A repeating decimal

Number Theory Level 2

Given that 0. 969 ‾ = a b 0.\overline{969}= \dfrac{a}{b} , where a a and b b are coprime positive integers . What is a + b a+b ?


The answer is 656.

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Geoff Pilling by Geoff Pilling , 53, USA

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2 solutions

0.96996969...=x (1) Multiply each side by 1000. => 969.969...=1000x (2) (2)-(1) => 969=999x x= 969/999= 323/333

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Geoff Pilling
Aug 2, 2016

0. 969 ‾ = 969 999 = 323 333 0.\overline{969}= \frac{969}{999} = \frac{323}{333} 323 + 333 = 656 323+333 = \boxed{656}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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