You are given the following information about the number of outstanding shares and prices for two stocks on $2$ days as shown below:

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Stock $1$: Number of shares outstanding $=3,000$ shares

Price per share on $1/1/2024=$ $$30$

Price per share on $2/1/2024=$ $$20$

Stock $2$: Number of shares outstanding $=6,000$ shares

Price per share on $1/1/2024=$ $$60$

Price per share on $2/1/2024=$ $$90$

$($a$).$ Assume that a stock price$-$weighted index consisted of the above two stocks with their prices. What are the index values on $1/1/2024$ and $2/1/2024$ and what is the percentage change in the index value?

$($b$).$ Now assume that the stocks described above are associated with a $2-$stock value$-$weighted index, and not a price weighted index as in part $($a$).$ Answer the following questions $($i$)-($v$)$ given this information:

i$.$ What is the holding period return on Stock $1?$

ii$.$ What is the holding period return on Stock $2,$ assuming that it paid a dividend of $$5$ per share during January $2024?$

iii. What is the value weight of each stock in this value weighted index?

iv$.$ What is the holding period return on this $2-$stock value weighted index?