Question

1- Determine a forma fracionária de :

a) 3,2444444...

b) 0,0588888....

c) 8,141414....

d) 2,123123123....

Answer 1

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$\boxed{\begin{array}{l}\rm a)~\sf x=3,2444....\cdot10\\\sf 10x=32,444...\cdot10\\\sf 100x=324,444...\\-\underline{\begin{cases}\sf100x=324,444...\\\sf 10x=32,444...\end{cases}}\\\sf90x=292\\\sf x=\dfrac{292}{90}\\\sf x=\dfrac{146}{45}\end{array}}$

$\boxed{\begin{array}{l}\rm b)~\sf z=0,05888...\cdot100\\\sf 100z=5,888...\cdot10\\\sf 1000z=58,888...\\-\underline{\begin{cases}\sf1000z=58,888...\\\sf 100z=5,888...\end{cases}}\\\sf 900z=53\\\sf z=\dfrac{53}{900}\end{array}}$

$\boxed{\begin{array}{l}\rm c)~\sf w=8,141414...\cdot100\\\sf 100w=814,141414...\\-\underline{\begin{cases}\sf100w=814,141414...\\\sf w=8,141414...\end{cases}}\\\sf 99w=806\\\sf w=\dfrac{806}{99}\end{array}}$

$\boxed{\begin{array}{l}\rm d)~\sf k=2,123123...\cdot1000\\\sf 1000k=2123,123123...\\-\underline{\begin{cases}\sf 1000k=2123,123\\\sf k=2,123123...\end{cases}}\\\sf999k=2121\\\sf k=\dfrac{2121\div3}{999\div3}\\\sf k=\dfrac{707}{333}\end{array}}$