L6 · Repeating decimals and fractions

Learning goal

“I can write a repeating decimal, turn fractions like 1/3, 5/9, 4/11 and 1/6 into repeating decimals by dividing, and name the fraction for the classic repeating families (ninths and elevenths).”

Repeating decimal — a decimal whose digits cycle forever in a fixed pattern: 0.333..., 0.3636...

Bar notation — a bar written over the repeating block to show it repeats forever: 0.333... is written with a bar over the 3; 0.3636... with a bar over 36.

Repetend — the block of digits that repeats: in 0.7272... the repetend is 72; in 0.1666... it is just 6.

Terminate — to end (from L5): a terminating decimal stops; a repeating decimal never does.

Worked example

Divide 1 by 9 and watch the remainders. Step 1: 1.000 / 9: 9 goes into 10 once, remainder 1. Bring down a 0: 10 again. Remainder 1 again. Step 2: The remainder is 1 forever, so the quotient digit is 1 forever: 1/9 = 0.111... Step 3: Bar notation writes this exactly, with a bar over the repeating 1. The whole ninths family follows the pattern: n/9 repeats the single digit n - so 2/9 = 0.222..., 5/9 = 0.555..., 7/9 = 0.777... Thirds are ninths in disguise: 1/3 = 3/9 = 0.333... and 2/3 = 6/9 = 0.666... On the number line, 0.333... sits a hair to the RIGHT of 0.3 - the endless 3s keep adding tiny amounts past 0.300.

Check yourself — explain it

The division 1 / 9 never ends because...

Paired problem — example still visible

Write 5/9 as a repeating decimal. -> 0.555...

Faded completion — last step is yours

"2/3 = 6/9, and 6/9 as a repeating decimal is ___" -> 0.666...

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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