In a particular reactor, polymerization appears to occur by two simultaneous mechanisms that give rise to two separate molecular weight distributions.

A high$-$molecular$-$weight portion $(62\mathrm{.}5$ wt $\%$ of the total batch$)$ can be resolved by solvent fractionation. It has $M_w=550,000$ and $M_n=210,000.$

The entire batch has $M_w=360,000,$ but $M_n$ cannot be measured unequivocally.

Assuming that the lower molecular$-$weight portion $(37\mathrm{.}5wt\%$ of the total batch$)$ has a most probable distribution:

a$)$ find the $M_n$ of the lower molecular$-$weight portion and

b$)$ find the $M_n$ of the entire batch.