Fractions, Decimals, and Percents - Quantitative Reasoning

SECTION 1

Historical Context & Motivation

The need to express parts of a whole is among the oldest problems in mathematics, predating even the concept of zero. Ancient civilizations developed distinct notational strategies for representing quantities between whole numbers, and these strategies evolved over millennia into the three interchangeable systems we use today: fractions, decimals, and percents. Understanding the historical trajectory of these representations not only provides intellectual context but also illuminates why certain conversions feel natural while others require deliberate calculation.

~1800 BCE • Egyptian Unit Fractions

The Rhind Papyrus reveals that ancient Egyptians expressed all fractional quantities as sums of unit fractions (fractions with numerator 1), such as writing ⅔ as ½ + ⅙. This system, while elegant, made arithmetic cumbersome.

~300 BCE • Babylonian Sexagesimal System

Babylonian mathematicians used a base-60 positional system that anticipated modern decimals. Their approach allowed fractional parts to be expressed as 'sexagesimal fractions,' and remnants survive in our 60-minute hours and 360-degree circles.

~500 CE • Indian Fraction Notation

Indian mathematicians, including Aryabhata and Brahmagupta, developed the numerator-over-denominator notation we use today, though without the horizontal bar. The bar itself was added by Arab scholars who transmitted these ideas westward.

1585 • Stevin's Decimal System

Simon Stevin published 'De Thiende' (The Tenth), systematically introducing decimal fractions to Europe. His work demonstrated that any fraction could be approximated by tenths, hundredths, and thousandths, dramatically simplifying commerce and science.

~1600s • Percent Notation Standardized

The percent symbol (%) evolved from the Italian 'per cento' used in trade manuscripts. By the 17th century, percent had become the universal language of commerce, taxation, and interest calculations—a role it retains on the GRE's quantitative section.

The central question driving this lesson is deceptively simple: if fractions, decimals, and percents all express the same underlying quantity, why does choosing the right representation matter? On the GRE, the answer is speed and accuracy. Certain problems become trivial when a fraction is converted to a decimal, while others demand the intuitive grasp of magnitude that percents provide. Mastering the fluid interconversion among all three forms is therefore not merely an academic exercise—it is a strategic advantage.

SECTION 2

Core Principles & Definitions

Before diving into conversion algorithms and problem-solving strategies, it is essential to establish precise definitions. A fraction, a decimal, and a percent are three syntactic representations of the same semantic object: a rational number (or, in certain cases, an approximation thereof). The following foundational ideas govern the entire lesson.

Fraction as Division

A fraction a/b literally means a ÷ b. The numerator counts parts; the denominator names the size of each part. Every fraction is a ratio, and every ratio can be written as a fraction.

Decimal as Positional Expansion

A decimal is a fraction whose denominator is a power of 10, expressed through place value. Thus 0.375 = 375/1000. Terminating decimals have denominators whose only prime factors are 2 and 5.

Percent as Hundredths

The word 'percent' derives from Latin 'per centum,' meaning 'per hundred.' Therefore 45% = 45/100 = 0.45. Percents normalize every comparison to a common denominator of 100.

Equivalence Principle

For any rational number, there exist infinitely many equivalent fractions (e.g., 1/2 = 2/4 = 50/100), exactly one reduced form, one decimal expansion (terminating or repeating), and one percent representation.

Strategic Representation

On the GRE, choosing the optimal form can save minutes. Fractions excel at exact arithmetic and cancellation. Decimals simplify ordering and addition. Percents clarify proportional comparisons.

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KEY TAKEAWAY

Think of fractions, decimals, and percents as three different languages describing the same scene. A bilingual speaker doesn't translate word-for-word; they think in whichever language fits the context. Similarly, a strong GRE performer doesn't mechanically convert every number—they think natively in the representation that makes the current problem simplest. Fluency in all three 'dialects' is the goal.

SECTION 3

Visual Explanation — The Conversion Triangle

The relationship among fractions, decimals, and percents can be visualized as a triangle of bidirectional conversions. Each vertex represents one form, and each edge represents the operation required to move between two forms. This diagram encapsulates the six core conversion pathways you must internalize for the GRE.

The Conversion Triangle shows all six pathways. To go from fraction → decimal, divide numerator by denominator. To go from decimal → percent, multiply by 100. To go from percent → fraction, place over 100 and reduce.

