A boring problem if you have learnt this.

Number Theory Level 2

Express the following recurring decimal in its simplest fraction form.

0.414414414414414414... 0.414414414414414414...

Find in the multiple choice selection the numerator of this fraction.

118118 73 253 The answer is irrational and cannot be expressed as a fraction. 414 71 46 23

Alfred Shen by Alfred Shen , 28, United Kingdom

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2 solutions

Alfred Shen
Aug 4, 2019

Call recurring decimal, X. Divide X by 1000 to find the tail and then "minus off the tail". You should find 999X=414. Find X and simplify to get 46/111. The answer is 46.

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0. 414414 ‾ = 414414 999999 = 2 × 3 2 × 7 × 11 × 13 × 23 3 3 × 7 × 11 × 13 × 37 = 2 × 23 3 × 37 = 46 111 \begin{aligned} 0.\overline{414414} & = \frac {414414}{999999} = \frac {2\times 3^2 \times 7 \times 11 \times 13 \times 23}{3^3 \times 7 \times 11 \times 13 \times 37} = \frac {2\times 23}{3\times 37} = \frac {\boxed{46}}{111} \end{aligned}

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