Profit & Loss, Discounts, Simple & Compound Interest

Commercial arithmetic forms an essential competency in healthcare facility administration, hospital procurement tenders, medical equipment lifecycle depreciation, and public health supply chain management. Whether calculating institutional trade discounts on bulk pharmaceutical purchases, computing the depreciation of intensive care ventilators, or evaluating credit society loans for healthcare personnel, quantitative proficiency in profit, loss, discount, and interest calculations ensures fiscal integrity.

1. Fundamentals of Profit and Loss

Commercial transactions involve three primary price parameters:

• Cost Price (CP): The total financial expenditure incurred to acquire or manufacture a commodity, including transportation, taxes, and handling overheads.
• Selling Price (SP): The final monetary value at which a commodity is transferred to a purchaser.
• Marked Price (MP) / List Price: The retail price printed on the packaging or catalog against which trade discounts are calculated.

The Mathematical Governing Rules

1. Profit (Gain): Occurs when the Selling Price is strictly greater than the Cost Price ($SP > CP$): $\text{Profit} = SP - CP$ $\text{Profit \%} = \left( \frac{\text{Profit}}{CP} \right) \times 100 = \left( \frac{SP - CP}{CP} \right) \times 100$
2. Loss: Occurs when the Cost Price is strictly greater than the Selling Price ($CP > SP$): $\text{Loss} = CP - SP$ $\text{Loss \%} = \left( \frac{\text{Loss}}{CP} \right) \times 100 = \left( \frac{CP - SP}{CP} \right) \times 100$
3. THE GOLDEN RULE OF COMMERCIAL ARITHMETIC: Profit percentage and Loss percentage are ALWAYS calculated on the Cost Price (CP) unless explicitly stated otherwise in the problem formulation.

Direct Formulas for Selling Price and Cost Price

• To Find Selling Price (SP): $\text{When Profit: } SP = CP \times \left( \frac{100 + \text{Profit \%}}{100} \right)$ $\text{When Loss: } SP = CP \times \left( \frac{100 - \text{Loss \%}}{100} \right)$
• To Find Cost Price (CP): $\text{When Profit: } CP = \frac{SP \times 100}{100 + \text{Profit \%}}$ $\text{When Loss: } CP = \frac{SP \times 100}{100 - \text{Loss \%}}$

Special Case: Two Items Sold at Identical Selling Price

When two separate commodities are sold at the identical Selling Price ($SP$), where one transaction yields a profit of $x\%$ and the other yields an identical loss of $x\%$, the vendor always suffers an overall loss on the combined transaction: $\text{Overall Loss \%} = \left( \frac{x}{10} \right)^2 = \frac{x^2}{100}\%$ (Important: The combined selling price does not alter the percentage loss, which depends exclusively on the common percentage figure $x$).

Worked Proof: A medical distributor sells two hemodialysis machines at the same selling price of ₹2,40,000 each. On the first machine, the distributor realizes a profit of 20%, while on the second machine, a loss of 20% is incurred. What is the net percentage outcome?

• Machine 1: $SP_1 = \text{INR } 2{,}40{,}000$ at $+20\%$. $CP_1 = \frac{2{,}40{,}000 \times 100}{120} = \text{INR } 2{,}00{,}000$.
• Machine 2: $SP_2 = \text{INR } 2{,}40{,}000$ at $-20\%$. $CP_2 = \frac{2{,}40{,}000 \times 100}{80} = \text{INR } 3{,}00{,}000$.
• Combined Cost Price: $CP = 2{,}00{,}000 + 3{,}00{,}000 = \text{INR } 5{,}00{,}000$.
• Combined Selling Price: $SP = 2{,}40{,}000 + 2{,}40{,}000 = \text{INR } 4{,}80{,}000$.
• Combined Loss: $5{,}00{,}000 - 4{,}80{,}000 = \text{INR } 20{,}000$.
• Overall Loss%: $\frac{20{,}000}{5{,}00{,}000} \times 100 = 4\%$.
• Using Formula: $\frac{20^2}{100}\% = \frac{400}{100}\% = \mathbf{4\%\text{ Loss}}$.

