Converting
Fractions & Decimals
Learn to fluently convert between fractions and decimals, understand why they are equivalent, and build number sense that lasts.
Pre-Algebra
What Are Fractions & Decimals?
Both fractions and decimals are ways to represent parts of a whole. A fraction like \(\frac{3}{4}\) and a decimal like \(0.75\) describe the exact same quantity: they just use different notation.
Fraction
A ratio of two integers: \(\frac{\text{numerator}}{\text{denominator}}\). The denominator tells you how many equal parts, the numerator tells you how many you have.
Decimal
A number expressed in base-10 place value, using a decimal point. Each digit after the point represents tenths, hundredths, thousandths, etc.
Seeing the Equivalence
Why Converting Matters
Real-world problems mix fractions and decimals constantly: recipes use \(\frac{1}{2}\) cup, but a calculator displays \(0.5\). Being able to switch fluently between the two forms lets you:
1
Compare numbers easily: Is \(\frac{3}{8}\) or \(0.4\) larger? Converting both to decimals makes it obvious.
2
Use calculators effectively: Most calculators give decimal answers, but your work may require fractions.
3
Build algebra skills: Fraction-decimal fluency is the foundation for ratios, percents, and proportional reasoning.
The Core Rule: Fraction to Decimal
The Core Rule: Decimal to Fraction
1
Identify the last decimal place: \(0.75\) ends in the hundredths place.
2
Write as a fraction over that power of 10: \(0.75 = \frac{75}{100}\).
3
Simplify by dividing by the GCF: \(\frac{75}{100} = \frac{75 \div 25}{100 \div 25} = \frac{3}{4}\).
Terminating vs Repeating Decimals
Terminating
The decimal ends after a finite number of digits. Example: \(\frac{1}{4} = 0.25\). This happens when the denominator's only prime factors are \(2\) and/or \(5\).
Repeating
One or more digits repeat forever. Example: \(\frac{1}{3} = 0.333\ldots = 0.\overline{3}\). This happens when the denominator has a prime factor other than \(2\) or \(5\).
The Place Value Bridge
Step-by-Step Conversion Guide
Fraction → Decimal
1. Divide numerator by denominator
2. Use long division or a calculator
3. Note if the result terminates or repeats
2. Use long division or a calculator
3. Note if the result terminates or repeats
Decimal → Fraction
1. Count decimal places
2. Write digits over the matching power of \(10\)
3. Find the GCF and simplify
2. Write digits over the matching power of \(10\)
3. Find the GCF and simplify
Example 1: Convert \(\frac{7}{8}\) to a Decimal
1
Set up the division: \(7 \div 8\)
2
Perform long division: \(8\) goes into \(70\) eight times (\(64\)), remainder \(6\). Then \(8\) into \(60\) seven times (\(56\)), remainder \(4\). Then \(8\) into \(40\) exactly \(5\) times.
3
Result: \(7 \div 8 = 0.875\). The decimal terminates because \(8 = 2^3\).
Example 2: Convert \(0.6\) to a Fraction
1
Identify place value: \(0.6\) has one decimal place, so it is in the tenths position.
2
Write as a fraction: \(0.6 = \frac{6}{10}\).
3
Simplify: The GCF of \(6\) and \(10\) is \(2\). So \(\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}\).
Example 3: Convert \(\frac{5}{6}\) to a Decimal
1
Set up the division: \(5 \div 6\)
2
Divide: \(6\) into \(50\) is \(8\) (remainder \(2\)); \(6\) into \(20\) is \(3\) (remainder \(2\)); the remainder \(2\) repeats, so the digit \(3\) repeats forever.
3
Result: \(\frac{5}{6} = 0.8333\ldots = 0.8\overline{3}\). The bar over the \(3\) means it repeats indefinitely.
Repeating Decimals → Fractions
To convert a repeating decimal like \(0.\overline{3}\) back to a fraction, use the algebraic method:
1
Let \(x\) equal the decimal: \(x = 0.333\ldots\)
2
Multiply by \(10\): \(10x = 3.333\ldots\)
3
Subtract: \(10x - x = 3.333\ldots - 0.333\ldots\), giving \(9x = 3\).
4
Solve: \(x = \frac{3}{9} = \frac{1}{3}\).
Must-Know Equivalents
| Fraction | Decimal | Type |
|---|---|---|
| \(\frac{1}{2}\) | \(0.5\) | Terminating |
| \(\frac{1}{3}\) | \(0.\overline{3}\) | Repeating |
| \(\frac{1}{4}\) | \(0.25\) | Terminating |
| \(\frac{1}{5}\) | \(0.2\) | Terminating |
| \(\frac{1}{6}\) | \(0.1\overline{6}\) | Repeating |
| \(\frac{1}{8}\) | \(0.125\) | Terminating |
| \(\frac{2}{3}\) | \(0.\overline{6}\) | Repeating |
| \(\frac{3}{4}\) | \(0.75\) | Terminating |
Common Mistakes to Avoid
✕ Forgetting to Simplify
Writing \(0.4 = \frac{4}{10}\) and stopping. You must reduce to \(\frac{2}{5}\) by dividing by the GCF.
✕ Wrong Place Value
Putting \(0.35\) over \(10\) instead of \(100\). Count the decimal places: two digits means hundredths.
✕ Division Direction
Computing \(8 \div 3\) instead of \(3 \div 8\) for \(\frac{3}{8}\). The numerator is always the dividend.
Question 1 of 5
What is \(\frac{3}{8}\) expressed as a decimal?
\(\frac{3}{8}\) means \(3 \div 8\). Performing the long division: \(8\) into \(30\) is \(3\) (remainder \(6\)), \(8\) into \(60\) is \(7\) (remainder \(4\)), \(8\) into \(40\) is \(5\) exactly. So \(\frac{3}{8} = 0.375\).
Question 2 of 5
Convert \(0.45\) to a fraction in simplest form.
\(0.45\) has two decimal places, so it equals \(\frac{45}{100}\). The GCF of \(45\) and \(100\) is \(5\): \(\frac{45 \div 5}{100 \div 5} = \frac{9}{20}\).
Question 3 of 5
Which fraction produces a repeating decimal?
A fraction in simplest form gives a repeating decimal when its denominator has a prime factor other than \(2\) or \(5\). Here, \(6 = 2 \times 3\), so the factor of \(3\) causes \(\frac{5}{6} = 0.8\overline{3}\) to repeat. The others (\(4 = 2^2\), \(8 = 2^3\), \(5 = 5\)) only have factors of \(2\) and \(5\).
Question 4 of 5
What is \(0.\overline{6}\) as a fraction in simplest form?
Let \(x = 0.666\ldots\). Then \(10x = 6.666\ldots\). Subtracting: \(10x - x = 6\), so \(9x = 6\) and \(x = \frac{6}{9} = \frac{2}{3}\). While \(\frac{6}{9}\) is technically correct, it must be simplified to \(\frac{2}{3}\).
Question 5 of 5
A student converts \(\frac{2}{5}\) by computing \(5 \div 2 = 2.5\). What did they do wrong?
To convert \(\frac{2}{5}\) to a decimal, you must divide the numerator by the denominator: \(2 \div 5 = 0.4\). The student reversed the order and computed \(5 \div 2 = 2.5\), which is actually \(\frac{5}{2}\), not \(\frac{2}{5}\).