Fractions, Decimals,
and Percents
Master the interconversions between fractions, decimals, and percents: the foundation of GRE Quantitative Reasoning.
GRE Quantitative
Three Ways to Write the Same Number
Every rational number can be expressed in three interchangeable forms. The GRE expects you to switch fluently between them to compare quantities and simplify calculations.
Fraction
A ratio of two integers: \(\frac{a}{b}\) where \(b \neq 0\).
Decimal
A number written in base-10 place value: \(0.75\).
Percent
"Per hundred": a fraction with denominator \(100\).
Seeing One Value in Three Forms
Why This Matters on the GRE
GRE Quantitative Comparison and Problem Solving questions frequently mix fractions, decimals, and percents in the same problem. You need to convert them into a common form quickly to compare or compute.
1
Quantitative Comparison: "Which is greater: \(\frac{7}{12}\) or \(58\%\)?" Requires conversion to compare.
2
Data Interpretation: Charts show percents while questions ask for fractional parts.
3
Word Problems: Discounts, tax, interest, and probability all mix these forms.
The Core Conversion Rules
Fraction → Decimal
Divide the numerator by the denominator. This is the most direct conversion and the one you will use most on the GRE.
\(\frac{3}{8}\)
\(3 \div 8 = \) \(0.375\)
\(\frac{5}{6}\)
\(5 \div 6 = \) \(0.8\overline{3}\)
\(\frac{7}{20}\)
\(7 \div 20 = \) \(0.35\)
Decimal ↔ Percent
→
Decimal → Percent: Multiply by \(100\) (move decimal two places right). Example: \(0.045 \times 100 = \) \(4.5\%\)
←
Percent → Decimal: Divide by \(100\) (move decimal two places left). Example: \(125\% \div 100 = \) \(1.25\)
Fraction → Percent
Two approaches: convert through a decimal or use the direct formula.
Method 1: Via Decimal
\(\frac{3}{8} \rightarrow 0.375 \rightarrow \) \(37.5\%\)
Method 2: Direct
\(\frac{3}{8} \times 100\% = \frac{300}{8}\% = \) \(37.5\%\)
Terminating vs. Repeating Decimals
| Terminating | Repeating | |
|---|---|---|
| Definition | Finite digits after decimal | Infinite repeating block |
| Example | \(\frac{3}{8} = 0.375\) | \(\frac{2}{3} = 0.\overline{6}\) |
| Denominator rule | Only factors of \(2\) and \(5\) | Has factors other than \(2, 5\) |
| GRE strategy | Convert freely | Keep as fraction when possible |
Must-Know Benchmark Values
| Fraction | Decimal | Percent |
|---|---|---|
| \(\frac{1}{2}\) | 0.5 | 50% |
| \(\frac{1}{3}\) | \(0.\overline{3}\) | \(33.\overline{3}\%\) |
| \(\frac{1}{4}\) | 0.25 | 25% |
| \(\frac{1}{5}\) | 0.2 | 20% |
| \(\frac{1}{8}\) | 0.125 | 12.5% |
| \(\frac{2}{3}\) | \(0.\overline{6}\) | \(66.\overline{6}\%\) |
| \(\frac{3}{4}\) | 0.75 | 75% |
| \(\frac{1}{6}\) | \(0.1\overline{6}\) | \(16.\overline{6}\%\) |
Step-by-Step Conversion Method
1
Identify the target form. What does the question want: fraction, decimal, or percent?
2
Choose the shortest path. If comparing, convert everything to the same form. Decimals are usually fastest for comparison.
3
Apply the conversion rule: Fraction → Decimal: divide. Decimal → Percent: \(\times 100\). Reverse to go backward.
4
Simplify. Reduce fractions to lowest terms. Round decimals only if answer choices are approximate.
5
Sanity-check. A percent > 100% means the value is greater than \(1\). A negative percent means the value is negative.
