Recurring decimals and fraction equivalents

Number • Fractions, decimals and percentages

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Describe the first step when converting a mixed repeating decimal to a fraction.

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Identify n, the number of non-repeating digits, and r, the length of the repeating block.

Key concepts

What you'll likely be quizzed about

Definition of recurring decimal

A recurring decimal contains one or more digits that repeat indefinitely after the decimal point. Pure repeating decimals start immediately after the decimal point (for example 0.333...). Mixed repeating decimals contain a finite non-repeating part followed by a repeating block (for example 1.2 34 repeating). Repeating decimals represent rational numbers.

Pure repeating decimal → fraction (algebra method)

Let x equal the recurring decimal. Multiply x by 10^r, where r equals the number of repeating digits, to shift one full repeat block left of the decimal point. Subtract the original x from the shifted value to cancel the repeating part. Solve the resulting integer equation for x. Example: x = 0.333...; 10x = 3.333...; 10x − x = 3; 9x = 3; x = 3/9 = 1/3. Cause: shifting creates identical repeating tails; Effect: subtraction removes the infinite tail and yields an equation with integers.

Mixed repeating decimal → fraction (general rule)

Let x equal the decimal with a non-repeating part of length n and a repeating block of length r. Multiply x by 10^n to move the non-repeating digits to the integer part. Multiply by 10^(n+r) to move one full repeating block past the decimal point. Subtract the two results to cancel the infinite repeat. The fraction equals (integer formed by all digits up to the end of the first repeat − integer formed by the non-repeating digits) divided by (10^(n+r) − 10^n). Example: x = 1.2 34 repeating; n = 1, r = 2; 1000x − 10x = 1234 − 12; 990x = 1222; x = 1222/990 = simplified.

Fraction → decimal (terminating or recurring)

Long division of numerator by denominator produces either a terminating decimal or a recurring decimal. Cause: denominator prime factors. After simplification, if the denominator contains only 2 and/or 5 as prime factors, the decimal terminates because powers of 10 eventually divide the denominator. If the simplified denominator contains any prime factor other than 2 or 5, long division produces a repeating cycle and therefore a recurring decimal.

Shortcut forms and limiting factors

Fractions with denominators 9, 99, 999, etc. produce repeating blocks equal to the numerator (scaled): for example 7/9 = 0.777..., 13/99 = 0.1313... Careful handling of repeating nines is necessary because 0.999... equals 1; recurring-nine representations correspond to terminating values. Simplification of the fraction before conversion prevents unnecessarily long repeating blocks.

Key notes

Important points to keep in mind

Identify repeating block and its length before any algebraic manipulation.

Use 10^r to shift pure repeating decimals; use 10^n and 10^(n+r) for mixed repeats.

Simplify fractions before or after conversion to avoid unnecessarily large numerators and denominators.

Check if the simplified denominator has only factors 2 and 5 to determine termination.

Recognise recurring nine patterns because 0.999... equals 1 and similar cases produce integer values.