Simplification of a complex fraction

$\dfrac{x}{x+\dfrac{y}{y+\dfrac{z}{2}}}$

Substitute: $x=3,y=4$ and $z=5$

$=\dfrac{3}{3+\dfrac{4}{4+\dfrac{5}{2}}}$

$=\dfrac{3}{3+\dfrac{4}{\dfrac{8+5}{2}}}$

$=\dfrac{3}{3+\dfrac{4}{\dfrac{13}{2}}}$

$=\dfrac{3}{3+\dfrac{4}{1}\times \dfrac{2}{13}}$

$=\dfrac{3}{3+\dfrac{8}{13}}$

$=\dfrac{3}{\dfrac{39+8}{13}}$

$=\dfrac{3}{\dfrac{47}{13}}$

$=\dfrac{3}{1} \times \dfrac{13}{47}$

$=\dfrac{39}{47}=\dfrac{a}{b}$

$100a+b=100(39)+47=\text{\boxed{3947}}$

$\begin{aligned} &= \dfrac{x}{x + \dfrac{y}{y + \dfrac{z}{2}}} \\ \\ &= \dfrac{x}{x + \dfrac{y}{\dfrac{\blue{2y + z}}{2}}} &\blue{\text{Take LCM of last fraction}} \\ \\ &= \dfrac{x}{x + \blue{\dfrac{2y}{2y + z}}} &\blue{\text{Flip the fraction}} \\ \\ &= \dfrac{x}{\blue{\dfrac{2xy + xz + 2y}{2y + z}}} &\blue{\text{Again Take LCM}} \\ \\ &= \dfrac{2xy + xz}{2xy + xz + 2y} &\blue{\text{Flip the fraction}} \\ \\ &= \dfrac{2(3)(4) + (3)(5)}{2(3)(4) + (3)(5) + 2(4)} &\blue{\text{Put in the values}} \\ \\ &= \dfrac{39}{47} \end{aligned}$

Answer is $100(39) + 47$ : $\boxed{3947}$