Ch 01 - Fractions in Disguise8th Ganita Prakash Part 2

• Percentage comes from the Latin 'per centum', meaning 'per hundred'.
• It is mathematically defined as a fraction where the denominator is strictly fixed at 100.
• Example: $x\%$ is mathematically equivalent to $\frac{x}{100}$.

Boundary Conditions / Domain:

• $x \in \mathbb{R}$.

• Easy Comparison: Fixing the base at 100 standardizes ratios, making it instantly clear which proportion is larger.
• Decimal Integration: Base 100 perfectly fits our Base-10 decimal system (e.g., $31\% = \frac{31}{100} = 0.31$).
• Cognitive Scale: 100 is large enough to show fine detail but small enough to grasp mentally.

• Percentages $> 100\%$ indicate a value strictly greater than the original whole or base.
• It acts as a multiplier strictly greater than 1.
• Examples:
  • $120\%$ = $1.2 \times \text{Original}$ (A $20\%$ increase over the base).
  • $200\%$ = $2.0 \times \text{Original}$ (Exactly double the base).
  • $250\%$ = $2.5 \times \text{Original}$.

• Symmetric Property: $x\% \text{ of } y = y\% \text{ of } x$.
• Algebraic Proof: $\frac{x}{100} \times y = \frac{xy}{100} = \frac{y}{100} \times x$.
• Application: Simplifies mental math. For example, calculating $5\%$ of $40$ is mathematically identical to calculating $40\%$ of $5$ (which is easily computed as $2$).

Component Anatomy:

• $a, b$: Fractional percentage components that sum up to a target percentage.
• $y$: The base quantity.

Boundary Conditions / Domain:

• Valid universally for all $a, b, y \in \mathbb{R}$.

Component Anatomy:

• $P$: The resulting percentage.
• $a$: Numerator of the fraction.
• $b$: Denominator of the fraction.

Boundary Conditions / Domain:

• $b \neq 0$.

Component Anatomy:

• $F$: The resulting fraction in simplest form.
• $z$: The given percentage value.

Boundary Conditions / Domain:

• $z \in \mathbb{R}$ (including decimals), but the final fraction should typically be rationalized to integer numerator and denominator.

Component Anatomy:

• $P_a$: The percentage share of component $a$.
• $a, b$: The individual terms (parts) in the ratio.

Boundary Conditions / Domain:

• $(a+b) \neq 0$ (The sum of ratio parts cannot be zero). Generally, $a, b > 0$.

Component Anatomy:

• $Q$: The calculated partial quantity.
• $y$: The given percentage value.
• $z$: The base quantity or whole.

Boundary Conditions / Domain:

• $z \in \mathbb{R}$. The resulting quantity $Q$ will share the exact same units as the base quantity $z$.

Component Anatomy:

• $W$: The whole or total original base quantity.
• $x$: The given percentage.
• $y$: The absolute partial value corresponding to $x\%$.

Boundary Conditions / Domain:

• $x \neq 0$ (Cannot divide by a zero percentage).

• Trap: Percentages are relative proportions, not absolute values. You cannot deduce absolute size from percentages alone if the base quantities differ.
• A smaller percentage of a significantly larger base can yield a higher absolute value than a high percentage of a small base.
• Rule: Always calculate the absolute quantity ($y\% \times \text{Base}$) before making a magnitude comparison.

Component Anatomy:

• $V_{\text{new}}$: The final quantity after the increase.
• $V_{\text{old}}$: The initial base quantity.

Boundary Conditions / Domain:

• $V_{\text{old}} > 0$ (Cannot calculate a valid percentage increase from an absolute zero base).
• $V_{\text{new}} > V_{\text{old}}$.

Component Anatomy:

• $V_{\text{old}}$: The initial base quantity.
• $V_{\text{new}}$: The final quantity after the decrease.

Boundary Conditions / Domain:

• $V_{\text{old}} > 0$.
• $V_{\text{new}} < V_{\text{old}}$.

• Both statements yield the exact same mathematical multiplier: $1.65$.
• '165% of Original' translates algebraically to $1.65 \times P$.
• 'Increased by 65%' translates to $P + (0.65 \times P) = 1.65 \times P$.
• Universal Rule / SOP: A percentage increase of $r\%$ creates a new multiplier equal to $(1 + \frac{r}{100})$ of the base.

• Cost Price (CP): The price at which the merchant/shopkeeper procures or manufactures the item.
• Marked Price (MP / MRP): The label price or initial quote shown to the customer before any discounts are applied.
• Selling Price (SP): The final transaction price the customer actually pays after all discounts.

• Gross Profit: The basic difference between Total Sales Revenue and the initial Cost of Goods Sold (COGS).
• $\text{Gross Profit} = \text{Total Sales} - \text{COGS}$.
• Net Profit: The actual bottom-line profit calculated after deducting all secondary operational overheads (e.g., employee salaries, transport, electricity) from the Gross Profit.
• $\text{Net Profit} = \text{Gross Profit} - \text{Operating Expenses}$.

