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Question Number 122649 by mathocean1 last updated on 18/Nov/20

By divising an integer a by   integer b we find the result:  0.285714285714... followed  by a group of 6 digits: 285714  which is repeated indefinited.  determinate the fraction (a/b)

$${By}\:{divising}\:{an}\:{integer}\:{a}\:{by}\: \\ $$$${integer}\:{b}\:{we}\:{find}\:{the}\:{result}: \\ $$$$\mathrm{0}.\mathrm{285714285714}...\:{followed} \\ $$$${by}\:{a}\:{group}\:{of}\:\mathrm{6}\:{digits}:\:\mathrm{285714} \\ $$$${which}\:{is}\:{repeated}\:{indefinited}. \\ $$$${determinate}\:{the}\:{fraction}\:\frac{{a}}{{b}}\: \\ $$$$ \\ $$$$ \\ $$

Commented by mr W last updated on 18/Nov/20

0.285714...=x  285714+x=100000x  285714=999999x  x=((285714)/(999999))=(2/7)=(a/b)

$$\mathrm{0}.\mathrm{285714}...={x} \\ $$$$\mathrm{285714}+{x}=\mathrm{100000}{x} \\ $$$$\mathrm{285714}=\mathrm{999999}{x} \\ $$$${x}=\frac{\mathrm{285714}}{\mathrm{999999}}=\frac{\mathrm{2}}{\mathrm{7}}=\frac{{a}}{{b}} \\ $$

Commented by mathocean1 last updated on 18/Nov/20

thank you very much sir

$${thank}\:{you}\:{very}\:{much}\:{sir} \\ $$

Answered by liberty last updated on 19/Nov/20

let p = 0.285714285714...  ⇒1000000p=285714.285714...  subtract (2) by(1)  999999p=285714 ⇒p=((285714)/(999999))=(2/7)

$${let}\:{p}\:=\:\mathrm{0}.\mathrm{285714285714}... \\ $$$$\Rightarrow\mathrm{1000000}{p}=\mathrm{285714}.\mathrm{285714}... \\ $$$${subtract}\:\left(\mathrm{2}\right)\:{by}\left(\mathrm{1}\right) \\ $$$$\mathrm{999999}{p}=\mathrm{285714}\:\Rightarrow{p}=\frac{\mathrm{285714}}{\mathrm{999999}}=\frac{\mathrm{2}}{\mathrm{7}} \\ $$

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