Question

Convert 0.345634563456…to a rational number in the form of ab, where b≠0.

Answer 1

A standard trick for finding the rational form of a repeated decimal:

$x=0.34563456\ldots=0.\overline{3456}$

Multiply by an appropriate power of 10 to move one "cycle" of the repeated pattern to the left side of the decimal point:

$\implies10^4x=3456.34563456\ldots=3456.\overline{3456}$

Now subtract $x$ from this number and we lose the fractional part:

$\implies10^4x-x=9999x=3456$

And from here you can solve for $x$.

$\implies x=\dfrac{3456}{9999}=\dfrac{384}{1111}$