Question

Determine a fração geratriz das seguintes dizimas periódicas: com cálculos a) 5,12323232... b) 1,3818181 c) 0.252525 d) 0,2333 e) 6,080808 f) 2,1888​

Answer 1

$\Large\boxed{\begin{array}{l}\tt a)~\sf x=5,12323232...\cdot100\\\sf 100x=512,3232...\cdot100\\\sf 10000 x=51232,323232....\\-\underline{\begin{cases}\sf 10000x=51232,3232...\\\sf 100x=512,3232...\end{cases}}\\\sf9900x=50720\\\sf x=\dfrac{50720\div20}{9900\div20}\\\\\sf x=\dfrac{2536}{495}\end{array}}$

$\Large\boxed{\begin{array}{l}\tt b)~\sf x=1,3818181...\cdot 10\\\sf 10x=13,8181...\cdot100\\\sf 1000x=1381,8181...\\-\underline{\begin{cases}\sf 1000x=1381,8181...\\\sf 10x=13,8181...\end{cases}}\\\sf 990x=1368\\\sf x=\dfrac{1368\div18}{990\div18}\\\\\sf x=\dfrac{76}{55}\end{array}}$

$\Large\boxed{\begin{array}{l}\tt c)~\sf x=0,2525...\cdot100\\\sf 100x=25,2525...\\-\underline{\begin{cases}\sf 100x=25,2525...\\\sf x=0,2525...\end{cases}}\\\sf 99x=25\\\sf x=\dfrac{25}{99}\end{array}}$

$\Large\boxed{\begin{array}{l}\tt d)~\sf x=0,2333...\cdot 10\\\sf 10x=2,333...\cdot10\\\sf 100x=23,333...\\-\underline{\begin{cases}\sf 100x=23,333...\\\sf 10x=2,333...\end{cases}}\\\sf 90x=21\\\sf x=\dfrac{21\div3}{90\div3}\\\\\sf x=\dfrac{7}{30}\end{array}}$

$\Large\boxed{\begin{array}{l}\tt e)~\sf x=6,0808...\cdot100\\\sf 100x=608,0808...\\-\underline{\begin{cases}\sf 100x=608,0808...\\\sf x=6,0808...\end{cases}}\\\sf 99x=602\\\sf x=\dfrac{602}{99}\end{array}}$

$\Large\boxed{\begin{array}{l}\tt f)~\sf x=2,18888...\cdot 10\\\sf 10x=21,888...\cdot 10\\\sf 100x=218,888...\\-\underline{\begin{cases}\sf 100x=218,888...\\\sf 10x=21,888...\end{cases}}\\\sf 90x=197\\\sf x=\dfrac{197}{90}\end{array}}$