8.2 Fractions, Ratios, Proportions & Percentage Mastery

8.2 Fractions, Ratios, Proportions & Percentage Mastery

Core Principle: Fractions, ratios, and percentages are not three separate mathematical subjects; they are three isomorphic notations for expressing the relative comparison of quantities. In competitive aptitude, fluency in instantly translating between fractions, ratios, and percentages is the single most powerful accelerator of calculation speed and conceptual precision.

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1. Fractions and Decimals: Taxonomy, Comparison & Recurring Forms

A fraction represents a part of a whole or the quotient of two quantities expressed in the form $\frac{a}{b}$, where $a$ is the numerator and $b$ is the non-zero denominator ($b \neq 0$).

1.1 Structural Classification of Fractions

• Proper Fraction: Numerator is strictly less than the denominator ($a < b$). The numerical value is strictly less than one (e.g., $\frac{3}{7}, \frac{5}{12}$). Proper fractions represent genuine fractional parts.
• Improper Fraction: Numerator is greater than or equal to the denominator ($a \ge b$). The numerical value is greater than or equal to one (e.g., $\frac{9}{4}, \frac{13}{5}$).
• Mixed Fraction: An improper fraction expressed as the combination of an integer quotient and a proper fraction remainder: $\frac{a}{b} = Q \frac{R}{b} = Q + \frac{R}{b} \quad \left(\text{e.g., } \frac{17}{5} = 3 \frac{2}{5}\right)$
• Equivalent Fractions: Fractions whose numerical values remain identical when both numerator and denominator are multiplied or divided by the same non-zero scalar $k$: $\frac{a}{b} = \frac{a \times k}{b \times k} = \frac{a \div k}{b \div k}$

1.2 High-Speed Fraction Comparison Techniques

In CG SET Paper I, questions frequently ask candidates to identify the largest, smallest, or ascending/descending order of fractions. Three standard methods exist:

Method 1: The Cross-Multiplication Method (Optimal for 2 Fractions)

To determine whether $\frac{a}{b} > \frac{c}{d}$ (where $b, d > 0$):

• Multiply $a \times d$ (left product) and $b \times c$ (right product).
• If $a \cdot d > b \cdot c$, then $\frac{a}{b} > \frac{c}{d}$.
• If $a \cdot d < b \cdot c$, then $\frac{a}{b} < \frac{c}{d}$.
• Demonstration: Compare $\frac{7}{11}$ and $\frac{9}{14}$. $7 \times 14 = 98 \quad \text{vs.} \quad 11 \times 9 = 99$ Since $98 < 99$, it follows immediately that $\frac{7}{11} < \frac{9}{14}$.

Method 2: The Lowest Common Multiple (LCM) Method (Optimal for 3 to 4 Fractions)

Convert all denominators to their LCM:

• Compare $\frac{2}{3}, \frac{3}{4}, \frac{5}{6}, \frac{7}{12}$.
• $\text{LCM}(3, 4, 6, 12) = 12$.
• Equivalent numerators: $\frac{8}{12}, \frac{9}{12}, \frac{10}{12}, \frac{7}{12}$.
• Ascending order: $\frac{7}{12} < \frac{2}{3} < \frac{3}{4} < \frac{5}{6}$.

Method 3: The Characteristic Difference Theorem

For positive proper fractions where the difference between denominator and numerator is identical ($(D - N) = k$):

• Rule for Proper Fractions ($N < D$): As the magnitude of the terms increases, the value of the fraction increases. $\text{If } \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{11}{12} \quad (D - N = 1) \implies \frac{3}{4} < \frac{5}{6} < \frac{7}{8} < \frac{11}{12}$
• Rule for Adding a Constant: If $x > 0$ is added to both numerator and denominator of a proper fraction ($\frac{N}{D} < 1$): $\frac{N + x}{D + x} > \frac{N}{D}$ (e.g., $\frac{2}{3} = 0.667 \implies \frac{2+5}{3+5} = \frac{7}{8} = 0.875 > 0.667$).

