Recurring Decimals As Geometric Series

Question

Express the recurring decimal 0.overline{36}, meaning 0.363636 repeating endlessly, as a fully reduced rational fraction in lowest terms.

RankGuru · Accepted Answer

A purely recurring decimal is the sum of an infinite geometric series, which JEE Advanced treats with the convergent-sum formula a/(1 - r). Here 0.\overline{36} = 36/100 + 36/100^2 + ..., with first term a = 36/100 and ratio r = 1/100. The sum equals (36/100)/(1 - 1/100) = (36/100)/(99/100) = 36/99. Reducing by the common factor 9 gives 4/11. Option 36/99 is correct in value but not in lowest terms, so it is rejected by the reduced-fraction requirement. Option 9/25 equals 0.36 exactly, a terminating decimal, not the repeating one. Option 2/5 equals 0.4, clearly different. Hence the lowest-terms answer is 4/11. A useful general rule emerges: a block of d repeating digits always equals that block divided by the number consisting of d nines, which is exactly the infinite geometric series in disguise, and JEE Advanced expects this rule to be both stated and justified. Plausibility check: dividing 4 by 11 yields 0.363636..., reproducing the original recurring decimal exactly and confirming both the value and the reduction.

Recurring Decimals As Geometric Series

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