2. Discounts, Successive Discounts & Dishonest Trader Math

Trade Discounts

A discount is a price reduction granted from the Marked Price (MP): $\text{Discount} = MP - SP$ $\text{Discount \%} = \left( \frac{\text{Discount}}{MP} \right) \times 100$ $SP = MP \times \left( \frac{100 - \text{Discount \%}}{100} \right)$ (Note: While profit/loss is anchored to CP, discount is strictly anchored to MP).

Successive Discounts Formula

Vendors frequently offer successive trade discounts (e.g., an institutional clearance discount followed by a prompt payment discount). If successive discounts of $d_1\%$ and $d_2\%$ are granted on a marked price:

• After first discount: Price becomes $MP \times \left(1 - \frac{d_1}{100}\right)$.
• After second discount: Final $SP = MP \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right)$.
• Equivalent Single Discount ($D_{\text{eq}}$): $D_{\text{eq}} = \left( d_1 + d_2 - \frac{d_1 \times d_2}{100} \right)\%$

Worked Example: A biomedical equipment vendor lists a physiological monitor at a marked price of ₹80,000 and offers two successive trade discounts of 15% and 10%. What is the final selling price?

• Method 1 (Formula): $D_{\text{eq}} = 15 + 10 - \frac{15 \times 10}{100} = 25 - 1.5 = 23.5\%$. $SP = 80{,}000 \times \left(1 - \frac{23.5}{100}\right) = 80{,}000 \times 0.765 = \mathbf{\text{INR } 61{,}200}$
• Method 2 (Stepwise): First discount $= 15\% \text{ of } 80{,}000 = \text{INR } 12{,}000 \implies 80{,}000 - 12{,}000 = \text{INR } 68{,}000$. Second discount $= 10\% \text{ of } 68{,}000 = \text{INR } 6{,}800 \implies 68{,}000 - 6{,}800 = \mathbf{\text{INR } 61{,}200}$.

Dishonest Dealer / False Weights

When a dishonest merchant professes to sell goods at Cost Price but uses an underweight measure (delivering less physical quantity than claimed): $\text{Gain \%} = \left( \frac{\text{Error}}{\text{True Value} - \text{Error}} \right) \times 100 = \left( \frac{\text{Claimed Measure} - \text{Actual Measure}}{\text{Actual Measure}} \right) \times 100$

• Example: If a distributor sells surgical cotton claiming 1,000 grams but delivers 900 grams using a false weight: $\text{Gain \%} = \frac{100}{900} \times 100 = 11.11\% = 11 \frac{1}{9}\%$.

3. Simple Interest (SI)

Simple Interest is a fixed percentage fee paid on the invariant initial borrowed capital (Principal) across the entire loan duration.

Mathematical Formulation

$\text{SI} = \frac{P \times R \times T}{100}$ Where:

• $P = \text{Principal sum borrowed or invested}$
• $R = \text{Annual rate of interest (\% per annum)}$
• $T = \text{Time duration in years}$
• $A = \text{Total accumulated maturity amount} = P + \text{SI} = P \left(1 + \frac{R \times T}{100}\right)$

Derived Equations for Problem Solving

$P = \frac{100 \times \text{SI}}{R \times T}, \quad R = \frac{100 \times \text{SI}}{P \times T}, \quad T = \frac{100 \times \text{SI}}{P \times R}$

Sum Multiplying Itself under Simple Interest

If a principal sum multiplies itself to become $n$ times its original value in $T$ years at simple interest:

• $\text{Amount } A = n \times P \implies \text{SI} = A - P = (n - 1)P$.
• Substituting into SI formula: $(n - 1)P = \frac{P \times R \times T}{100}$.
• Therefore, the required annual rate is: $R = \frac{100(n - 1)}{T}\%$
• Similarly, the time required is: $T = \frac{100(n - 1)}{R} \text{ years}$.
• Quick Deduction: A sum doubles itself ($n = 2$) in $T$ years when $R = \frac{100}{T}\%$. A sum triples itself ($n = 3$) when $R = \frac{200}{T}\%$.

