Fractions, Decimals,
and Percents
Master the art of converting between the three core representations of rational numbers: fractions, decimals, and percents.
ISEE Upper Level
Three Ways to Write the Same Number
Every rational number can be expressed in three equivalent forms. The value never changes: only the notation does.
Fraction
A ratio of two integers: \(\frac{a}{b}\) where \(b \neq 0\).
Decimal
A base-ten representation using a decimal point, e.g. \(0.75\).
Percent
A fraction with denominator \(100\): "per centum" means per hundred.
Visualizing Equivalence
Why Conversions Matter on the ISEE
The ISEE Quantitative Reasoning section frequently requires you to compare quantities that are presented in different forms. You cannot compare \(\frac{7}{8}\) to \(85\%\) unless you convert one to the other.
1
Comparing columns: Column A might be a fraction while Column B is a decimal.
2
Word problems: "What percent of 60 is 45?" requires converting between forms.
3
Ordering numbers: Arrange \(\frac{2}{3}\), \(0.67\), and \(66\%\) from least to greatest.
Fraction → Decimal
Decimal → Percent
Percent → Fraction
1
Write: \(62.5\% = \frac{62.5}{100}\)
2
Clear the decimal: multiply top and bottom by \(10\): \(\frac{625}{1000}\)
3
Find GCF (\(125\)) and simplify: \(\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}\)
Terminating vs. Repeating Decimals
Terminating
The division ends. The denominator (in lowest terms) has only factors of \(2\) and \(5\).
\(\frac{3}{8} = 0.375\)
\(\frac{3}{8} = 0.375\)
Repeating
The division cycles forever. The denominator has prime factors other than \(2\) or \(5\).
\(\frac{1}{3} = 0.\overline{3}\)
\(\frac{1}{3} = 0.\overline{3}\)
Step-by-Step Conversion Checklist
1
Identify the form: Is the given value a fraction, decimal, or percent?
2
Choose the target form: What does the question need? Match the other quantity's format.
3
Apply the rule: Fraction → Decimal: divide. Decimal → Percent: \(\times 100\). Percent → Fraction: put over 100.
4
Simplify: Reduce fractions to lowest terms. Check for repeating decimals.
5
Verify: Convert back to confirm the answer makes sense.
The Conversion Triangle
Example 1: Basic Conversion
Convert \(\frac{7}{20}\) to a decimal and a percent.
1
Fraction → Decimal: Divide \(7 \div 20 = 0.35\)
2
Decimal → Percent: Multiply \(0.35 \times 100 = 35\%\)
Example 2: Repeating Decimal
Convert \(0.\overline{45}\) to a fraction.
1
Let \(x = 0.\overline{45} = 0.454545\ldots\)
2
Multiply by \(100\): \(100x = 45.454545\ldots\)
3
Subtract: \(100x - x = 45\), so \(99x = 45\)
4
Solve: \(x = \frac{45}{99} = \frac{5}{11}\)
Example 3: ISEE-Style Comparison
Which is greater: \(\frac{5}{6}\) or \(83\%\)?
1
Convert \(\frac{5}{6}\) to a decimal: \(5 \div 6 = 0.8\overline{3}\)
2
Convert \(83\%\) to a decimal: \(83\% = 0.83\)
3
Compare: \(0.8\overline{3} = 0.83333\ldots > 0.83000\ldots\)
Common Mistakes to Avoid
✕ Moving the Decimal Wrong
Converting \(0.06\) to a percent: students write \(60\%\) instead of the correct \(6\%\). Count exactly two places.
✕ Forgetting to Simplify
Writing \(\frac{45}{100}\) as the final answer instead of reducing to \(\frac{9}{20}\). Always find the GCF.
✕ Rounding Repeating Decimals
Treating \(0.\overline{3}\) as \(0.33\) in comparisons. The exact value \(\frac{1}{3}\) is greater than \(0.33\).
Quick Reference Table
| Fraction | Decimal | Percent |
|---|---|---|
| \(\frac{1}{2}\) | \(0.5\) | \(50\%\) |
| \(\frac{1}{3}\) | \(0.\overline{3}\) | \(33.\overline{3}\%\) |
| \(\frac{1}{4}\) | \(0.25\) | \(25\%\) |
| \(\frac{1}{5}\) | \(0.2\) | \(20\%\) |
| \(\frac{1}{8}\) | \(0.125\) | \(12.5\%\) |
| \(\frac{2}{3}\) | \(0.\overline{6}\) | \(66.\overline{6}\%\) |
| \(\frac{3}{4}\) | \(0.75\) | \(75\%\) |
| \(\frac{7}{8}\) | \(0.875\) | \(87.5\%\) |
Question 1 of 5
What is \(\frac{3}{8}\) expressed as a percent?
\(\frac{3}{8} = 3 \div 8 = 0.375\), and \(0.375 \times 100 = 37.5\%\).
Question 2 of 5
Which fraction is equivalent to \(0.\overline{27}\)?
For a 2-digit repeating block, place the digits over \(99\): \(\frac{27}{99} = \frac{3}{11}\). Note that \(\frac{9}{33} = \frac{3}{11}\) as well, but \(\frac{3}{11}\) is already in lowest terms and is the conventional answer.
Question 3 of 5
Arrange from least to greatest: \(\frac{2}{5}\), \(0.39\), \(41\%\).
Convert all to decimals: \(\frac{2}{5} = 0.40\), \(0.39 = 0.39\), \(41\% = 0.41\). Order: \(0.39 < 0.40 < 0.41\).
Question 4 of 5
What percent of \(80\) is \(56\)?
Set up the fraction: \(\frac{56}{80}\). Simplify: \(\frac{56}{80} = \frac{7}{10} = 0.70 = 70\%\).
Question 5 of 5
A store marks an item at \(125\%\) of its wholesale cost. If the wholesale cost is \(\$48\), what is the store's price?
Convert \(125\%\) to a decimal: \(1.25\). Multiply: \(1.25 \times 48 = 60\). The store's price is \(\$60\).