1.3 Terminating vs. Recurring Decimals

A rational fraction $\frac{p}{q}$ in irreducible form generates a terminating decimal if and only if the prime factorization of the denominator $q$ contains no prime factors other than 2 or 5 ($q = 2^m \times 5^n$). If $q$ contains any other prime factor (such as 3, 7, 11, 13), the decimal expansion is non-terminating and recurring.

Pure Recurring Decimals to Vulgar Fractions

In pure recurring decimals, repetition begins immediately after the decimal point. The equivalent fraction is the repeating block divided by as many 9s as there are repeating digits:

$0.\overline{a} = \frac{a}{9}, \quad 0.\overline{ab} = \frac{ab}{99}, \quad 0.\overline{abc} = \frac{abc}{999}$

• Examples: $0.\overline{7} = \frac{7}{9}$; $0.\overline{45} = \frac{45}{99} = \frac{5}{11}$; $0.\overline{123} = \frac{123}{999} = \frac{41}{333}$.

Mixed Recurring Decimals to Vulgar Fractions

In mixed recurring decimals, non-repeating digits appear between the decimal point and the repeating cycle. To convert to a vulgar fraction:

$\text{Fraction} = \frac{\text{Full Number (Non-repeating and repeating digits)} - \text{Non-repeating digits}}{\text{Denominator formed by as many 9s as repeating digits, followed by as many 0s as non-repeating digits}}$

$\text{Algebraic Formula: } 0.a\overline{b} = \frac{ab - a}{90}, \quad 0.ab\overline{cd} = \frac{abcd - ab}{9900}, \quad 0.a\overline{bcd} = \frac{abcd - a}{9990}$

• Worked Demonstration: Convert $0.4\overline{16}$ to a vulgar fraction: $\text{Fraction} = \frac{416 - 4}{990} = \frac{412}{990} = \frac{206}{495}$

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2. Ratios, Proportions & Partnership Mathematics

2.1 Foundational Concept and Properties of Ratios

A ratio is a mathematical comparison of two quantities of the same kind and measured in the same units, expressed as $a : b$ or $\frac{a}{b}$. In the ratio $a : b$, $a$ is called the antecedent and $b$ is called the consequent.

• Unit Independence: Because a ratio represents relative comparison through division, it is a dimensionless pure number ($a : b$ has no physical units).
• Scalar Invariance: Multiplying or dividing both antecedent and consequent by a non-zero constant $k$ leaves the ratio unchanged ($a : b = ka : kb$).

Derived Ratios

• Duplicate Ratio: $a^2 : b^2$
• Sub-duplicate Ratio: $\sqrt{a} : \sqrt{b}$
• Triplicate Ratio: $a^3 : b^3$
• Sub-triplicate Ratio: $\sqrt[3]{a} : \sqrt[3]{b}$
• Compound Ratio: The compound ratio of $(a : b)$ and $(c : d)$ is $(a \times c : b \times d) = ac : bd$.
• Inverse (Reciprocal) Ratio: The inverse of $a : b$ is $\frac{1}{a} : \frac{1}{b} = b : a$.

2.2 Proportions and Algebraic Theorems

An equality of two ratios constitutes a proportion. The expression $a : b :: c : d$ (or $\frac{a}{b} = \frac{c}{d}$) asserts that the ratio $a : b$ equals $c : d$. Here, $a$ and $d$ are called the extremes, while $b$ and $c$ are called the means.

$\text{Fundamental Law of Proportion: } a \cdot d = b \cdot c \iff \text{Product of Extremes} = \text{Product of Means}$

Continued Proportion & Proportional Quantities

Three quantities $a, b, c$ are in continued proportion if $\frac{a}{b} = \frac{b}{c}$.

