How to Convert Fractions to Decimals and Decimals to Fractions
In mathematics, fractions and decimals are two different notations used to express identical numerical quantities. A fraction represents a quotient or ratio of two integers ($a/b$), whereas a decimal expresses fractional parts using place values based on powers of 10 (tenths, hundredths, thousandths).
1. How to Convert a Fraction to a Decimal Without a Calculator
To convert any fraction into a decimal manually using pencil and paper:
- Set up a standard long division problem with the denominator on the outside (divisor) and the numerator on the inside (dividend).
- Place a decimal point directly above the division bar and add trailing zeros after the numerator (e.g. convert $3$ into $3.000$).
- Divide sequentially from left to right, bringing down zeros after each subtraction until the remainder is $0$ (terminating decimal) or begins an identical repeating loop (repeating decimal).
• 3 ÷ 8 = 0 with remainder 3
• Bring down 0: 30 ÷ 8 = 3 (remainder 6)
• Bring down 0: 60 ÷ 8 = 7 (remainder 4)
• Bring down 0: 40 ÷ 8 = 5 (remainder 0)
Result = 0.375
2. How to Turn Mixed Fractions with Whole Numbers into Decimals
A mixed fraction consists of an integer whole number alongside a proper fraction (such as $2\frac{3}{4}$):
- Method 1 (Separate Addition): Keep the whole number $2$, convert $3/4$ into $0.75$, then add: $2 + 0.75 = 2.75$.
- Method 2 (Improper Fraction Conversion): Multiply the whole number by the denominator and add the numerator: $(2 \times 4) + 3 = 11$, giving $11/4$. Then divide $11 \div 4 = 2.75$.
3. How to Convert Repeating Decimals to Fractions (Algebraic Method)
When a decimal repeats indefinitely (such as $0.1666...$ or $0.333...$), use the system of equations technique:
• Let x = 0.1666...
• Multiply by 10 (shift non-repeating part): 10x = 1.666...
• Multiply by 100 (shift one repeating period): 100x = 16.666...
• Subtract equations: 100x - 10x = 16.666... - 1.666... → 90x = 15
• Simplify: x = 15 / 90 = 1/6