$($a$)$ Olivia would like to go on a world tour in $3$ years. She saves RM$5,500$ every $6$ months$$ starting at the end of the first period to provide for that purpose. If money is worth $7$ percent per annum, calculate the saving amount that Olivia will have by the time she is$$ ready for her trip. $(3$ marks$)$

$($b$)$ Benjamin would like to start his own coffee shop. While preparing his business plan, he$$ estimated that his coffee equipments and furniture would cost around RM$60,000,$ including setup expenses. If he starts saving today a sum of RM$6,000$ every quarter at a$10$ percent compounding rate, determine if Benjamin would be able to start the coffee$$ shop after $3$ years.$(4$ marks$)$

$($c$)$ Zara has just won a songwriting competitions. Suppose that Zara has an opportunity cost$$ of capital of $7$ percent per annum paid quarterly, and she must choose between the$$ following two alternatives of prizes. Decide which is the better option for Zara.$$

i$.$ Instalment of RM$25,000$ paid every end of $6$ months over a $5-$year period.$$

ii$.$ Lump sum of RM$300,000$ paid $5$ years later compounded annually. $(6$ marks$)$