1. Mean Proportional (Geometric Mean): $\frac{a}{x} = \frac{x}{b} \implies x^2 = ab \implies x = \sqrt{ab}$
2. Third Proportional: The third proportional to $a$ and $b$ is the value $x$ such that: $a : b :: b : x \implies x = \frac{b^2}{a}$
3. Fourth Proportional: The fourth proportional to three quantities $a, b, c$ is the value $x$ such that: $a : b :: c : x \implies x = \frac{b \cdot c}{a}$

Proportional Transformation Theorems

• Invertendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{b}{a} = \frac{d}{c}$.
• Alternando: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a}{c} = \frac{b}{d}$.
• Componendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a + b}{b} = \frac{c + d}{d}$.
• Dividendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a - b}{b} = \frac{c - d}{d}$.
• Componendo & Dividendo (C&D Rule): If $\frac{a}{b} = \frac{c}{d}$, then: $\frac{a + b}{a - b} = \frac{c + d}{c - d}$

2.3 Partitioning Quantities in Given Ratios

When a total quantity $Q$ is partitioned into parts according to the ratio $a : b : c$:

$\text{Total Proportional Parts} = a + b + c$ $\text{Part } 1 = \left(\frac{a}{a + b + c}\right) \times Q, \quad \text{Part } 2 = \left(\frac{b}{a + b + c}\right) \times Q, \quad \text{Part } 3 = \left(\frac{c}{a + b + c}\right) \times Q$

2.4 Partnership Mathematics

In business partnerships, profit is distributed based on both financial capital and time duration of the investment:

$\text{Profit } (P) \propto \text{Capital } (C) \times \text{Time } (T)$

1. Simple Partnership: All partners invest capital for identical durations ($T_1 = T_2 = T_3$): $P_1 : P_2 : P_3 = C_1 : C_2 : C_3$
2. Compound Partnership: Partners invest varying capitals for varying periods ($T_1, T_2, T_3$): $P_1 : P_2 : P_3 = (C_1 \times T_1) : (C_2 \times T_2) : (C_3 \times T_3)$
3. Active vs. Sleeping Partners: An active partner manages the enterprise and receives a specified salary or percentage of gross profit before the remaining profit is partitioned according to the investment-time ratio.

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3. Percentages: The Universal Quantitative Bridge

The word percentage originates from the Latin per centum ("by the hundred"). A percentage is simply a fraction with a denominator of 100:

$x\% = \frac{x}{100} = x \times 0.01$

3.1 Benchmark Fraction-to-Percentage Equivalence Table

Memorizing benchmark equivalents is essential for rapid paper solving in CG SET Paper I:

3.2 Percentage Change vs. Percentage Points

• Percentage Change: The relative change compared against the original base value: $\text{Percentage Change} = \left(\frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}}\right) \times 100\%$
• Percentage Points: The arithmetic difference between two percentage values: $\Delta_{\text{points}} = P_2\% - P_1\%$
  • Example: If an institution's pass percentage rises from $40\%$ to $50\%$, the absolute increase is 10 percentage points, but the relative percentage increase is $\frac{50 - 40}{40} \times 100\% = 25\%$. Confusing these two concepts is a frequent exam pitfall.

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4. Advanced Percentage Formulations & Economic Applications

4.1 Successive Percentage Changes

When a quantity undergoes two successive percentage changes of $a\%$ and $b\%$:

$\text{Net Percentage Change} = \left(a + b + \frac{a \cdot b}{100}\right)\%$

(Where positive values denote increases and negative values denote decreases).

• Two Successive Increases ($+a\%, +b\%$): Net increase $= a + b + \frac{ab}{100}$.
  • Demonstration: Two successive salary hikes of $20\%$ and $10\%$: $\text{Net Hike} = 20 + 10 + \frac{20 \times 10}{100} = 30 + 2 = 32\%$
• Equal Successive Increase and Decrease ($+x\%, -x\%$): $\text{Net Change} = +x - x + \frac{(+x)(-x)}{100} = -\frac{x^2}{100}\%$ Key Invariant: When a quantity is increased by $x\%$ and then decreased by $x\%$, there is always an overall net decrease of $\frac{x^2}{100}\%$.