4. Compound Interest (CI) & The CI–SI Divergence

Compound Interest incorporates accrued interest into the principal at defined compounding intervals, creating exponential or geometric capital growth.

Compounding Interval Formulas

1. Compounded Annually (Once per year): $A = P \left(1 + \frac{R}{100}\right)^T$ $\text{CI} = A - P = P \left[ \left(1 + \frac{R}{100}\right)^T - 1 \right]$
2. Compounded Semi-Annually / Half-Yearly (Twice per year): Rate is halved ($\frac{R}{2}$) and compounding periods are doubled ($2T$): $A = P \left(1 + \frac{R}{200}\right)^{2T}$
3. Compounded Quarterly (Four times per year): Rate is quartered ($\frac{R}{4}$) and compounding periods are quadrupled ($4T$): $A = P \left(1 + \frac{R}{400}\right)^{4T}$

The Mathematical Difference Between CI and SI

Because Simple Interest remains strictly linear while Compound Interest generates "interest on interest", the two values diverge systematically over time:

• For a Duration of 1 Year (compounded annually): $\text{CI}_1 = \text{SI}_1 \implies \text{Difference} = 0$
• For a Duration of 2 Years: $\text{Difference}_{2\text{ yrs}} = \text{CI} - \text{SI} = P \left( \frac{R}{100} \right)^2$
• For a Duration of 3 Years: $\text{Difference}_{3\text{ yrs}} = P \left( \frac{R}{100} \right)^2 \left( \frac{300 + R}{100} \right)$

Worked Example: A cooperative credit society provides a medical staff development loan of ₹50,000 at an annual interest rate of 8% for 2 years. What is the difference between the Compound Interest (compounded annually) and Simple Interest accrued?

• Use the 2-year difference formula: $\text{Difference} = P \left( \frac{R}{100} \right)^2$
• $\text{Difference} = 50{,}000 \times \left( \frac{8}{100} \right)^2 = 50{,}000 \times \frac{64}{10{,}000} = 5 \times 64 = \mathbf{\text{INR } 320}$.
• Verification: $\text{SI} = \frac{50{,}000 \times 8 \times 2}{100} = \text{INR } 8{,}000$. $A_{\text{CI}} = 50{,}000 \times (1.08)^2 = 50{,}000 \times 1.1664 = \text{INR } 58{,}320 \implies \text{CI} = \text{INR } 8{,}320$. $\text{CI} - \text{SI} = 8{,}320 - 8{,}000 = \mathbf{\text{INR } 320}$.

5. Hospital Capital Finance & Asset Depreciation

Capital assets in hospitals, such as CT scanners, mechanical ventilators, and automated blood analyzers, lose monetary value over time due to operational wear and technical obsolescence.

Diminishing Balance Depreciation Formula

If an asset with an initial purchase value $P$ depreciates at an annual rate of $R\%$ per annum, its depreciated value $V_n$ after $n$ years is calculated using the negative compounding formula: $V_n = P \left(1 - \frac{R}{100}\right)^n$

Worked Clinical Example: A district hospital purchases a computerized tomography (CT) scanner for ₹50,00,000. If the equipment depreciates at a rate of 10% per annum under the diminishing balance method, what will be the book value of the CT scanner at the end of 3 years?

• Identify parameters: $P = \text{INR } 50{,}00{,}000$, $R = 10\%$, $n = 3$.
• Apply depreciation formula: $V_3 = 50{,}00{,}000 \times \left(1 - \frac{10}{100}\right)^3 = 50{,}00{,}000 \times (0.9)^3$ $V_3 = 50{,}00{,}000 \times 0.729 = \mathbf{\text{INR } 36{,}45{,}000}$
• Total accumulated depreciation over 3 years: $50{,}00{,}000 - 36{,}45{,}000 = \text{INR } 13{,}55{,}000$ (₹13,55,000).