8.3 Profit, Loss, Marked Price, Discount & Dishonest Seller Traps

8.3 Profit, Loss, Marked Price, Discount & Dishonest Seller Traps

Commercial arithmetic problems evaluating Profit, Loss, Marked Price, and Trade Discounts represent a foundational pillar of the SBI Clerk Numerical Ability syllabus. Beyond standalone arithmetic word problems, these principles feature prominently in commercial caselet Data Interpretation sets (e.g., evaluating retail merchant loan portfolios, micro-enterprise profitability, or inventory financing). Success requires an absolute command of price transition mechanics and the ability to detect subtle deceptive wording in dishonest merchant and retail promotion questions.

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The Price Hierarchy: CP, SP, and MP

Commercial transactions follow a defined progression of three financial benchmarks:

1. Cost Price (CP): The total expenditure incurred to acquire, manufacture, or procure an asset, including overheads, freight, and servicing charges.
2. Marked Price (MP): Also known as List Price, Printed Price, or Maximum Retail Price (MRP). This is the nominal value displayed before offering concessions.
3. Selling Price (SP): The actual net monetary consideration received from the buyer upon concluding the transaction.

   [Cost Price (CP)]  ---(+ Markup %)---->  [Marked Price (MP)]
           |                                        |
           |                                  (- Discount %)
           |                                        |
           v                                        v
   [Profit / Loss %] <---------------------- [Selling Price (SP)]

Master Commercial Arithmetic Formula Reference Table

[!IMPORTANT] The Base Allocation Rule: Profit and Loss are always computed upon Cost Price (CP) unless the problem explicitly states "profit calculated on selling price". Conversely, Discount is strictly computed upon Marked Price (MP).

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The Central Bridge Equation: Connecting MP and CP Directly

In many SBI Clerk problems, the candidate is given the Markup/Discount percentage and asked to find the net Profit percentage, or vice versa. Solving for intermediate Selling Price ($SP$) wastes critical seconds. We equate the two independent expressions for $SP$:

$SP = CP \times \left(1 + \frac{P}{100}\right) = MP \times \left(1 - \frac{D}{100}\right)$

Rearranging terms establishes the Universal MP/CP Ratio Equation:

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

If the transaction results in a net loss of $L%$, the formula adjusts to:

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

Practical Application of the Bridge Equation

A retail merchant financed by an SBI SME loan wishes to earn a net profit of $20%$ after allowing a festive discount of $10%$ on the Marked Price. What markup above Cost Price must be applied? $\frac{MP}{CP} = \frac{100 + 20}{100 - 10} = \frac{120}{90} = \frac{4}{3}$

• The ratio of $MP$ to $CP$ is $4 : 3$.
• Markup $= 4 - 3 = 1$ unit on a base of $CP = 3$ units.
• $\text{Markup } % = \frac{1}{3} \times 100% = 33.33%$.
• The merchant must mark the goods $33.33%$ above cost.

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Successive Discounts & Single Equivalent Discount (SED)

Retail establishments frequently promote multiple successive discounts (e.g., "30% off plus an additional 20% clearance discount"). Successive discounts operate on diminishing intermediate balances, meaning their cumulative effect is strictly less than their arithmetic sum.

Two Successive Discounts Formula

For two successive discount rates $d_1%$ and $d_2%$:

$\text{Single Equivalent Discount (SED)} = \left(d_1 + d_2 - \frac{d_1 \times d_2}{100}\right)\%$

Example: Two successive discounts of $20%$ and $10%$: $\text{SED} = 20 + 10 - \frac{20 \times 10}{100} = 30 - 2 = 28\%$

Three or More Successive Discounts (Fractional Multiplier Method)

When evaluating three discounts ($d_1, d_2, d_3$), applying the two-stage formula repeatedly is cumbersome. Instead, compute the surviving Selling Price fraction:

$\frac{SP}{MP} = \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \times \left(1 - \frac{d_3}{100}\right)$ $\text{SED} = \left(1 - \frac{SP}{MP}\right) \times 100\%$

For discounts of $20%$, $25%$, and $10%$: $\frac{SP}{MP} = \frac{4}{5} \times \frac{3}{4} \times \frac{9}{10} = \frac{27}{50} = 0.54$ $\text{SED} = (1 - 0.54) \times 100\% = 46\%$

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The Dishonest Merchant / False Weight Traps

Questions involving dishonest shopkeepers represent one of the most reliable question archetypes in banking exams. Candidates frequently falter because they benchmark profit against the nominal quantity requested by the customer rather than the physical inventory actually parted with by the merchant.

Conceptual Core: The Real Cost Incurred

A dishonest trader professes to sell goods at Cost Price ($SP = CP$ per kg) but uses a fraudulent weight that measures only $800\text{ g}$ instead of a true $1000\text{ g}$ ($1\text{ kg}$).

• The customer pays for $1000\text{ g}$ worth of goods.
• The merchant actually surrenders only $800\text{ g}$ of inventory.
• The merchant's cost is strictly the cost of $800\text{ g}$.
• The merchant's profit is the value of the $200\text{ g}$ withheld.