Example 1: Basic Conversion
Problem: Express \(\frac{7}{20}\) as a decimal and as a percent.
1
Fraction → Decimal: \(7 \div 20 = \) \(0.35\)
2
Decimal → Percent: \(0.35 \times 100 = \) \(35\%\)
Example 2: Percent to Fraction
Problem: Convert \(12.5\%\) to a fraction in lowest terms.
1
Write as a fraction over 100: \(\frac{12.5}{100}\)
2
Eliminate the decimal: Multiply numerator and denominator by \(2\): \(\frac{25}{200}\)
3
Reduce: \(\frac{25}{200} = \frac{1}{8}\)
Example 3: GRE-Style Comparison
Problem: Which is greater: \(\frac{5}{7}\) or \(72\%\)?
1
Convert to decimals. \(\frac{5}{7} = 0.71428\ldots \approx\) \(0.7143\)
2
Convert percent to decimal. \(72\% = \) \(0.72\)
3
Compare: \(0.7143 < 0.72\), so \(72\%\) is greater.
Common Mistakes to Avoid
✕ Decimal Point Shift
Moving the decimal only one place instead of two when converting between decimal and percent. \(0.6 \neq 6\%\); it equals \(60\%\).
✕ Forgetting to Reduce
Writing \(\frac{75}{100}\) instead of \(\frac{3}{4}\). GRE answers expect lowest terms.
✕ Percent of vs. Percent
Confusing "what is \(25\%\) of \(80\)?" with "what percent of \(80\) is \(25\)?" These are different calculations.
Summary of Conversion Rules
| Conversion | Method | Example |
|---|---|---|
| Fraction → Decimal | Divide: \(a \div b\) | \(\frac{3}{4} = 0.75\) |
| Decimal → Fraction | Place-value, then reduce | \(0.6 = \frac{3}{5}\) |
| Decimal → Percent | Multiply by \(100\) | \(0.45 = 45\%\) |
| Percent → Decimal | Divide by \(100\) | \(8\% = 0.08\) |
| Fraction → Percent | \(\frac{a}{b} \times 100\%\) | \(\frac{2}{5} = 40\%\) |
| Percent → Fraction | \(\frac{p}{100}\), then reduce | \(65\% = \frac{13}{20}\) |
Question 1 of 5
What is \(\frac{5}{8}\) expressed as a percent?
\(\frac{5}{8} = 5 \div 8 = 0.625\), and \(0.625 \times 100 = 62.5\%\). This is one of the benchmark values worth memorizing.
Question 2 of 5
Convert \(0.0375\) to a fraction in lowest terms.
\(0.0375 = \frac{375}{10000} = \frac{75}{2000} = \frac{15}{400} = \frac{3}{80}\). Divide numerator and denominator by their GCD (\(125\)) to reduce.
Question 3 of 5
Which of the following is the largest?
\(\frac{5}{9} \approx 0.5556\). Comparing: (A) \(\approx 0.5556\), (B) \(0.54\), (C) \(0.55\), (D) \(\approx 0.5656\). Choice D is the largest because adding \(0.01\) to \(\frac{5}{9}\) makes it greater than all others.
Question 4 of 5
A store marks up an item by \(40\%\), then offers a \(25\%\) discount. What percent of the original price does the customer pay?
After a \(40\%\) markup, the price is \(1.40 \times P\). A \(25\%\) discount takes it to \(0.75 \times 1.40P = 1.05P\). The customer pays \(105\%\) of the original price. The markup and discount do not cancel because they apply to different base amounts.
Question 5 of 5
Which fraction produces a terminating decimal?
A fraction in lowest terms terminates if and only if the denominator's prime factors are only \(2\) and \(5\). \(40 = 2^3 \times 5\), so \(\frac{9}{40}\) terminates: \(9 \div 40 = 0.225\). The other denominators (\(12, 30, 6\)) all contain \(3\) as a prime factor.