Suppose a bond is purchased with a settlement date of October $15$ and the next coupon payment

is on December $1.$ The par amount purchased on the bond is $$100,000,$ and its annual coupon rate

is $4\%$ paid semiannually.

$(1)$ What is the accrued interest using the $\frac{30}{360}$ day count convention?

$(2)$ What is the accrued interest using the actual$/$actual day count convention?