Notice that every conversion can be derived from one fundamental relationship: a fraction is a division. Once you perform the division, you have a decimal. Once you have a decimal, multiplying by 100 yields a percent. The reverse pathways simply invert these operations. On the GRE, the most frequent conversion you will perform is fraction to decimal (for comparison problems) and percent to fraction (for discount and interest problems). Memorizing the common equivalences in Section 5 will allow you to skip the computation entirely.

SECTION 4

Mathematical Framework — Conversion Formulas

While the conversion triangle provides a visual mnemonic, the underlying mathematics can be stated with precision. The following formulas govern every conversion between fractions, decimals, and percents. Each formula is algebraically derivable from the others, confirming that these three representations are merely notational variants of the same quantity.

FRACTION → DECIMAL

d = a ÷ b

Given fraction a/b, perform long division or recognize a memorized equivalence. The result d is the decimal form. If b has prime factors other than 2 and 5, the decimal will repeat.

DECIMAL → PERCENT

p = d × 100

Multiply the decimal d by 100 and append the % symbol. Equivalently, shift the decimal point two places to the right. So 0.375 becomes 37.5%.

PERCENT → FRACTION

a/b = p / 100 (then reduce to lowest terms)

Place the percent value over 100 and simplify by dividing both numerator and denominator by their greatest common divisor (GCD). For example, 45% = 45/100 = 9/20.

REPEATING DECIMAL → FRACTION

If d = 0.ā̄b̄ (repeating block of n digits), then a/b = repeating block ÷ (10ⁿ − 1)

For a purely repeating decimal, the numerator is the repeating block and the denominator is a string of n nines. So 0.272727… = 27/99 = 3/11. For mixed repeating decimals (e.g., 0.1666…), the algebraic approach of multiplying by powers of 10 and subtracting is required.

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GRE Strategy Note

The GRE often tests your ability to compare quantities. When comparing a fraction to a decimal or percent, convert everything to the same form first. For Quantitative Comparison questions, fractions are often easiest to cross-multiply; for Data Interpretation, decimals or percents let you work directly with the graph values.

SECTION 5

Common Equivalences You Must Memorize

One of the highest-yield time investments for GRE preparation is memorizing the common fraction-decimal-percent equivalences. These values appear with extraordinary frequency in both Quantitative Comparison and Problem Solving questions. Rather than computing 3 ÷ 8 during the exam, you should recognize instantly that 3/8 = 0.375 = 37.5%. The table below organizes these essential equivalences by denominator family.

Fraction	Decimal	Percent	Notes
1/2	0.5	50%	Halves family
1/3	0.333…	33.3̄%	Repeating; 2/3 = 0.666… = 66.6̄%
1/4	0.25	25%	3/4 = 0.75 = 75%
1/5	0.2	20%	2/5 = 0.4 = 40%; 3/5 = 0.6 = 60%
1/6	0.1666…	16.6̄%	5/6 = 0.833… = 83.3̄%
1/8	0.125	12.5%	3/8 = 0.375; 5/8 = 0.625; 7/8 = 0.875
1/9	0.111…	11.1̄%	n/9 = 0.nnn… for n < 9
1/10	0.1	10%	Tenths align directly with decimal place value
1/11	0.0909…	9.09̄%	Less common but tested; 2-digit repeating block
1/12	0.0833…	8.3̄%	Know at least 1/12 and 5/12 = 0.4166…

Common Fraction-Decimal-Percent Equivalences for GRE

The three parallel number lines display fractions (top, purple), decimals (middle, cyan), and percents (bottom, pink). Vertical dashed lines connect equivalent values across all three representations.

The aligned number-line diagram above reinforces a crucial insight: the vertical dashed lines connect equivalent values. Notice how 1/3 and 2/3 do not align with 'nice' decimal values—their repeating decimals (0.333… and 0.666…) reflect the fact that 3 is not a factor of any power of 10. This is precisely why thirds are among the most common traps on the GRE: test-takers who round 0.333 to 0.33 can introduce errors in comparison problems. Always use the exact fraction form when precision matters.

SECTION 6

Worked Example — GRE-Style Multi-Step Conversion

Consider a GRE Problem Solving question that tests conversions in the context of a real scenario. This worked example demonstrates the full conversion pipeline, the strategic choice of representation, and the precision required for a correct answer.

A store offers a 15% discount on a jacket originally priced at $80. A sales tax of 1/12 of the discounted price is then applied. What is the final price?

1
Step 1 — Convert the percent discount to a decimal

The discount is 15%, which equals 15 ÷ 100 = 0.15. Therefore the fraction of the original price you actually pay is 1 − 0.15 = 0.85.

