$1.(15$ Points$)$

$($a$)$ Determine the annual interest rates for the past and upcoming years given the following conditions: If

an amount of $$260$ at Time $0$ will appreciate to $$300$ after one year and was $$230$ a year ago.

$($b$)$ In a scenario where $$200$ invested a year ago is expected to yield $$300$ a year from now, calculate the

corresponding annual interest rate.