Comprehensive Study Guide: Commercial Arithmetic, Profit and Loss, and Percentage Fluctuations

Core Formulas and Definitions in Commercial Arithmetic

• Cost Price ($\text{C.P.}$): The original expenditure or total amount spent to acquire, produce, or manufacture a commodity or raw materials.

• Selling Price ($\text{S.P.}$): The price at which a good, product, or service is sold to an end consumer or client.

• Profit ($\text{Gain}$): The positive financial gain realized when the selling price exceeds the cost price:   $\text{Profit} = \text{S.P.} - \text{C.P.}$

• Profit Percentage ($\text{Profit } \%$): The relative profit calculated strictly as a proportion of the original cost price:   $\text{Profit } \% = \left(\frac{\text{Profit}}{\text{C.P.}}\right) \times 100$

• Revenue Equation: The total proceeds generated from selling an asset, combining the baseline cost and accrued profit:   $\text{Revenue} = \text{C.P.} + \text{Profit}$

• Marked Price ($\text{M.P.}$): The stated list price, sticker price, or tag price placed on an article prior to applying any promotional reductions.

• Discount: A monetary deduction allowed from the marked price to determine the actual realized selling price:   $\text{Discount} = \text{Discount Rate} \times \text{M.P.}$   $\text{S.P.} = \text{M.P.} - \text{Discount}$

• Percentage Increase: The relative expansion of an economic indicator or commodity price benchmark over time:   $\text{Percentage Increase} = \left(\frac{\text{Amount of Increase}}{\text{Original Amount}}\right) \times 100$

Classwork Exercises: Figure It Out (Page 19)

• Context and Date:

  • Session label: $\text{CW}$

  • Session record date: $28\text{-}10\text{-}26$

  • Reference: Figure it out, Page $19$

• Problem 1: Profit and Profit Percentage Computation

  • Given Values:

  • Selling Price ($\text{S.P.}$) = $\text{₹}\,110$

  • Cost Price ($\text{C.P.}$) = $\text{₹}\,75$

  • Profit Calculation:     $\text{Profit} = \text{S.P.} - \text{C.P.}$     $\text{Profit} = 110 - 75 = \text{₹}\,35$

  • Profit Percentage Evaluation:     $\text{Profit } \% = \left(\frac{\text{Profit}}{\text{C.P.}}\right) \times 100$     $\text{Profit } \% = \left(\frac{35}{75}\right) \times 100$

  • Arithmetic Simplification Steps:

  • Dividing both numerator and denominator by the common divisor $5$:       $\frac{35 \div 5}{75 \div 5} = \frac{7}{15}$

  • Direct multiplication of the reduced fraction:       $\frac{7 \times 100}{15} = \frac{700}{15} = \frac{140}{3}$

  • Long division conversion to decimal form ($75 \overline{) 35.0000}$):       $\frac{35}{75} = 0.4666\dots$

  • Multiplying by the percentage factor:       $0.4666 \times 100 = 46.66\%$

  • Final Value:     $\text{Profit } \% = 46.66\%$

Classwork Exercises: Multi-Tiered Profit and Margin Analysis (Page 13)

• Context and Date:

  • Notebook register: Page No. $13$

  • Date: $28/10/26$

  • Session designation: $\text{CW}$

• Problem 2: Determining Selling Price from Material Costs and Profit Margin

  • Given Parameters:

  • Cost of materials ($\text{C.P.}$) = $\text{₹}\,475$

  • Designated profit rate = $50\% \text{ of } 475$

  • Step-by-Step Profit Evaluation:     $\text{Profit} = \left(\frac{50}{100}\right) \times 475$     $\text{Profit} = \left(\frac{1}{2}\right) \times 475$     $\text{Profit} = \text{₹}\,237.5$

  • Selling Price Calculation:     $\text{S.P.} = \text{C.P.} + \text{Profit}$     $\text{S.P.} = 475 + 237.5$     $\text{S.P.} = \text{₹}\,712.5$

• Problem 3: Deriving Base Production Cost from Gross Revenue

  • Objective: Find the baseline cost given a target percentage profit margin and final accrued enterprise revenue.

  • Algebraic Modeling:

  • Let the original baseline cost be designated as variable $x$.

  • Set profit rate: $25\% \text{ of } x$     $\text{Profit} = \left(\frac{25}{100}\right) \times x = 0.25x$

  • Revenue Expression Formulation:     $\text{Revenue} = \text{Cost} + \text{Profit}$     $\text{Revenue} = x + 0.25x = 1.25x$

  • Given Revenue Magnitude:     $\text{Revenue} = 2.5\,\text{cr}$

  • Equation Setup and Resolution:     $1.25x = 2.5\,\text{cr}$     $x = \frac{2.5}{1.25}$

  • Normalizing decimal points by expanding the fraction by $1000$:       $x = \frac{2500}{1250} = 2$

  • Conclusion:     $\therefore \text{Cost is } \text{₹}\,2\,\text{cr}$

Marked Price, Discount Structures, and Commercial Retail Exercises (Page 14)

• Context and Date:

  • Notebook register: Page No. $14$

  • Date: $28/10/26$

  • Session designation: $\text{CW}$ and $\text{HW}$

• Problem 4: Retail Discount and Net Selling Price for an Apparel Item

  • Item Profile: Garment designated as a shirt.

  • Given Parameters:

  • Marked Price ($\text{M.P.}$) of shirt = $\text{₹}\,300$

  • Discount specification = $25\% \text{ of } \text{₹}\,300$

  • Discount Computation:     $\text{Discount} = \left(\frac{25}{100}\right) \times 300$     $\text{Discount} = 25 \times 3 = \text{₹}\,75$

  • Net Selling Price Determination:     $\text{S.P.} = \text{M.P.} - \text{Discount Amount}$     $\text{S.P.} = 300 - 75$     $\text{S.P.} = \text{₹}\,225$

• Problem 6: Algebraic Formulation of Proportional Markdowns

  • Task Context: Homework assignment ($\text{HW}$).

  • Initial Conditions:

  • Let the Marked Price ($\text{M.P.}$) be variable $x$.

  • Discount rate applied = $15\% \text{ of } x$

  • Proportional Value:     $\text{Discount} = \left(\frac{15}{100}\right) \times x$     $\text{Discount} = 0.15x$

Commodity Price Inflation and Percentage Increase Analysis (Page 15)

• Context and Date:

  • Notebook register: Page No. $15$

  • Date: $28/10/26$

  • Work category: $\text{HW}$

• Problem 5: Longitudinal Petrol Cost Analysis ($2015$ vs. $2025$)

  • Historical and Future Price Benchmarks:

  • Price of petrol in fiscal baseline year $2015$ = $\text{₹}\,60$

  • Price of petrol in projected year $2025$ = $\text{₹}\,100$

  • Absolute Increase Calculation:     $\text{Amount of Increase} = 100 - 60 = \text{₹}\,40$

  • Relative Expansion Formula:     $\text{Percentage Increase} = \left(\frac{\text{Amount of Increase}}{\text{Original Amount}}\right) \times 100$

  • Substitution and Numerical Calculation:     $\text{Percentage Increase} = \left(\frac{40}{60}\right) \times 100$

  • Simplifying the rational coefficient by cancelling factors of $20$:       $\frac{40 \div 20}{60 \div 20} = \frac{2}{3}$

  • Multiplying to compute the percentage:       $\text{Percentage Increase} = \frac{200}{3} = 66.66\%$

  • Final Deduction and Multiple-Choice Answer:

  • The total price of petrol has increased by $66.66\%$.

  • Resulting classification: Option $\text{(iv)}$ is confirmed correct.