3.1 Fractions, Decimals, and Real-Number Operations

3.1 Fractions, Decimals, and Real-Number Operations

On the GMAT Focus Edition, the Quantitative Reasoning section consists of 21 Problem Solving questions with a 45-minute time limit—averaging roughly 2 minutes and 8 seconds per question. Critically, no calculator is permitted during the Quantitative Reasoning section (an on-screen calculator is available only during Data Insights). Candidates who rely on long division or tedious manual arithmetic squander precious time and suffer mental fatigue. Achieving an elite quantitative score demands instantaneous recognition of number structures, mastery of fraction-decimal-percentage conversions, and rapid mental manipulation of real-number operations.

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The Real-Number System and Decimal Characteristics

The real numbers tested on the GMAT partition into two foundational sets:

1. Rational Numbers ($\mathbb{Q}$): Numbers that can be expressed as a ratio of two integers, $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$. Rational numbers encompass integers, finite (terminating) decimals, and infinite repeating decimals.
2. Irrational Numbers: Numbers that cannot be expressed as a ratio of integers; their decimal expansions are infinite and non-repeating (e.g., $\sqrt{2} \approx 1.414$, $\sqrt{3} \approx 1.732$, and $\pi \approx 3.14159$).

The Terminating Decimal Theorem

A classic GMAT problem type asks whether a given fraction yields a terminating decimal or an infinite repeating decimal. Because our numerical system is base 10, whose prime factors are exclusively $2$ and $5$ ($10 = 2 \times 5$), a rational number produces a terminating decimal if and only if its denominator can be transformed into a power of 10.

The Terminating Decimal Rule: Let $\frac{p}{q}$ be a fraction reduced to its lowest terms (such that $\gcd(p, q) = 1$). The fraction terminates in a finite number of decimal places if and only if the prime factorization of $q$ contains no prime factors other than $2$ and $5$: $q = 2^a \cdot 5^b$ where $a$ and $b$ are non-negative integers.

If the reduced denominator contains any prime factor other than 2 or 5—such as 3, 7, 11, or 13—the decimal expansion is strictly non-terminating and periodic (repeating).

Consider the following diagnostic examples:

• $\frac{7}{80}$: In lowest terms already. The denominator factorizes as $80 = 2^4 \times 5^1$. Because the only prime factors are 2 and 5, it terminates: $\frac{7}{80} = \frac{7 \times 5^3}{80 \times 5^3} = \frac{875}{10,000} = 0.0875$.
• $\frac{21}{336}$: Before inspecting the denominator, you must reduce! $\gcd(21, 336) = 21$, since $336 / 21 = 16$. Thus $\frac{21}{336} = \frac{1}{16} = \frac{1}{2^4}$. The only prime factor is 2, so it terminates: $\frac{1}{16} = 0.0625$.
• $\frac{14}{105}$: Reducing by dividing numerator and denominator by 7 yields $\frac{2}{15}$. The reduced denominator factorizes as $15 = 3 \times 5$. The presence of the prime factor 3 guarantees a repeating decimal: $\frac{2}{15} = 0.1333\dots = 0.1\overline{3}$.

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Converting Infinite Repeating Decimals to Fractions

When a repeating decimal appears in a Problem Solving question, converting it to an algebraic fraction is mandatory for exact calculation. GMAT problems use repeating decimals to test your grasp of base-10 algebra.

Pure Repeating Decimals

For a decimal whose repeating sequence begins immediately after the decimal point, the fraction takes the repeating block over an equal number of nines:

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

Mixed Repeating Decimals (Algebraic Elimination Method)

When non-repeating digits precede the repeating block, use a system of two equations scaled by powers of 10 to eliminate the infinite repetend:

Example: Convert $x = 0.2\overline{7} = 0.2777\dots$ to a fraction in simplest form.

