3.1 Percentages, Fractional Equivalents, Successive Variations & Population Formulas

Core Percentage Concepts & Fractional Equivalents

A percentage is a dimensionless ratio expressing a fraction of 100. Mathematically, $x\%$ represents $\frac{x}{100}$. To achieve speed on the exam, you must eliminate manual long division by converting percentage expressions directly into benchmark fractions.

Decimal and percentage benchmarks below are rounded approximations when the exact expansion repeats. Use the displayed exact fractions or mixed-number percentages for precise calculations.

Percentage-to-Fraction Memory Table ($1/2$ to $1/20$)

The following table forms the foundational memory grid required for rapid mental computation on the RRB CBT:

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Percentage Change vs. Percentage Points

A critical conceptual trap on railway examinations is confusing an absolute difference in percentages with a relative percentage change.

Distinguishing Definitions

1. Percentage Points: The simple arithmetic difference between two percentage values:

$$\Delta_{\text{points}} = P_2 - P_1$$

1. Percentage Change: The relative change normalized against the original baseline value:

$$\text{Percentage Change} = \left(\frac{P_2 - P_1}{P_1}\right) \times 100$$

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Successive Percentage Change Formula

When a base quantity undergoes two consecutive percentage changes of $a\%$ and $b\%$, the cumulative net percentage change is given by:

$$\text{Net Change}\% = a + b + \frac{ab}{100}$$

Algebraic Sign Conventions

• Increase / Inflation / Surcharge: $+a$ or $+b$
• Decrease / Reduction / Concession: $-a$ or $-b$

Extension to Three Successive Changes

For three consecutive percentage changes $a\%$, $b\%$, and $c\%$, compute the net effect in two sequential stages:

1. Find the intermediate rate $R_1 = a + b + \frac{ab}{100}$.
2. Compute the final rate: $R_{\text{final}} = R_1 + c + \frac{R_1 c}{100}$.

Alternatively, use the decimal multiplier product:

$$\text{Multiplier} = \left(1 + \frac{a}{100}\right) \left(1 + \frac{b}{100}\right) \left(1 + \frac{c}{100}\right)$$ $$\text{Net Change}\% = (\text{Multiplier} - 1) \times 100$$

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Product Constancy & Expenditure Invariance ($A \times B = C$)

In railway operational logistics, many core variables exist as two-factor products: $\text{Expenditure} = \text{Price} \times \text{Consumption}$, $\text{Distance} = \text{Speed} \times \text{Time}$, and $\text{Work} = \text{Manpower} \times \text{Hours}$. When the total product is held constant, an increase in one factor necessitates a reciprocal reduction in the other.

The Reciprocal Shift Principle

Let a factor increase by the fraction $\frac{a}{b}$:

$$X_{\text{new}} = X \left(1 + \frac{a}{b}\right) = X \left(\frac{b + a}{b}\right)$$

To keep the product invariant ($X_{\text{new}} \times Y_{\text{new}} = X \times Y$):

$$Y_{\text{new}} = Y \left(\frac{b}{b + a}\right) = Y \left(1 - \frac{a}{b + a}\right)$$ $$\mathbf{\text{Required Reduction} = \frac{a}{b + a}}$$

Conversely, if a positive factor decreases by $\frac{a}{b}$ with $0<a<b$:

$$\mathbf{\text{Allowable Increase} = \frac{a}{b - a}}$$

Reciprocal Conversion Matrix

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Population Growth & Asset Depreciation Models

Compounding growth and depreciation follow identical geometric trajectories across discrete time intervals.

Governing Mathematical Formulas

1. Population / Asset Growth after $n$ periods:

$$P_n = P_0 \left(1 + \frac{r}{100}\right)^n$$

1. Asset Depreciation / Value Decay after $n$ periods:

$$V_n = V_0 \left(1 - \frac{r}{100}\right)^n$$

1. Value / Population $n$ periods AGO:

$$P_{\text{past}} = \frac{P_n}{\left(1 \pm \frac{r}{100}\right)^n}$$

1. Variable Rates across successive years ($r_1, r_2, r_3$):

$$P_3 = P_0 \left(1 \pm \frac{r_1}{100}\right) \left(1 \pm \frac{r_2}{100}\right) \left(1 \pm \frac{r_3}{100}\right)$$

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Examination Passing Marks & Electoral Margin Models

Examination Threshold Mechanics

When two candidates obtain different scores and variances relative to the passing score:

• Candidate 1 scores $x\%$ and fails by $m$ marks: $\text{Pass Mark} = \frac{x}{100} T + m$.
• Candidate 2 scores $y\%$ and secures $n$ marks above pass mark: $\text{Pass Mark} = \frac{y}{100} T - n$.

