Verified answer

[tex]\Large\boxed{\begin{array}{l}\rm a)~\sf x=0,1515\dotsc\cdot100\\\sf 100x=15,1515\dotsc\\-\underline{\begin{cases}\sf 100x=15,1515\dotsc\\\sf x=0,151515\dotsc\end{cases}}\\\sf 99x=15\\\sf x=\dfrac{15\div3}{99\div3}\\\\\sf x=\dfrac{5}{33}\end{array}}[/tex]

[tex]\Large\boxed{\begin{array}{l}\rm b)~\sf x=2,141414\dotsc\cdot100\\\sf 100x=214,1414\dotsc\\-\underline{\begin{cases}\sf 100x=214,1414\\\sf x=2,141414\dotsc\end{cases}}\\\sf 99x=212\\\sf x=\dfrac{212}{99}\\\\\sf -2,141414\dotsc=-\dfrac{212}{99}\end{array}}[/tex]

[tex]\Large\boxed{\begin{array}{l}\rm c)~\sf x=0,23777\dotsc\cdot100\\\sf 100x=23,777\dotsc\cdot10\\\sf 1000x=237,777\dotsc\\-\underline{\begin{cases}\sf 1000x=237,777\dotsc\\\sf 100x=23,777\dotsc\end{cases}}\\\sf 900x=214\\\sf x=\dfrac{214\div2}{900\div2}\\\\\sf x=\dfrac{107}{450}\end{array}}[/tex]

[tex]\Large\boxed{\begin{array}{l}\rm d)~\sf x=0,87656565\dotsc\cdot100\\\sf 100x=87,656565\dotsc\cdot100\\\sf 10000x=8765,6565\dotsc\\-\underline{\begin{cases}\sf 10000x=8765,6565\dotsc\\\sf 100x=87,656565\dotsc\end{cases}}\\\sf 9900x=8678\\\sf x=\dfrac{8678\div2}{9900\div2}\\\\\sf x=\dfrac{4339}{4950}\end{array}}[/tex]