The following chart shows the prices for five companies on Jan $1^{st}$ of $2022,2023$ and $2024$:

a$)$ Suppose that on Jan $1,2022,$ the K$9$ stock index consists of four companies:

Labrador Limited, Malamute Manufacturing, Collie Corporation and Akita

Automotive. Further, suppose the K$9$ index is price weighted, and that the Jan $1,$

$2022$ divisor is $0\mathrm{.}565.$ Calculate the K$9$ index on Jan $1,2022.$

b$)$ Now suppose that, on Jan $1,2023,$ Akita Automobile was removed from the index,

and Dalmatian Designs was added to the index. Calculate the new divisor.

c$)$ Now $($after the replacement in part b$),$ calculate the value of the K$9$ index in $2024.$

d$)$ What was the rate of return on the K$9$ index for $2022,$ before the replacement $($from

Jan $1,2022$ to Jan $1,2023)?$

e$)$ What was the rate of return on the $K9$ index for $2023,$ after the replacement $($from

Jan $1,2023$ to Jan $1,2024)?$