Decimals, Fractions, and Percentages — Definition, Formula & Examples
Decimals, fractions, and percentages are three different ways to represent the same quantity. A fraction like 3/4, the decimal 0.75, and 75% all express the same value — just written in different forms.
Any rational number can be expressed as a fraction (where ), as a terminating or repeating decimal, or as a percentage (a ratio per 100). Converting among these forms preserves the value: (decimal form) and (percent form).
Key Formula
\frac{a}{b} = a \div b \quad\text{(decimal)}$$
$$\frac{a}{b} \times 100 = \text{percentage}$$
$$\text{decimal} = \frac{\text{percentage}}{100}
Where:
- = The numerator of the fraction
- = The denominator of the fraction (must not be zero)
How It Works
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100 and add the percent sign. To go the other direction — percentage to decimal — divide by 100, and decimal to fraction, write the decimal over the appropriate power of 10 and simplify. These three conversions form a cycle: fraction → decimal → percentage → fraction. Mastering this cycle lets you move freely between forms depending on which is most convenient for a given problem.
Worked Example
Problem: Convert the fraction 3/8 to a decimal and then to a percentage.
Fraction to Decimal: Divide the numerator by the denominator.
Decimal to Percentage: Multiply the decimal by 100 and attach the percent sign.
Verify: Check by converting back: 37.5% means 37.5 out of 100, which simplifies to 3/8.
Answer: 3/8 = 0.375 = 37.5%
Another Example
Problem: Convert 45% to a fraction in simplest form and to a decimal.
Percentage to Fraction: Write the percentage over 100.
Simplify the Fraction: Find the greatest common factor of 45 and 100, which is 5, and divide both by it.
Percentage to Decimal: Divide the percentage by 100 (move the decimal point two places left).
Answer: 45% = 9/20 = 0.45
Visualization
Why It Matters
Converting between decimals, fractions, and percentages is essential in pre-algebra and appears constantly in everyday life — calculating sales tax, interpreting statistics, and reading nutrition labels all require switching between these forms. In science courses, you often convert measured data from decimals to percentages for lab reports. Careers in finance, healthcare, and engineering rely on fluent conversion between all three representations.
Common Mistakes
Mistake: Moving the decimal point the wrong direction when converting between decimals and percentages.
Correction: Remember: decimal to percent, multiply by 100 (move right two places). Percent to decimal, divide by 100 (move left two places). For example, 0.05 is 5%, not 50%.
Mistake: Forgetting to simplify the fraction after converting from a decimal or percentage.
Correction: Always reduce to lowest terms. For instance, 60% = 60/100 should be simplified to 3/5 by dividing numerator and denominator by their GCF of 20.