$.$ The Omega Index is a price weighted stock index based on the $5$ largest skateboard

manufacturers in the nation. The stock prices for the five stocks are $$10,$$$20,$$$80,$$$50$

and $$40.$ The price of the last stock was just split $2$ for $1$ and the stock price was halved

from $$40$ to $$20.$

$($a$).$ What is the initial value of the index?

$($b$).$ What is the new divisor for a price weighted index?

$($c$).$ what would have been the daily return on this index had the price of the last stock

changed from $$40$ to $$20$ the next day due to a disappointing earnings announcement by

the firm? Assume that the prices of the other four stocks remain the same $,$

and $$50)$