A retailer receives an invoice dated October $7$ for $$1200$ with terms $\frac{5}{10},\frac{2}{20},\frac{n}{30}.$ The retailer wishes to reduce his balance due by $$900$ within the first cash discount period.

a$)$ Determine the retailer's partial payment on or before October $17.$

b$)$ Determine the balance due on October $27$ if this retailer makes another partial payment on this date of $$200\mathrm{.}00.$