$:

$$\mathbf{\text{Absolute Net Loss} = \frac{2 \times x^2 \times SP}{100^2 - x^2}}$$

Equal SP vs. Equal CP Comparison

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Comprehensive Worked Operational Problems

Worked Problem 1: Railway Catering Menu Pricing via Master Bridge

Scenario: A zonal railway catering licensee intends to realize a net profit of $20\%$ on premium thali meal boxes after granting an advertised promotional discount of $10\%$ on the printed menu price. If the actual cost price ($CP$) of preparing each meal box is ₹$180$, calculate:

1. The marked menu price ($MP$).
2. The percentage markup over cost price.

Step-by-Step Mathematical Resolution:

1. Apply the Master Unified Bridge:

$$\frac{MP}{CP} = \frac{100 + P}{100 - D}$$ $$\frac{MP}{180} = \frac{100 + 20}{100 - 10} = \frac{120}{90} = \frac{4}{3}$$

1. Solve for Marked Price ($MP$):

$$MP = 180 \times \frac{4}{3} = \text{INR } 240$$

1. Compute the Percentage Markup:

$$\text{Markup}\% = \frac{MP - CP}{CP} \times 100 = \frac{240 - 180}{180} \times 100 = \frac{60}{180} \times 100 = 33.33\%$$

1. Verification via Intermediate Selling Price:

$$SP = MP \times (1 - 0.10) = 240 \times 0.90 = \text{INR } 216$$ $$\text{Profit} = 216 - 180 = \text{INR } 36$$ $$\text{Profit}\% = \frac{36}{180} \times 100 = 20.0\%$$

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Worked Problem 2: Symmetrical Gain/Loss on Equal Selling Price Locomotives

Scenario: A divisional railway administration auctions two retired diesel shunting tractors for ₹$1,98,000$ each. On the first tractor, the administration realizes a profit of $10\%$, while on the second tractor, it incurs a loss of $10\%$. Calculate the overall net gain or loss percentage and the total absolute monetary loss on the dual sale.

Step-by-Step Mathematical Resolution:

1. Apply the Symmetrical Loss Theorem:

$$\text{Overall Loss}\% = \frac{x^2}{100}\% = \frac{10^2}{100}\% = 1.0\%$$

1. Evaluate Individual Cost Prices:

$$CP_1 = \frac{SP}{1 + 0.10} = \frac{198000}{1.10} = \text{INR } 1,80,000$$ $$CP_2 = \frac{SP}{1 - 0.10} = \frac{198000}{0.90} = \text{INR } 2,20,000$$

1. Calculate Aggregate Financials:

$$\text{Total CP} = 1,80,000 + 2,20,000 = \text{INR } 4,00,000$$ $$\text{Total SP} = 1,98,000 + 1,98,000 = \text{INR } 3,96,000$$ $$\text{Absolute Net Loss} = 4,00,000 - 3,96,000 = \text{INR } 4,000$$

1. Verify via the Direct Formula:

$$\text{Absolute Loss} = \frac{2 \times 10^2 \times 198000}{100^2 - 10^2} = \frac{200 \times 198000}{9900} = 200 \times 20 = \text{INR } 4,000$$

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Worked Problem 3: Multi-Factor Tampered Weighbridge Siding

Scenario: A siding contractor supplying locomotive coal advertises a price markup of $20\%$ above base procurement cost price and offers a nominal concession discount of $10\%$ on billing. However, inspection reveals that the electronic weighbridge sensor is calibrated so that when 800 kg of coal is loaded into a wagon hopper, the digital terminal records 1,000 kg. Calculate the contractor's actual net profit percentage.

Step-by-Step Mathematical Resolution:

1. Identify the Operational Multipliers:

   • Markup factor: $1 + \frac{20}{100} = \frac{6}{5} = 1.20$
   • Concession discount factor: $1 - \frac{10}{100} = \frac{9}{10} = 0.90$
   • False weight multiplier: $\frac{W_{\text{true}}}{W_{\text{false}}} = \frac{1000}{800} = \frac{5}{4} = 1.25$

2. Compute Composite Net Transaction Multiplier:

$$\text{Net Multiplier} = \frac{6}{5} \times \frac{9}{10} \times \frac{5}{4} = \frac{6 \times 9 \times 5}{5 \times 10 \times 4} = \frac{54}{40} = \frac{27}{20} = 1.35$$

1. Extract Actual Net Profit Percentage:

$$\text{Net Profit}\% = (1.35 - 1) \times 100 = 35.0\%$$

The contractor realizes an actual net gain of $35.0\%$.

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Common Exam Traps & Conceptual Pitfalls

In these shortcuts, $P,D,x,a,b$ are numerical percentage rates, such as $20$ for $20\%$. The equal-selling-price loss shortcut assumes $0<x<100$; at $x=0$ there is no loss. For false-weight problems the multiplier is billed quantity divided by delivered quantity; the delivered quantity determines the actual cost.