Equating both pass mark formulations:

$$\frac{y - x}{100} T = m + n \implies \mathbf{T = \frac{(m + n) \times 100}{y - x}}$$ $$\mathbf{\text{Pass Percentage}} = x + \left(\frac{m}{T} \times 100\right) = y - \left(\frac{n}{T} \times 100\right)$$

Election Problem Architecture

In two-candidate elections with invalid votes:

1. Total Enrolled Voters $= V$.
2. Votes Cast $= V \times (1 - \text{Voter Absenteeism}\%)$.
3. Valid Votes $= \text{Votes Cast} \times (1 - \text{Invalid}\%)$.
4. If the winner secures $W\%$ of valid votes, the runner-up secures $(100 - W)\%$ of valid votes.
5. Winning Margin: $\Delta V = (W - (100 - W))\% \times \text{Valid Votes} = (2W - 100)\% \times \text{Valid Votes}$.

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Comprehensive Worked Operational Problems

Worked Problem 1: Electric Traction Power Consumption Invariance

Scenario: Due to peak summer tariff revisions, the unit rate of traction electricity for an electric locomotive operating shed increases by $33\frac{1}{3}\%$. In this fictional calculation, assume energy use is proportional to train-kilometers and all other factors stay constant. If the energy budget is unchanged, by what percentage must train-kilometers decrease?

Step-by-Step Mathematical Resolution:

1. Identify the Fractional Price Shift:

$$33\frac{1}{3}\% = \frac{1}{3}$$

1. Apply the Product Constancy Reciprocal Shift: Here $a = 1$ and $b = 3$. An increase of $+a/b$ requires a reduction of $\frac{a}{b + a}$:

$$\text{Required Reduction} = \frac{1}{3 + 1} = \frac{1}{4}$$

1. Convert to Percentage:

$$\frac{1}{4} \times 100 = 25.0\%$$

1. Verification via Baseline Values: Let initial rate $= \text{INR } 3.00$ per unit, consumption $= 300$ units. Total expenditure $= 3 \times 300 = \text{INR } 900$. New rate $= 3 + \frac{1}{3}(3) = \text{INR } 4.00$. New consumption $= \frac{900}{4} = 225$ units. Reduction $= 300 - 225 = 75$ units. Percentage reduction $= \frac{75}{300} \times 100 = 25.0\%$.

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Worked Problem 2: Multi-Stage Asset Depreciation of Maintenance Consist

Scenario: A newly procured mechanized track-laying maintenance train is capitalized at ₹$80,00,000$. Its asset valuation depreciates by $10\%$ during its first year of heavy service, by $15\%$ during the second year, and by $20\%$ in the third year. Calculate the book salvage value of the equipment at the end of the three-year period.

Step-by-Step Mathematical Resolution:

1. Formulate the Annual Multipliers:

   • Year 1 Multiplier: $1 - \frac{10}{100} = 0.90 = \frac{9}{10}$
   • Year 2 Multiplier: $1 - \frac{15}{100} = 0.85 = \frac{17}{20}$
   • Year 3 Multiplier: $1 - \frac{20}{100} = 0.80 = \frac{4}{5}$

2. Compute Composite Depreciation Multiplier:

$$\text{Composite Multiplier} = \frac{9}{10} \times \frac{17}{20} \times \frac{4}{5} = \frac{9 \times 17 \times 4}{1000} = \frac{612}{1000} = 0.612$$

1. Calculate Final Salvage Book Value ($V_3$):

$$V_3 = 80,00,000 \times 0.612 = \text{INR } 48,96,000$$

1. Determine Cumulative Depreciation Incurred:

$$\text{Total Depreciation} = 80,00,000 - 48,96,000 = \text{INR } 31,04,000 \quad (38.8\%\text{ cumulative decay})$$

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Worked Problem 3: Railway Recognized Union Election Analysis

Scenario: In an election for the Railway Recognized Employees' Union, $10\%$ of the voters on the electoral roll did not cast their ballots. Of the total votes cast, $10\%$ were declared invalid due to ballot marking discrepancies. The successful candidate polled $54\%$ of the total valid votes and defeated the sole rival by a margin of 1,620 votes. Determine the total number of eligible voters enrolled.

Step-by-Step Mathematical Resolution:

1. Define the Base Variable: Let the total number of enrolled voters be $V$.

2. Evaluate Votes Cast and Valid Votes:

$$\text{Votes Cast} = V \times (1 - 0.10) = 0.90 V$$ $$\text{Valid Votes} = 0.90 V \times (1 - 0.10) = 0.90 V \times 0.90 = 0.81 V$$

1. Formulate the Winning Majority Margin:

   • Winner's share of valid votes $= 54\%$
   • Rival's share of valid votes $= 100\% - 54\% = 46\%$
   • Majority margin percentage $= 54\% - 46\% = 8\%$

2. Set Up the Algebraic Equality:

$$8\% \text{ of Valid Votes} = 1,620$$ $$\frac{8}{100} \times 0.81 V = 1,620$$ $$0.0648 V = 1,620$$

1. Isolate $V$:

$$V = \frac{1620}{0.0648} = \frac{16200000}{648} = 25,000$$

There were exactly 25,000 registered voters on the electoral roll.

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Common Exam Traps & Conceptual Pitfalls