Question: A sum of money becomes 16 times of itself in 2 years if compounded half yearly. How much time it will take to becomes 27 times if compounded yearly.
(A) 2 years (B) 3 years (C) 4 years (D) 6 years
My approach:-
$$P(1+R/200)^4 =16P$$ which gives us the Rate of interest as 200% per annum compounded semi annually. From here I found the Effective annual rate(EAR) as $$P(1+EAR/100)= P(1+ 200/200)^2 =300 percent $$ , So now we have $$P(1+300/100)^t=27P$$, giving us $4^t=27$ => $t=log(27/4)= 2.37 years$
What the book's solution does is that it takes 200% as yearly rate and calculates as follows , $$P(1+200/100)^t=27P$$, and gets the answer as $t=3 years$
Where am I going wrong ?
The wording of the problem, at least as it's reported here, is a bit awkward, but it sounds like they want you to take that $200\%$ interest rate that's compounded half yearly (to produce a doubling every six months) and only compound it annually, which is an annual tripling. Understood that way, it does take three years for the sum of money to become $27$ times itself.