Question

Convert 11.42424242 … to a rational expression in the form of a over b, where b ≠ 0.

Answer 1

We could consider this sum to be \[11+\frac{42}{100}+\frac{42}{10,000}+\frac{42}{1,000,000}+\ldots\\
=11+\frac{42}{100}\left( 1+\frac{1}{100}+\frac{1}{10,000}+\ldots\right)\\
=11+\frac{42}{100}\left( \left[\frac{1}{100}\right]^0+\left[\frac{1}{100}\right]^1+\left[\frac{1}{100}\right]^2+\ldots\right)\\
=11+\frac{42}{100}\sum_{i=0}^{\infty}\frac{1}{100^i}\\
=11+\frac{42}{100}\frac{1}{1-\frac{1}{100}}\\=11+\frac{42}{100}\frac{100}{99}\\=\frac{377}{33} \]

Answer 2

u could remember it as 42 which is of two digits is repeating here, hence there will be 2 nines ie (99) in denominator of the fraction(11 +(42/99))