Multiplier = 0.85
2
Step 2 — Compute the discounted price

Multiply the original price by the multiplier: $80 × 0.85 = $68.00. Alternatively, 15% of $80 = 0.15 × 80 = $12, so the discounted price is $80 − $12 = $68. Both methods confirm the same result.

Discounted price = $68.00
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Step 3 — Convert the tax fraction to a decimal

The sales tax rate is 1/12. From our memorized equivalences, 1/12 = 0.08333… ≈ 8.33%. However, we need the exact value, so it is more efficient to work directly with the fraction.

Tax rate = 1/12
4
Step 4 — Compute the tax amount

Tax = (1/12) × $68 = $68/12. Simplify: 68 ÷ 4 = 17, 12 ÷ 4 = 3, so tax = $17/3 = $5.666… ≈ $5.67 (rounded to the nearest cent).

Tax = $5.67
5
Step 5 — Compute the final price

Add the tax to the discounted price: $68.00 + $5.67 = $73.67. Note that the GRE may ask for the exact fraction: $68 + $17/3 = $204/3 + $17/3 = $221/3 = $73.666…, which rounds to $73.67.

Final price = $73.67

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Strategic Insight

Notice how we switched between representations strategically. We converted 15% to a decimal for multiplication, but kept 1/12 as a fraction to avoid repeating-decimal errors. On the GRE, representation flexibility is the key skill being tested. The ability to recognize when to stay in fraction form versus converting to a decimal can mean the difference between a clean answer and a computational mess.

SECTION 7

Strengths & Limitations of Each Representation

Each representation has distinct advantages and pitfalls. Understanding these trade-offs is essential for choosing the optimal form on GRE problems where time pressure is significant. The following comparison table highlights when each representation excels and when it falls short.

Criterion	Fractions	Decimals	Percents
Exact Arithmetic	Excellent — no rounding errors with rationals	Good for terminating decimals; repeating decimals lose precision if truncated	Poor — intermediate percent calculations accumulate rounding errors
Ordering / Comparison	Requires cross-multiplication or common denominator	Excellent — align decimal places and compare digit by digit	Good — normalized to common base of 100
Multiplication	Strong — cancellation simplifies before computing	Moderate — straightforward but tedious for long decimals	Requires conversion to decimal first
Addition / Subtraction	Requires common denominator (can be slow)	Excellent — align decimal points and add	Excellent — add/subtract directly when base is the same
Proportional Reasoning	Natural — ratios are literally fractions	Less intuitive for 'part of a whole' reasoning	Excellent — designed for 'out of 100' thinking
GRE Data Interpretation	Rarely used in graph labels	Common in tables and bar charts	Dominant in pie charts and growth rates

When to use each representation on the GRE

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KEY TAKEAWAY

Think of representation choice like choosing the right tool in an engineering workshop. A wrench, a socket driver, and pliers can all turn a bolt, but the right tool for the right job saves time and prevents damage. Fractions are your precision tool for exact arithmetic and cancellation. Decimals are your quick-comparison instrument. Percents are your proportional-reasoning communicator. The GRE rewards test-takers who switch fluently among all three.

SECTION 8

Connections to Advanced GRE Topics

Fraction-decimal-percent fluency is not an isolated skill; it is a prerequisite for numerous advanced GRE topics. Recognizing these connections helps you see why the test emphasizes conversions so heavily. The table below maps this foundational skill to the higher-order topics that depend on it.

Advanced GRE Topic	How Fractions/Decimals/Percents Appear
Percent Change	Percent increase/decrease formulas require converting between absolute differences (decimals) and relative changes (percents): Change = (New − Old)/Old × 100%.
Compound Interest	The formula A = $P(1 + r/n)^{nt}$ uses the rate r as a decimal, but problems often state it as a percent (e.g., 5% → 0.05). Misplacing the decimal is the #1 error.
Probability	Probabilities are computed as fractions (favorable/total) but often compared as decimals or reported as percents in Data Interpretation sets.
Ratios and Proportions	Ratios are fractions in disguise. Converting a ratio to a percent (e.g., 3:5 → 3/5 → 60%) is essential for mixture and work-rate problems.
Data Interpretation	Pie charts display percents; bar/line charts display decimals or integers. Answering questions across chart types requires on-the-fly conversions.

