How do you know if a fraction will be a terminating or repeating decimal?

Simplify the fraction fully, then look at the denominator's prime factors. If the denominator has only factors of 2 and 5, the decimal terminates. If it has any other prime factor (like 3, 7, or 11), the decimal repeats. For example, $\frac{1}{8}$ terminates because $8 = 2^3$, but $\frac{1}{6}$ repeats because $6 = 2 \times 3$.

Converting Fractions to Decimals — Definition, Formula & Examples

Converting fractions to decimals means rewriting a fraction as a decimal number by dividing the numerator (top number) by the denominator (bottom number). For example, the fraction

\frac{3}{4}

becomes

0.75

A fraction $\frac{a}{b}$ , where $b \neq 0$ , is expressed in decimal form by computing the quotient $a \div b$ . The result is either a terminating decimal (finitely many digits) or a repeating decimal (a block of digits that cycles indefinitely).

Key Formula

\frac{a}{b} = a \div b

Where:

$a$ = The numerator (top number of the fraction)
$b$ = The denominator (bottom number of the fraction), where $b \neq 0$

How It Works

To convert any fraction to a decimal, divide the numerator by the denominator using long division or a calculator. If the division ends with a remainder of zero, you get a terminating decimal like

0.25

. If the remainder starts repeating, you get a repeating decimal like

0.333\ldots

, which is written as

0.\overline{3}

. For mixed numbers, convert the fractional part separately and add it to the whole number. You can also convert some fractions by finding an equivalent fraction whose denominator is 10, 100, or 1000 — for instance,

\frac{3}{5} = \frac{6}{10} = 0.6

Worked Example

Problem: Convert

\frac{7}{8}

to a decimal.

Step 1: Set up the division: numerator divided by denominator.

7 \div 8

Step 2: Since 8 does not go into 7, place a decimal point and add zeros. 8 goes into 70 eight times (64), leaving a remainder of 6.

70 \div 8 = 8 \text{ remainder } 6

Step 3: Bring down another zero. 8 goes into 60 seven times (56), leaving a remainder of 4.

60 \div 8 = 7 \text{ remainder } 4

Step 4: Bring down another zero. 8 goes into 40 exactly 5 times with no remainder.

40 \div 8 = 5 \text{ remainder } 0

Answer:

\frac{7}{8} = 0.875

Another Example

Problem: Convert

\frac{2}{3}

to a decimal.

Step 1: Divide the numerator by the denominator.

2 \div 3

Step 2: 3 goes into 20 six times (18), remainder 2. Then 3 goes into 20 six times again. The remainder keeps repeating.

2 \div 3 = 0.666\ldots

Step 3: Since the digit 6 repeats forever, write it with a bar over the repeating part.

\frac{2}{3} = 0.\overline{6}

Answer:

\frac{2}{3} = 0.\overline{6}

(a repeating decimal)

Visualization

Why It Matters

Converting fractions to decimals is essential in pre-algebra and appears constantly on standardized tests. It is also a practical skill for everyday tasks — comparing prices, reading measurements, and interpreting data all require moving fluently between fractions and decimals. Science courses rely on decimal form for calculations with formulas and lab measurements.

Common Mistakes

Mistake: Dividing the denominator by the numerator instead of the numerator by the denominator.

Correction: Always divide top by bottom:

\frac{3}{4}

means

3 \div 4 = 0.75

, not

4 \div 3 = 1.\overline{3}

Mistake: Stopping long division too early and rounding a repeating decimal as if it terminates.

Correction: If the remainder keeps cycling, the decimal repeats. Use bar notation (e.g.,

0.\overline{3}

) to show the repeating block, or state that you are rounding.

Related Terms

Fraction — The starting form you convert from
Decimal — The result after converting a fraction
Numerator — The dividend in the conversion division
Denominator — The divisor in the conversion division
Mixed Number — Whole-number-plus-fraction form to convert
Improper Fraction — Fraction greater than 1 converted similarly
Fraction Rules — Core rules for simplifying before converting
Ratio — Ratios can be written as fractions then decimals