GRE Fractions Decimals and Percentages

Mastering the interplay between fractions, decimals, and percentages is non-negotiable for GRE Quantitative success. These concepts are not isolated topics but the fundamental language of nearly every problem type, from algebra and arithmetic to data interpretation and real-world word problems. Achieving fluency here means you spend less time on basic calculations and more time on the logical reasoning the GRE is designed to test.

Core Conversions and Key Equivalences

The first step to fluency is seamless conversion between the three forms. You must be able to see $\frac{1}{8}$, 0.125, and 12.5% as interchangeable expressions of the same value. This skill allows you to choose the most convenient form for any calculation, a critical test-day strategy.

Converting a fraction to a decimal is straightforward: perform the division. $\frac{3}{5}=3÷5=0.6$. For converting a fraction to a percentage, first convert to a decimal and then multiply by 100. $\frac{3}{5}=0.6=60\%$. To go from a percentage to a decimal, divide by 100 (move the decimal two places left). 62.5% becomes 0.625. To convert that decimal to a fraction, write it over the appropriate power of ten and simplify: $0.625=\frac{625}{1000}=\frac{5}{8}$.

Memorizing common equivalences is a powerful time-saver. You should instantly recognize:

Fraction Arithmetic and Comparison

GRE problems often require quick, accurate manipulation of fractions without a calculator. Key operations include simplification, addition/subtraction, multiplication/division, and comparison.

For addition and subtraction, you need a common denominator. To compare $\frac{5}{12}$ and $\frac{7}{18}$, find the least common denominator (LCD). The LCM of 12 and 18 is 36: $\frac{5}{12}=\frac{15}{36}$ and $\frac{7}{18}=\frac{14}{36}$. Clearly, $\frac{5}{12}>\frac{7}{18}$. Multiplication is direct: $\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$. Division is multiplying by the reciprocal: $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$.

A critical shortcut for comparing fractions is cross-multiplication. To compare $\frac{7}{9}$ and $\frac{4}{5}$, cross-multiply upwards: $7\times 5=35$ and $9\times 4=36$. Since $35<36$, the fraction linked to 35 ($\frac{7}{9}$) is less than the fraction linked to 36 ($\frac{4}{5}$). This method avoids finding a common denominator.

Percentage Calculations and Change

The percentage formula $\text{Part}=\text{Percent (as decimal)}\times \text{Whole}$ is the cornerstone of most problems. The GRE's challenge lies in correctly identifying which quantity is the "Part" and which is the "Whole." In the phrase "x is what percent of y?", "x" is the Part and "y" is the Whole.

Percent increase and decrease follow a standard process: 1) Find the amount of change, 2) Divide the change by the original value, and 3) Convert to a percentage. If a price increases from $80to$100, the change is $20.$\frac{20}{80} = 0.25$,soi{t}^{\prime }sa25$(1 + 0.15) = 1.15$.Fora15$(1 - 0.15) = 0.85$. This is especially efficient for successive changes.

Successive Percentages and Compound Interest

Problems involving multiple percentage changes in sequence are common. The key is to apply each change to the new value, not the original one. Simply adding or subtracting the percentages will lead to an incorrect answer. The multiplier method is essential here.

For example, if a population increases by 10% one year and then decreases by 10% the next, the net change is not 0%. Apply the multipliers: Starting value $\times 1.10\times 0.90=\text{Starting value}\times 0.99$. This is a net 1% decrease. This principle extends directly to compound interest, which is essentially a series of identical percentage increases. The formula is $$A=P(1+\frac{r}{n}{)}^{nt}$$ where $A$ is the final amount, $P$ is principal, $r$ is annual rate, $n$ is compounding periods per year, and $t$ is time in years. For annual compounding, it simplifies to $A=P(1+r{)}^t$.

Common Pitfalls

1. Misidentifying the "Whole" in Percent Change: The most frequent error is using the new value as the denominator when calculating percent change. Remember, percent change is always $\frac{\text{New}-\text{Original}}{\text{Original}}$. If something drops from 100 to 80, the percent decrease is $\frac{20}{100}=20\%$, not $\frac{20}{80}=25\%$.
2. Adding Percentages of Different Wholes: You cannot simply add percentage changes that apply to different bases. A 50% increase followed by a 30% decrease does not yield a net 20% increase, as demonstrated in the successive percentages section. Always use the multiplier method.
3. Confusing "Percent More Than" vs. "Percent Of": The phrasing is critical. "x is 300% of y" means $x=3y$. "x is 300% greater than y" means $x=y+3y=4y$. The word "greater than" or "more than" adds the original 100%.
4. Calculation Errors with Complex Fractions: When a problem involves nested fractions (e.g., $\frac{1}{1+\frac{1}{2}}$), work step-by-step from the innermost fraction outward. For $\frac{1}{1+\frac{1}{2}}$, first solve $1+\frac{1}{2}=\frac{3}{2}$, then $\frac{1}{\frac{3}{2}}=\frac{2}{3}$. Rushing leads to mistakes.

Summary

• Automatic conversions between fractions, decimals, and percentages for common values (eighths, thirds, fifths) is a primary time-saving strategy on the GRE.
• Use the multiplier method $(1\pm r)$ for all percentage increase/decrease problems, especially successive changes. Never simply add or subtract percentages.
• In percent change formulas, the denominator is always the original value, not the final value. Correctly identifying the "Part" and the "Whole" is paramount.
• For fraction comparison, cross-multiplication is faster than finding a common denominator.
• Understand the precise language: "percent greater than" means adding to 100% of the original base.
• Compound interest and successive percentage problems are applications of the same core principle: apply sequential multipliers to the changing value.