  By making use of the multiplicate form of the extended Carlitz inverse series
relations, we establish two general `dual' theorems of Jackson's summation
formula for well--poised $_8\phi_7$-series. Their duplicate forms under the
partition pattern $n=\lfloor{\frac{n}2}\rfloor+\lfloor{\frac{n+1}2}\rfloor$ are
explored and yield numerous $q$-series identities whose limiting cases as
$q\to1$ result in classical $\pi$-related Ramanujan--like series of convergence
rate ``$\frac1{16}$" including one for $1/\pi^2$ discovered by Guillera (2003).
The triplicate dual formulae under the partition pattern
$n=\lfloor{\frac{n}3}\rfloor+\lfloor{\frac{n+1}3}\rfloor+\lfloor{\frac{n+2}3}\rfloor$
are examined via the ``reverse bisection method", which leads us to twenty new
$q$-series identities together with their classical counterparts of convergence
rate ``$\frac{-1}{27}$" when $q\to1$.