1. Multiply by $10^1$ to shift the non-repeating transient digit left of the decimal point: $10x = 2.777\dots$
2. Multiply by $10^2$ to shift one full repeating cycle left of the decimal point: $100x = 27.777\dots$
3. Subtract the first equation from the second equation to cancel the infinite decimal tail: $100x - 10x = 27.777\dots - 2.777\dots$ $90x = 25$
4. Solve for $x$ and reduce to lowest terms: $x = \frac{25}{90} = \frac{5}{18}$

Example 2: Convert $y = 0.4\overline{16} = 0.4161616\dots$ to a fraction in simplest form.

1. Align before repetend: $10y = 4.161616\dots$
2. Align after one cycle: $1000y = 416.161616\dots$
3. Subtract: $1000y - 10y = 416.1616\dots - 4.1616\dots \implies 990y = 412$
4. Reduce: $y = \frac{412}{990} = \frac{206}{495}$

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Benchmark Fraction-Decimal-Percent Equivalence Table

Memorizing benchmark equivalents is non-negotiable for high-speed calculation. On test day, recognizing that $0.375$ is $\frac{3}{8}$ or that $\frac{5}{6}$ is $83\frac{1}{3}%$ saves up to 45 seconds per question.

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Complex and Nested Fractions

Nested fractions (fractions within fractions) are a favorite GMAT device for creating algebraic clutter. There are two primary techniques to dismantle them swiftly:

Method 1: The Global LCD Multiplier (Top-Down Clearing)

Instead of simplifying the numerator and denominator independently through multiple steps, multiply the overarching numerator and denominator by the Least Common Denominator (LCD) of all internal secondary fractions.

Example: Simplify $\frac{\frac{3}{4} - \frac{2}{5}}{\frac{5}{6} + \frac{1}{3}}$

The internal denominators are 4, 5, 6, and 3. Their LCD is $\text{LCM}(4, 5, 6, 3) = 60$. Multiply numerator and denominator by 60: $\frac{60 \left(\frac{3}{4} - \frac{2}{5}\right)}{60 \left(\frac{5}{6} + \frac{1}{3}\right)} = \frac{45 - 24}{50 + 20} = \frac{21}{70} = \frac{3}{10} = 0.3$ This single operation eliminates all fractional layers in under 15 seconds.

Method 2: Bottom-Up Sequential Collapse (Continued Fractions)

For continued fractions of the form $1 / (a - 1 / (b - 1/c))$, work strictly from the deepest layer upwards:

$\frac{1}{2 - \frac{1}{2 - \frac{1}{3}}}$

1. Lowest layer: $2 - \frac{1}{3} = \frac{5}{3}$.
2. Invert: $\frac{1}{5/3} = \frac{3}{5}$.
3. Next layer: $2 - \frac{3}{5} = \frac{7}{5}$.
4. Invert: $\frac{1}{7/5} = \frac{5}{7}$.

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Rapid Fraction Comparison and Ordering

When ordering a set of fractions on the GMAT, never convert every fraction to decimals using long division. Use one of four targeted strategies:

1. The Cross-Multiplication (Bowtie) Method

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

• Multiply numerator $a$ by denominator $d$ to get the left product $ad$.
• Multiply numerator $c$ by denominator $b$ to get the right product $bc$.
• Compare: $\frac{a}{b} > \frac{c}{d} \iff ad > bc$.

Example: Compare $\frac{7}{11}$ and $\frac{9}{14}$. $7 \times 14 = 98 \quad \text{vs.} \quad 9 \times 11 = 99$ Because $99 > 98$, $\frac{9}{14} > \frac{7}{11}$.

2. Matching Numerators or Matching Denominators

• Identical Numerators: The fraction with the smaller denominator is larger (e.g., $\frac{5}{17} > \frac{5}{19}$).
• Scaling to Match: To compare $\frac{4}{15}$ and $\frac{3}{11}$, scale numerators to 12: $\frac{4}{15} = \frac{12}{45}$ and $\frac{3}{11} = \frac{12}{44}$. Since $44 < 45$, $\frac{12}{44} > \frac{12}{45}$, so $\frac{3}{11} > \frac{4}{15}$.