Downstream dependencies of fraction-decimal-percent fluency

As you advance through GRE preparation, you will find that nearly every arithmetic and algebra topic circles back to the interplay of fractions, decimals, and percents. The percent change formula, for instance, is simply a structured application of the conversion pipeline: compute the difference (subtraction in decimal or fraction form), divide by the base (yielding a fraction), then multiply by 100 to express the result as a percent. Mastering the foundational layer now will pay exponential dividends in more complex problem types.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why the fraction 7/40 produces a terminating decimal, while 7/12 produces a repeating decimal. What property of the denominator determines the outcome?

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A fraction in lowest terms produces a terminating decimal if and only if the denominator's prime factorization contains no primes other than 2 and 5. The denominator 40 = 2³ × 5, which satisfies this condition, so 7/40 = 0.175. In contrast, 12 = 2² × 3, and the presence of the prime factor 3 means that no power of 10 is divisible by 12 (since 10ⁿ = 2ⁿ × 5ⁿ never contains a factor of 3). Therefore 7/12 = 0.58333… with a repeating block. This insight lets you instantly determine on the GRE whether a decimal terminates without performing division.

Question 2: BASIC CALCULATION

Convert 0.0625 to a fraction in lowest terms, and then express it as a percent.

Show answer

Write 0.0625 as 625/10000. Factor: 625 = 5⁴ and 10000 = 10⁴ = 2⁴ × 5⁴. Dividing both by 625 yields 1/16. To convert to a percent, compute 1 ÷ 16 = 0.0625, then multiply by 100 to get 6.25%. Alternatively, you could note that 1/16 = (1/4) × (1/4) = 0.25 × 0.25 = 0.0625. This confirms 0.0625 = 1/16 = 6.25%.

Question 3: INTERMEDIATE

Which is greater: 5/7 or 71%? Do not use a calculator.

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Convert 71% to a fraction: 71/100. Now cross-multiply to compare 5/7 and 71/100: compute 5 × 100 = 500 and 71 × 7 = 497. Since 500 > 497, we conclude that 5/7 > 71/100. Equivalently, 5/7 ≈ 0.7142857…, which exceeds 0.71. The difference is small (about 0.43%), which is precisely the kind of close comparison the GRE favors. Memorizing that 5/7 ≈ 0.714 or recognizing the cross-multiplication shortcut are both efficient strategies.

Question 4: APPLIED

A researcher's sample of 240 participants includes 45% women. Of the women, 2/9 are left-handed. How many left-handed women are in the sample?

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First, find the number of women: 45% of 240 = 0.45 × 240 = 108 women. Next, find 2/9 of 108: (2/9) × 108 = 216/9 = 24. So there are 24 left-handed women. Notice the strategic value of keeping 2/9 as a fraction: 108 is divisible by 9 (since 1 + 0 + 8 = 9), making the arithmetic clean. Converting 2/9 to 0.222… and multiplying would yield the same result but invite rounding errors. This problem illustrates the GRE's habit of combining percents and fractions in multi-step word problems.

Question 5: CRITICAL THINKING

A quantity increases by 20%, then decreases by 20%. Is the final value equal to, greater than, or less than the original value? Prove your answer algebraically and express the net percent change as an exact fraction.

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Let the original value be x. After a 20% increase: x × 1.20 = 1.2x. After a subsequent 20% decrease: 1.2x × 0.80 = 0.96x. The final value is 0.96x, which is less than x by 0.04x. The net percent change is (0.96x − x)/x × 100% = −4%. Expressed as a fraction, the final value is (24/25)x, meaning you lose 1/25 of the original. This result generalizes: for any percent p, an increase of p% followed by a decrease of p% yields a net loss of (p/100)², which is always negative. The GRE frequently tests whether students fall for the trap of assuming symmetric percent changes cancel out.

SUMMARY

Lesson Summary

Summary

This lesson established that fractions, decimals, and percents are three interchangeable representations of rational numbers, each with strategic strengths. The Conversion Triangle encodes six bidirectional pathways: divide to go from fraction to decimal, multiply by 100 for decimal to percent, place over 100 and reduce for percent to fraction, and their reverses. Fractions excel at exact arithmetic and cancellation, decimals are optimal for ordering and comparison, and percents are the natural language of proportional reasoning and data interpretation.

Key memorization targets include the common equivalence table (halves through twelfths) and the terminating-decimal criterion (denominator factors limited to 2 and 5). These skills form the foundation for percent change, compound interest, probability, and virtually every other GRE Quantitative topic. Fluent interconversion—choosing the right representation for the right context—is the single most impactful arithmetic skill you can bring to the exam.