, 2026

Written as a mixed fraction, $\dfrac{90}{11}=8\dfrac{2}{11}$, and so
$a+\dfrac{1}{b+\frac1c}=8+\dfrac{2}{11}$.

If $a=8$, then $\dfrac{1}{b+\frac1c}=\dfrac{2}{11}$.

Since $\dfrac{2}{11}$ has numerator
$2$, we rewrite the previous equation
as $\dfrac{2\times1}{2\times\left(b+\frac1c\right)}=\dfrac{2}{11}$.

Both numerators are now equal to $2$,
and so $2\times\left(b+\frac1c\right)=11$ or $b+\dfrac1c=\dfrac{11}{2}$.

Written as a mixed fraction, $\dfrac{11}{2}=5\dfrac{1}{2}$, and so $b+\dfrac1c=5+\dfrac{1}{2}$.

If $b=5$, then $\dfrac{1}{c}=\dfrac{1}{2}$, and so $c=2$.

Therefore, $a+\dfrac{1}{b+\frac1c}=8+\dfrac{1}{5+\frac12}$
and $a+b+c=8+5+2=15$.