3. Benchmarking Against Standard Reference Values ($1/2$, $1$)

To compare $\frac{11}{23}$ and $\frac{17}{32}$:

• Half of 23 is 11.5, so $\frac{11}{23} < \frac{1}{2}$.
• Half of 32 is 16, so $\frac{17}{32} > \frac{1}{2}$.
• Therefore, $\frac{17}{32} > \frac{11}{23}$ without any arithmetic.

4. The Positive Addition Transformation Property

The Distance-from-One Principle: Let $a, b, k > 0$ with $a < b$ (a proper positive fraction, so $\frac{a}{b} < 1$). Adding a positive constant $k$ to both numerator and denominator increases the fraction's value by moving it closer to 1: $\frac{a+k}{b+k} > \frac{a}{b}$ Conversely, if $a > b > 0$ (an improper fraction, so $\frac{a}{b} > 1$), adding $k > 0$ decreases the value toward 1: $\frac{a+k}{b+k} < \frac{a}{b}$

Illustration: $\frac{3}{4} = 0.75$. Add 5 to both terms: $\frac{3+5}{4+5} = \frac{8}{9} \approx 0.888$. Since $0.888 > 0.75$, the value increased toward 1.

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Common GMAT Pitfalls & Cognitive Traps

Trap 1: Illegal Denominator Splitting

$\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c} \quad (\text{Valid})$ $\frac{c}{a + b} \neq \frac{c}{a} + \frac{c}{b} \quad (\text{INVALID!})$ Test-takers frequently attempt to cancel terms across an addition or subtraction in the denominator. For example, $\frac{x}{x + y}$ cannot be reduced to $1 + \frac{x}{y}$ or $\frac{1}{1 + y}$.

Trap 2: Testing Denominator Before Complete Reduction

Evaluating the terminating decimal status of $\frac{6}{75}$ by inspecting 75 ($75 = 3 \times 5^2$) would lead to the false conclusion that it repeats because of the factor 3. But reducing first yields $\frac{6}{75} = \frac{2}{25} = \frac{2}{5^2}$, which terminates ($0.08$)!

Trap 3: Sign Errors in Reciprocals of Negative Fractions

Remember that the reciprocal of a negative fraction preserves the negative sign: the reciprocal of $-\frac{3}{7}$ is $-\frac{7}{3}$. Squaring flips the sign ($(-\frac{3}{7})^2 = \frac{9}{49}$), but taking reciprocals never changes signs.

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Worked Problem Solving Example

Question: Which of the following fractions is strictly between $\frac{2}{5}$ and $\frac{5}{8}$, and has a terminating decimal expansion?

• (A) $\frac{7}{18}$
• (B) $\frac{19}{40}$
• (C) $\frac{11}{24}$
• (D) $\frac{13}{30}$

Step-by-step Solution:

1. Convert bounds to decimals: $\frac{2}{5} = 0.400, \quad \frac{5}{8} = 0.625$ We need a terminating fraction in the open interval $(0.400, 0.625)$.
2. Filter by the Terminating Decimal Theorem:
   • Denominator of (A) is $18 = 2 \times 3^2$ (contains 3, repeats: $\frac{7}{18} \approx 0.388\dots$). Eliminate.
   • Denominator of (B) is $40 = 2^3 \times 5^1$ (only 2 and 5, terminates!).
   • Denominator of (C) is $24 = 2^3 \times 3$ (contains 3, repeats). Eliminate.
   • Denominator of (D) is $30 = 2 \times 3 \times 5$ (contains 3, repeats). Eliminate.
3. Verify the remaining fraction (B): $\frac{19}{40} = \frac{19 \times 25}{40 \times 25} = \frac{475}{1,000} = 0.475$ $0.400 < 0.475 < 0.625$. The value falls cleanly inside the target range and terminates.