The Universal False Weight Profit Formula

$\text{Profit } \% = \left(\frac{\text{Error}}{\text{True Value} - \text{Error}}\right) \times 100\% = \left(\frac{\text{Error}}{\text{Weight Actually Dispensed}}\right) \times 100\%$

Substituting the values: $\text{Profit } \% = \left(\frac{1000 - 800}{800}\right) \times 100\% = \frac{200}{800} \times 100\% = \frac{1}{4} \times 100\% = 25.00\%$

[!WARNING] The 20% Distractor Trap: Untrained candidates compute $\frac{200}{1000} \times 100% = 20%$. This option is intentionally placed as a prominent distractor. Profit must be calculated on the goods actually surrendered by the seller ($800\text{ g}$), never on the nominal weight requested by the buyer ($1000\text{ g}$). The correct profit is $25%$.

Dual Deception: Markup Coupled with False Weight

If a merchant marks up the goods by $m%$ above cost AND additionally uses a false weight that is $w%$ short, the net multiplier is compound:

$\text{Overall Multiplier} = \left(1 + \frac{m}{100}\right) \times \left(\frac{\text{Claimed Weight}}{\text{Actual Weight Dispensed}}\right)$

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Promotional Schemes: "Buy $x$ Get $y$ Free"

Retail marketing schemes offering free inventory units are mathematically equivalent to trade discounts:

Derivation of Effective Discount

Suppose a store advertises: "Buy 4, Get 1 Free".

1. Let the Marked Price of each individual item be Rs. 100.
2. The customer takes home a total of $4 + 1 = 5$ items.
3. The total Marked Value of goods transferred is $5 \times 100 = \text{Rs. } 500$.
4. The customer pays cash for only 4 items: $4 \times 100 = \text{Rs. } 400$.
5. The effective monetary concession is Rs. 100.

$\text{Discount } \% = \left(\frac{\text{Free Items}}{\text{Total Items Transferred}}\right) \times 100\% = \left(\frac{y}{x + y}\right) \times 100\%$

For "Buy 4 Get 1 Free": $\text{Discount } \% = \frac{1}{4 + 1} \times 100\% = \frac{1}{5} \times 100\% = 20\%$

Hybrid Retail Schemes (Free Items Plus Cash Discount)

When a merchant offers "Buy 4 Get 1 Free AND an additional 20% discount on cash payment", calculate the compound effect using successive discounts:

• First Discount ($d_1$ from free items): $\frac{1}{4+1} = 20%$.
• Second Discount ($d_2$ from cash discount): $20%$.
• Single Equivalent Discount: $\text{SED} = d_1 + d_2 - \frac{d_1 \times d_2}{100} = 20 + 20 - \frac{400}{100} = 40 - 4 = 36\%$

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Step-by-Step Worked Commercial Examination Problems

Problem 1: Combined Markup, Discount, and Net Profit

Problem: A shopkeeper marks an article $40%$ above its Cost Price and sells it after granting a discount of $15%$ on the Marked Price. If the shopkeeper realizes a net profit of Rs. 380, determine the Cost Price of the article.

Solution:

1. Let the Cost Price ($CP$) be $100x$.
2. The Marked Price ($MP$) is marked $40%$ above cost: $MP = 100x \times 1.40 = 140x$.
3. Discount of $15%$ is deducted from $MP$: $\text{Discount} = 140x \times 0.15 = 21x$
4. Compute the Selling Price ($SP$): $SP = MP - \text{Discount} = 140x - 21x = 119x$
5. Compute the net profit: $\text{Profit} = SP - CP = 119x - 100x = 19x$
6. Equate to the actual profit of Rs. 380: $19x = 380 \implies x = \frac{380}{19} = 20$
7. Cost Price $= 100x = 100 \times 20 = \text{Rs. } 2,000$.

Problem 2: Dishonest Merchant with Compound Deception

Problem: A grocery dealer claims to sell pulses at a loss of $5%$ on Cost Price, but uses a fraudulent weight measuring $900\text{ grams}$ in place of $1\text{ kilogram}$ ($1000\text{ grams}$). What is the dealer's actual overall percentage gain or loss?

Solution:

1. Let the Cost Price of $1000\text{ g}$ be Rs. 1000 (i.e., Re. 1 per gram).
2. The dealer professes to sell $1000\text{ g}$ at a $5%$ loss: $\text{Nominal Selling Price for } 1000\text{ g} = 1000 \times (1 - 0.05) = \text{Rs. } 950$
3. However, the dealer physically dispenses only $900\text{ g}$ of pulses.
4. The actual cost incurred by the dealer for the dispensed goods is: $\text{Actual Cost} = 900 \times 1 = \text{Rs. } 900$
5. The dealer receives Rs. 950 from the customer while incurring a cost of only Rs. 900.
6. Compute the net profit: $\text{Net Profit} = 950 - 900 = \text{Rs. } 50$
7. Calculate percentage profit on the actual cost: $\text{Profit } \% = \frac{50}{900} \times 100\% = \frac{5}{90} \times 100\% = \frac{1}{18} \times 100\% = 5\frac{5}{9}\% \approx 5.56\%$
8. Despite advertising a $5%$ loss, the fraudulent scale yields an actual net gain of $5.56%$.