4.2 Expenditure, Price, and Consumption Elasticity

Total expenditure ($E$) is the product of price per unit ($P$) and total consumption quantity ($C$):

$\text{Expenditure } (E) = \text{Price } (P) \times \text{Consumption } (C)$

If total expenditure must remain strictly constant when the price changes:

1. Price Increases by $r\%$: $\text{Required Reduction in Consumption} = \left(\frac{r}{100 + r}\right) \times 100\%$
2. Price Decreases by $r\%$: $\text{Possible Increase in Consumption} = \left(\frac{r}{100 - r}\right) \times 100\%$

• Demonstration: If the price of petrol rises by $25\%$ ($r = 25$), to keep expenditure constant: $\text{Reduction in Consumption} = \left(\frac{25}{100 + 25}\right) \times 100\% = \frac{25}{125} \times 100\% = \frac{1}{5} \times 100\% = 20\%$

4.3 Demographic & Depreciation Exponential Models

• Population Growth: If initial population $P_0$ grows at annual rate $r\%$ for $n$ years: $P_n = P_0 \left(1 + \frac{r}{100}\right)^n$
• Asset Depreciation: If machinery value $V_0$ depreciates at annual rate $r\%$ for $n$ years: $V_n = V_0 \left(1 - \frac{r}{100}\right)^n$

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5. Worked Exemplars with Step-by-Step Algebraic Breakdowns

Exemplar 1: Continued Proportion & Ratios

Problem: If $A : B = 3 : 4$ and $B : C = 8 : 9$, find $A : C$ and the unified compound ratio $A : B : C$.

• Solution Step 1 (Equalize Intermediate Term B): $B$ corresponds to 4 in the first ratio and 8 in the second. Multiply the first ratio by 2: $A : B = (3 \times 2) : (4 \times 2) = 6 : 8$
• Solution Step 2 (Unify Ratios): Since $B = 8$ in both representations, $A : B : C = 6 : 8 : 9$.
• Solution Step 3 (Extract $A : C$): $A : C = 6 : 9 = 2 : 3$

Exemplar 2: Mixture and Ratio Replacement

Problem: A 60-liter vessel contains a mixture of milk and water in the ratio $2 : 1$. How many liters of water must be added to make the ratio $1 : 2$?

• Solution Step 1 (Calculate Initial Quantities): $\text{Milk} = \frac{2}{3} \times 60 = 40\text{ liters}, \quad \text{Water} = \frac{1}{3} \times 60 = 20\text{ liters}$
• Solution Step 2 (Formulate Equation for Added Water $w$): Milk volume remains invariant at 40 liters. Water becomes $20 + w$: $\frac{40}{20 + w} = \frac{1}{2} \implies 40 \times 2 = 20 + w \implies 80 = 20 + w \implies w = 60\text{ liters}$ Answer: Exactly 60 liters of water must be added.

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6. Exam Traps & Common Calculation Blunders

• Trap 1: The Base Inversion Trap in Percentages: Calculating percentage change against the final value rather than the initial base value. Remember: $\text{Percentage Change} = \frac{\text{Difference}}{\text{INITIAL Value}} \times 100\%$.
• Trap 2: Linear Summation of Successive Changes: Assuming a $+20\%$ increase followed by a $+10\%$ increase equals $+30\%$. The true value is $32\%$ because the second percentage acts upon the enlarged base.
• Trap 3: Unit Mismatch in Ratio Formulation: Comparing quantities measured in different units without normalization (e.g., forming a ratio between 45 minutes and 2 hours as $45 : 2$). Always convert to uniform units ($45 : 120 = 3 : 8$).
• Trap 4: Forgetting the Constant Term in C&D: Applying Componendo and Dividendo to only one side of an equation. The transformation must be applied simultaneously to both sides: $\frac{a+b}{a-b} = \frac{c+d}{c-d}$.