Fractions, Decimals
& Percents
Master the art of converting between fractions, decimals, and percents: three different ways to express the same value.
SSAT Middle Level: Quantitative
Three Representations, One Value
Fractions, decimals, and percents are three different ways to describe the same part of a whole. Learning to convert between them is one of the most useful skills in math.
Fraction
A ratio of two integers: \(\frac{\text{part}}{\text{whole}}\)
Decimal
A number using place value and a decimal point: \(0.75\)
Percent
A fraction out of \(100\): the symbol \(\%\) means "per hundred"
Seeing the Same Value
Why Conversions Matter
On the SSAT and in everyday math, you need to compare, add, or evaluate values that appear in different forms. Converting everything to the same form makes the problem solvable.
Comparing
Which is bigger: \(\frac{2}{5}\) or \(0.45\)? Convert to compare.
Computing
Finding \(20\%\) of \(80\) is easier as \(0.20 \times 80 = 16\).
SSAT Questions
Many problems mix all three forms in one question on purpose.
Fraction → Decimal
Decimal → Percent
Percent → Fraction
Terminating vs Repeating Decimals
Not every fraction gives a "nice" decimal. Some fractions produce terminating decimals (they end), while others produce repeating decimals (a pattern repeats forever).
Terminating
\(\frac{1}{4} = 0.25\) — the decimal stops. The denominator's only prime factors are \(2\) and \(5\).
Repeating
\(\frac{1}{3} = 0.333\ldots = 0.\overline{3}\) — the digit \(3\) repeats forever. The denominator has a prime factor other than \(2\) or \(5\).
The Conversion Cycle
1
Fraction → Decimal: Divide numerator by denominator.
2
Decimal → Percent: Multiply by \(100\) (shift decimal right 2 places).
3
Percent → Fraction: Write over \(100\), then simplify.
4
Decimal → Fraction: Read the place value, write as a fraction, simplify.
Example 1: Convert \(\frac{2}{5}\)
Convert \(\frac{2}{5}\) to a decimal and a percent.
1
Fraction → Decimal: \(2 \div 5 = \) \(0.4\)
2
Decimal → Percent: \(0.4 \times 100 = \) \(40\%\)
Example 2: Convert \(85\%\) to a Fraction
1
Write over 100: \(85\% = \frac{85}{100}\)
2
Find the GCF: The greatest common factor of \(85\) and \(100\) is \(5\).
3
Simplify: \(\frac{85 \div 5}{100 \div 5} = \) \(\frac{17}{20}\)
Example 3: Convert \(0.125\)
Convert \(0.125\) to a fraction and a percent.
1
Read the place value: \(0.125\) means \(125\) thousandths, so write \(\frac{125}{1000}\).
2
Simplify: GCF of \(125\) and \(1000\) is \(125\). So \(\frac{125 \div 125}{1000 \div 125} = \) \(\frac{1}{8}\)
3
Decimal → Percent: \(0.125 \times 100 = \) \(12.5\%\)
Must-Know Conversions
| Fraction | Decimal | Percent |
|---|---|---|
| \(\frac{1}{2}\) | \(0.5\) | \(50\%\) |
| \(\frac{1}{3}\) | \(0.\overline{3}\) | \(33.\overline{3}\%\) |
| \(\frac{1}{4}\) | \(0.25\) | \(25\%\) |
| \(\frac{1}{5}\) | \(0.2\) | \(20\%\) |
| \(\frac{1}{8}\) | \(0.125\) | \(12.5\%\) |
| \(\frac{2}{3}\) | \(0.\overline{6}\) | \(66.\overline{6}\%\) |
| \(\frac{3}{4}\) | \(0.75\) | \(75\%\) |
| \(\frac{3}{5}\) | \(0.6\) | \(60\%\) |
Common Mistakes
✕ Moving Decimal Wrong Way
Going from percent to decimal? Move the point left 2 places, not right. \(45\% = 0.45\), not \(4500\).
✕ Forgetting to Simplify
Writing \(\frac{60}{100}\) instead of \(\frac{3}{5}\). Always reduce fractions to lowest terms.
✕ Dividing Backwards
For \(\frac{3}{8}\), divide \(3 \div 8\), not \(8 \div 3\). The numerator goes inside the division.
Conversion Quick Reference
| From → To | Method | Example |
|---|---|---|
| Fraction → Decimal | Divide top by bottom | \(\frac{7}{8} = 0.875\) |
| Decimal → Percent | Multiply by \(100\) | \(0.875 = 87.5\%\) |
| Percent → Fraction | Over \(100\), simplify | \(87.5\% = \frac{7}{8}\) |
| Decimal → Fraction | Read place value, simplify | \(0.6 = \frac{3}{5}\) |
| Percent → Decimal | Divide by \(100\) | \(45\% = 0.45\) |
| Fraction → Percent | Divide, then \(\times 100\) | \(\frac{3}{10} = 30\%\) |
Question 1 of 5
What is \(\frac{3}{8}\) expressed as a percent?
First convert to a decimal: \(\frac{3}{8} = 3 \div 8 = 0.375\). Then multiply by \(100\): \(0.375 \times 100 = 37.5\%\).
Question 2 of 5
Which fraction is equivalent to \(0.65\)?
\(0.65 = \frac{65}{100}\). The GCF of \(65\) and \(100\) is \(5\), so \(\frac{65 \div 5}{100 \div 5} = \frac{13}{20}\).
Question 3 of 5
Which of the following is the largest?
Convert all to decimals: \(\frac{3}{5} = 0.6\), \(0.59\), \(58\% = 0.58\), \(\frac{5}{8} = 0.625\). The largest is \(\frac{5}{8} = 0.625\).
Question 4 of 5
What is \(175\%\) expressed as a fraction?
\(175\% = \frac{175}{100}\). The GCF of \(175\) and \(100\) is \(25\): \(\frac{175 \div 25}{100 \div 25} = \frac{7}{4}\). Note: percents greater than \(100\%\) give improper fractions (greater than \(1\)).
Question 5 of 5
A store marks a shirt as \(\frac{1}{3}\) off. What percent discount is that?
\(\frac{1}{3} = 1 \div 3 = 0.333\ldots\). Multiply by \(100\): \(0.\overline{3} \times 100 = 33.\overline{3}\%\). This is a repeating decimal, so the exact answer is \(33.\overline{3}\%\), not \(33\%\).