6.2 Fractions, Decimals, Percentages, and Percentage Change

6.2 Fractions, Decimals, Percentages, and Percentage Change

Fractions, decimals, and percentages form the backbone of diplomatic budgeting, quota allocation, trade balance reporting, and consular fee accounting. Whether evaluating the percentage increase in overseas passport applications or distributing foreign development grants, mastering base identification and fractional shortcuts is indispensable.

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The Tripartite Equivalence: High-Frequency Benchmark Values

To eliminate manual conversion delays during testing, memorize this table of benchmark equivalents:

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Advanced Fraction Arithmetic in Public Administration

Administrative problems in civil service examinations commonly present nested fractions, mixed quantities, and sequential subtractions representing fund depletion.

1. Addition and Subtraction: The Least Common Denominator (LCD)

To add or subtract fractions with unequal denominators, determine the Least Common Multiple of the denominators (the LCD):

$$\frac{5}{12} + \frac{7}{18}$$

• Prime factorizations: $12 = 2^2 \times 3$, $18 = 2 \times 3^2$.
• $\text{LCD} = 2^2 \times 3^2 = 4 \times 9 = 36$.
• Rescale numerators: $\frac{5 \times 3}{36} + \frac{7 \times 2}{36} = \frac{15 + 14}{36} = \frac{29}{36}$.

When working with mixed numbers (e.g., $4\frac{2}{3} - 1\frac{5}{6}$), you can either convert everything to improper fractions ($\frac{14}{3} - \frac{11}{6} = \frac{28 - 11}{6} = \frac{17}{6} = 2\frac{5}{6}$) or operate on integer and fractional components separately, borrowing $1 = \frac{6}{6}$ from the whole number when required.

2. Multiplication and Cross-Cancellation

Never multiply large numerators and denominators directly before reducing. Cancel common factors diagonally:

$$\frac{14}{27} \times \frac{45}{49} = \frac{14 \div 7}{27 \div 9} \times \frac{45 \div 9}{49 \div 7} = \frac{2}{3} \times \frac{5}{7} = \frac{2 \times 5}{3 \times 7} = \frac{10}{21}$$

3. Division and Complex Fractions

Dividing by a fraction is mathematically identical to multiplying by its reciprocal:

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Complex fractions (fractions within fractions) are resolved using this principle:

$$\frac{\frac{3}{8}}{\frac{9}{16}} = \frac{3}{8} \times \frac{16}{9} = \frac{3 \times 16}{8 \times 9} = \frac{1 \times 2}{1 \times 3} = \frac{2}{3}$$

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Converting Repeating Decimals to Common Fractions

Civil service word problems occasionally provide financial allocations or operational rates as repeating decimals. Two primary algebraic patterns govern these conversions:

1. Pure Repeating Decimals

If the repeating block starts immediately after the decimal point, place the repeating digits over an equal number of nines:

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

2. Mixed Repeating Decimals

When a non-repeating digit prefix precedes the repeating period, apply the standard subtraction formula:

$$\text{Fraction} = \frac{\text{All Digits up to end of first period} - \text{Non-repeating prefix}}{\text{A nine for each repeating digit, followed by a zero for each non-repeating digit}}$$

Algebraic Derivation for $0.41\bar{6}$:

1. Let $x = 0.41666\dots$
2. Multiply by $100$ to bring the non-repeating part in front: $100x = 41.666\dots$
3. Multiply by $1000$ to bring one repeating period in front: $1000x = 416.666\dots$
4. Subtract the two equations:

$$1000x - 100x = 416.666\dots - 41.666\dots$$ $$900x = 375 \implies x = \frac{375}{900}$$

1. Reduce by dividing numerator and denominator by 75: $\frac{375 \div 75}{900 \div 75} = \frac{5}{12}$.

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The Core Percentage Mechanics: Part, Rate, and Base

All basic percentage problems reduce to a single relationship connecting three variables:

$$\text{Part} = \text{Rate} \times \text{Base} \quad (P = R \times B)$$

• Base ($B$): The entire whole, baseline reference, or starting population. In English prose, the base almost always follows the preposition "of".
• Rate ($R$): The percentage expressed as a decimal or fractional proportion (e.g., $15\% = 0.15 = \frac{3}{20}$).
• Part ($P$): The portion, subset, or change derived from the base. In English prose, the part is typically associated with the verb "is".

Formulations:

• To find Part: $P = R \times B$
• To find Rate: $R = \frac{P}{B}$
• To find Base: $B = \frac{P}{R}$

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Percentage Change and the Asymmetric Base Trap

Percentage change measures the relative difference between a final value and an original baseline:

$$\text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\% = \frac{\Delta V}{V_0} \times 100\%$$

• If $\text{New} > \text{Original}$, the percentage change is positive (increase).
• If $\text{New} < \text{Original}$, the percentage change is negative (decrease).

The Asymmetric Base Trap

A pervasive trap in the CSE-FSO is assuming that if quantity $A$ is $x\%$ greater than quantity $B$, then $B$ must be $x\%$ less than $A$. This is mathematically false because the reference base changes between the two statements.

Example: If Post Manila's visa output is $25\%$ greater than Post Tokyo's, by what percentage is Post Tokyo's output less than Post Manila's?

• Let Post Tokyo's output ($B$) be $100$.
• Then Post Manila's output ($A$) is $100 + (0.25 \times 100) = 125$.
• Now calculate how much less $B$ is than $A$. The base is now $A$ ($125$):

$$\text{Percentage Less} = \frac{125 - 100}{125} \times 100\% = \frac{25}{125} \times 100\% = \frac{1}{5} \times 100\% = 20\%$$

Standard Compensating Asymmetries:

• $+20\% \iff -16\frac{2}{3}\%$
• $+25\% \iff -20\%$
• $+33\frac{1}{3}\% \iff -25\%$
• $+50\% \iff -33\frac{1}{3}\%$
• $+100\% \iff -50\%$

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Successive Percentage Changes and Multiplicative Compounding

When a quantity undergoes multiple percentage adjustments sequentially, percentages cannot be added linearly. Successive percentage changes are multiplicative because each subsequent change applies to the newly updated base.

Compound multipliers

Let $r$ denote the percentage rate written as a decimal: 20% means $r=0.20$. An increase has multiplier $1+r$ and a decrease has multiplier $1-r$. For multiple changes, multiply the successive factors; if the input is a whole-number percentage p, use $r=p/100$.

$$\text{Final Value} = \text{Initial Value} \times (1 \pm r_1) \times (1 \pm r_2) \times \dots \times (1 \pm r_n)$$

The Classical "+20% then -20%" Paradox

If an embassy's emergency fund is increased by $20\%$ and subsequently cut by $20\%$, what is the net overall effect?

• Multiplier for $+20\%$: $1 + 0.20 = 1.20$
• Multiplier for $-20\%$: $1 - 0.20 = 0.80$
• Combined Multiplier: $1.20 \times 0.80 = 0.96$
• Net Change: $(0.96 - 1.00) \times 100\% = -4\%$

The fund does not return to its starting balance; it suffers a net decrease of 4%. In general, an increase of $x\%$ followed by a decrease of $x\%$ always results in a net decrease of $\frac{x^2}{100}\%$.