5.7 Fraction and Decimal Conversion

Fraction and Decimal Conversion covers the procedures for rewriting a fraction as its equivalent decimal and rewriting a decimal as its equivalent fraction, describing the division process that converts a fraction to a decimal, the denominator characteristics that predict whether that decimal will terminate or repeat, the place-value-based process that converts a terminating decimal to a fraction, the specialized process required for repeating decimals, and the verification techniques used to confirm a conversion was performed correctly.

Fraction-to-Decimal Division

Fraction-to-decimal division converts a fraction into its decimal equivalent by dividing the numerator by the denominator, following the fraction-as-a-quotient interpretation directly, such as converting 3/4 to a decimal by computing 3 ÷ 4 = 0.75.

Terminating Fraction Decimal Conversion

Terminating fraction decimal conversion describes the outcome when fraction-to-decimal division produces a remainder of exactly zero after a finite number of steps, resulting in a terminating decimal such as 0.75.

Repeating Fraction Decimal Conversion

Repeating fraction decimal conversion describes the outcome when fraction-to-decimal division never produces a remainder of zero, instead cycling through a repeating sequence of remainders indefinitely, resulting in a repeating decimal such as 1/3 = 0.333...

Denominator Factors and Decimal Termination

Denominator factors and decimal termination describes the predictive rule that a fraction in simplest form produces a terminating decimal if and only if its denominator's only prime factors are 2 and 5; any other prime factor present in the denominator guarantees a repeating decimal instead. The fraction 3/8 terminates, since 8 factors only into 2s, while the fraction 1/3 repeats, since 3 is a prime factor other than 2 or 5.

Terminating Decimal-to-Fraction Conversion

Terminating decimal-to-fraction conversion rewrites a terminating decimal as a fraction by placing its digits (without the decimal point) over the appropriate power of ten matching its number of decimal places, then reducing to simplest form, such as converting 0.75 into 75/100, which simplifies to 3/4.

Decimal Place Value as a Fractional Denominator

Decimal place value as a fractional denominator identifies the specific power of ten used as the denominator during terminating decimal-to-fraction conversion, determined directly by the number of digits in the decimal's fractional part—10 for one decimal place, 100 for two decimal places, 1000 for three decimal places, and so on.

Repeating Decimal-to-Fraction Conversion

Repeating decimal-to-fraction conversion uses an algebraic technique involving multiplying the repeating decimal by an appropriate power of ten to shift the repeating block, then subtracting the original decimal from that shifted value to eliminate the infinite repetition, leaving a solvable equation for the equivalent fraction. Converting 0.333... to a fraction involves setting x = 0.333..., multiplying by 10 to get 10x = 3.333..., and subtracting to obtain 9x = 3, giving x = 1/3.

Mixed Repeating Decimal Conversion

Mixed repeating decimal conversion handles a decimal containing both a non-repeating portion and a repeating portion, requiring a modified version of the repeating decimal-to-fraction technique that uses two different powers of ten to isolate and shift only the repeating block before subtracting.

Exact Fraction and Rounded Decimal Distinction

Exact fraction and rounded decimal distinction clarifies that converting a repeating decimal fraction into a truncated decimal approximation, such as writing 1/3 as 0.33, produces a value close to but not exactly equal to the original fraction, and this distinction must be tracked carefully whenever an exact fractional value is temporarily approximated for convenience.

Conversion Result Verification

Conversion result verification confirms a fraction-to-decimal or decimal-to-fraction conversion by reversing the process—dividing the converted fraction's numerator by its denominator to check it reproduces the original decimal, or converting the resulting decimal back into a fraction to check it reproduces the original fraction.

Together, these conversion procedures establish a complete two-way bridge between fractions and decimals: division converts a fraction to a decimal, with the denominator's prime factors predicting termination or repetition in advance, while place-value-based and algebraic techniques convert terminating and repeating decimals back into their exact fractional equivalents, with verification confirming the conversion in either direction was carried out correctly.