Question

Express $0.\overline{1}+0.\overline{01}+0.\overline{0001}$ as a common fraction.

Answer 1

Answer:

$\dfrac{1213}{9999}$

Step-by-step explanation:

We are required to express $0.\overline{1}+0.\overline{01}+0.\overline{0001}$ as a common fraction.

The bar on top of the decimal part indicates the decimal number is a repeating decimal.

Therefore:

$0.\overline{1}=\dfrac{1}{10-1}= \dfrac{1}{9}\\\\0.\overline{01}=\dfrac{1}{100-1}= \dfrac{1}{99}\\\\0.\overline{0001}=\dfrac{1}{10000-1}= \dfrac{1}{9999}\\\\\\ Therefore :\\0.\overline{1}+0.\overline{01}+0.\overline{0001} \\=\dfrac{1}{9}+\dfrac{1}{99}+\dfrac{1}{9999}\\\\=\dfrac{1213}{9999}$