Fractions, Decimals
& Percents
Master the art of converting between fractions, decimals, and percents: the essential skill for quantitative reasoning.
ISEE Middle Level
Three Ways to Show the Same Number
Fractions, decimals, and percents are three different notations for the same value. Every fraction can be written as a decimal, and every decimal can be written as a percent.
Fraction
A ratio of two integers: \(\frac{a}{b}\) where \(b \neq 0\).
Decimal
A number written using place value and a decimal point, like \(0.75\).
Percent
A number out of \(100\). The symbol \(\%\) means "per hundred."
One Value, Three Faces
Why Does Converting Matter?
The ISEE frequently asks you to compare quantities expressed in different forms. You cannot compare \(\frac{3}{8}\) with \(0.4\) or \(38\%\) until they share the same format.
1
Quantitative comparison questions give one column as a fraction and another as a decimal.
2
Word problems mix percents and fractions in the same scenario.
3
Ordering & ranking questions ask you to arrange mixed forms from least to greatest.
Fraction → Decimal
Decimal → Percent
Percent → Fraction
The Conversion Cycle
Terminating vs. Repeating Decimals
Terminating
The decimal ends. Happens when the denominator (in lowest terms) has only factors of \(2\) and \(5\).
\(\frac{3}{8} = 0.375\)
\(\frac{3}{8} = 0.375\)
Repeating
One or more digits repeat forever. Happens when the denominator has other prime factors.
\(\frac{1}{3} = 0.333\ldots = 0.\overline{3}\)
\(\frac{1}{3} = 0.333\ldots = 0.\overline{3}\)
Step-by-Step Conversion Method
1
Identify the starting form: Is it a fraction, a decimal, or a percent?
2
Pick the target form that the question asks for (or the one that makes comparison easiest).
3
Apply the correct operation: divide (frac → dec), multiply by 100 (dec → %), or place over 100 (% → frac).
4
Simplify if you land on a fraction: find the GCF of numerator and denominator and divide both.
5
Double-check by converting your answer back to the original form.
Example 1: Fraction to Percent
Convert \(\frac{7}{20}\) to a percent.
1
Divide: \(7 \div 20 = 0.35\)
2
Multiply by 100: \(0.35 \times 100 = 35\)
3
Attach the symbol: \(35\%\)
Example 2: Percent to Fraction
Convert \(62.5\%\) to a fraction in simplest form.
1
Place over 100: \(\frac{62.5}{100}\)
2
Clear the decimal: Multiply top and bottom by \(10\): \(\frac{625}{1000}\)
3
Simplify: \(\gcd(625, 1000) = 125\), so \(\frac{625 \div 125}{1000 \div 125} =\) \(\frac{5}{8}\)
Example 3: Ordering Mixed Forms
Arrange from least to greatest: \(\quad 0.6, \quad \frac{5}{8}, \quad 58\%\)
1
Convert all to decimals:
\(0.6 = 0.600\)
\(\frac{5}{8} = 0.625\)
\(58\% = 0.580\)
\(0.6 = 0.600\)
\(\frac{5}{8} = 0.625\)
\(58\% = 0.580\)
2
Compare: \(0.580 < 0.600 < 0.625\)
3
Answer: \(58\%\), \(0.6\), \(\frac{5}{8}\)
Memorize These Benchmarks
| Fraction | Decimal | Percent |
|---|---|---|
| \(\frac{1}{2}\) | \(0.5\) | \(50\%\) |
| \(\frac{1}{3}\) | \(0.\overline{3}\) | \(33.\overline{3}\%\) |
| \(\frac{1}{4}\) | \(0.25\) | \(25\%\) |
| \(\frac{1}{5}\) | \(0.2\) | \(20\%\) |
| \(\frac{1}{8}\) | \(0.125\) | \(12.5\%\) |
| \(\frac{2}{3}\) | \(0.\overline{6}\) | \(66.\overline{6}\%\) |
| \(\frac{3}{4}\) | \(0.75\) | \(75\%\) |
| \(\frac{4}{5}\) | \(0.8\) | \(80\%\) |
Common Mistakes
✕ Moving the Decimal the Wrong Way
Going from percent to decimal? Move the point left two places, not right. \(45\% = 0.45\), not \(4500\).
✕ Forgetting to Simplify
\(\frac{25}{100}\) is correct but incomplete. Always reduce: \(\frac{25}{100} = \frac{1}{4}\).
✕ Confusing \(\frac{1}{3}\) with \(0.3\)
\(\frac{1}{3} = 0.\overline{3} \approx 0.333\), not \(0.3\). On the ISEE, this small difference can change the answer.
Question 1 of 5
What is \(\frac{3}{5}\) expressed as a percent?
\(\frac{3}{5} = 3 \div 5 = 0.6\), and \(0.6 \times 100 = 60\%\).
Question 2 of 5
Which fraction is equal to \(0.125\)?
\(0.125 = \frac{125}{1000}\). Dividing both by \(125\) gives \(\frac{1}{8}\).
Question 3 of 5
Arrange from least to greatest:
\(\frac{2}{5},\quad 0.45,\quad 38\%\)
Convert all to decimals: \(38\% = 0.38\), \(\frac{2}{5} = 0.40\), and \(0.45\). So the order is \(0.38 < 0.40 < 0.45\).
Question 4 of 5
What is \(87.5\%\) as a fraction in simplest form?
\(87.5\% = \frac{87.5}{100} = \frac{875}{1000}\). The GCF is \(125\): \(\frac{875 \div 125}{1000 \div 125} = \frac{7}{8}\).
Question 5 of 5
Which value is greatest?
Convert to decimals: \(\frac{5}{9} \approx 0.556\), \(0.54\), \(56\% = 0.56\), \(\frac{11}{20} = 0.55\). The greatest is \(0.56 = 56\%\).