Component Anatomy:

• $SP$: Selling Price.
• $CP$: Cost Price.

Boundary Conditions / Domain:

• $CP > 0$.
• $SP > CP$ (Strict mathematical condition for a profit to exist).

Component Anatomy:

• $CP$: Cost Price.
• $SP$: Selling Price.

Boundary Conditions / Domain:

• $CP > 0$.
• $CP > SP$ (Strict mathematical condition for a loss to exist).

Component Anatomy:

• $SP$: Selling Price.
• $CP$: Cost Price.
• $P\%, L\%$: Profit or Loss Percentage.

Boundary Conditions / Domain:

• $CP > 0$.
• $L\% \leq 100\%$ (Cannot lose more than 100% of the cost).

Component Anatomy:

• $MP$: Marked Price (Label Price).
• $SP$: Selling Price.

Boundary Conditions / Domain:

• $MP > 0$.
• $SP \leq MP$.

• Taxes like GST are always calculated on the Selling Price (SP) (i.e., the discounted price, not the Marked Price).
• Final Price Formula: $\text{Final Price} = SP + \left(\frac{\text{GST } \%}{100} \times SP\right)$.
• Using a multiplier, this is rapidly computed as: $\text{Final Price} = SP \times \left(1 + \frac{\text{GST } \%}{100}\right)$.

• Successive discounts compound. The second discount is applied strictly to the newly reduced price, not the original price.
• Example on ₹100:
  • $30\%$ off ₹100 = ₹70.
  • $20\%$ off ₹70 = ₹14 (not ₹20!).
• Final Price = ₹56. Total effective discount = $44\%$.
• Trap: Adding $30\% + 20\% = 50\%$ falsely assumes both discounts operate on the same $100\%$ base.

Component Anatomy:

• $D_{\text{effective}}$: The single equivalent discount percentage.
• $d_1, d_2$: The individual successive discount percentages.

Boundary Conditions / Domain:

• $0 \leq d_1, d_2 \leq 100$.

Component Anatomy:

• $Y$: The number of free items received.
• $X$: The number of items actually paid for.
• $X + Y$: The total number of items the customer takes home.

Boundary Conditions / Domain:

• $X > 0, Y \geq 0$.
• Strict assumption: All individual items have the exact same Marked Price (MP).

Component Anatomy:

• $L_{\%}$: The net loss percentage.
• $x$: The percentage by which the item is marked up and subsequently discounted.

Boundary Conditions / Domain:

• $x > 0$.
• Critical Edge-Case Note: This sequence always results in a net mathematical loss. It never results in a break-even, which is a common conceptual trap.

Component Anatomy:

• $a$: The numerical quantity of items sold.
• $b$: The numerical quantity of items bought.

Boundary Conditions / Domain:

• $a, b \in \mathbb{Z}^+$ (Must be strictly positive integers).
• If the final computed result is strictly positive ($>0$), the transaction is a Profit. If strictly negative ($<0$), it is a Loss.

• Simple Interest (Linear Growth): Interest is calculated only on the original Principal. The principal remains permanently constant, yielding the exact same absolute interest amount every period.
• Compound Interest (Exponential Growth): Accumulated interest is added back to the Principal after each period. The principal continuously increases, causing the interest earned to grow exponentially over time.

Component Anatomy:

• $SI$: Simple Interest generated.
• $A$: Total Amount at maturity.
• $P$: Principal (Initial deposit/loan).
• $r$: Rate of interest per annum (percentage).
• $t$: Time period in years.

Boundary Conditions / Domain:

• $P \geq 0$.
• $r \geq 0$.
• $t \geq 0$ (Time cannot be negative).

Component Anatomy:

• $A$ / $V_t$: Total Amount (or Depreciated Value) after $t$ periods.
• $P$ / $V_0$: Initial Principal (or Original Value).
• $r$: Rate per compounding/depreciation period.
• $t$: Number of periods.

Boundary Conditions / Domain:

• $P, r \geq 0$.
• $t \in \mathbb{Z}^+$ (for discrete periodic compounding, $t$ must be a positive integer).

Component Anatomy:

• $r$: The given percentage increase of the first variable (e.g., length).
• $D_{\%}$: The required percentage decrease of the second variable (e.g., breadth) needed to keep the product (e.g., area) perfectly constant.

Boundary Conditions / Domain:

• $r > 0$.
• This theorem applies strictly to any mathematical scenario governed by the equation $A \times B = \text{Constant}$ (e.g., Length $\times$ Breadth = Area, Price $\times$ Consumption = Expenditure, Speed $\times$ Time = Distance).

• The Cognitive Trap: In dynamic mixture problems where one specific component leaves the system (e.g., water evaporating from a fruit, or 99% left-handed people dropping to 98% because some left the room), calculating based on the shifting total volume leads to immediate algebraic errors.
• The Solution: You must shift your mathematical focus entirely to the Constant Component (the item that does NOT change in absolute quantity during the event, such as the pulp in the fruit).