In __________ years a sum will double at $5\%$ per annum compound interest.
Options given are:
a. 15 years 3 months
b. 14 years 2 months
c. 14 years 3 months
d. 15 years 2 months
The way to solve is as below: \begin{align*} A & = P[1+(r/100)]^n\\ 2P & = P [1+(5/100)]^n\\ 2 & = [1+(5/100)]^n\\ \log 2 & = n \cdot \log(1.05)\\ 0.3010 & = n \cdot 0.02118 \end{align*}
Therefore $n = 14.2069$ years $= 14$ years and $(0.2069 \cdot 12)$ months $= 14$ years and $2.48$ months.
Now the question is whether $2.48$ months should be rounded to $2$ months or $3$ months?
This is a academic question where I need to choose from the 4 options as given above. The book says answer is 14 years 2 months, but conceptually, before 2.48 months, the money does not double, so answer needs to be 14 years 3 months. Just asked this on forum to get to know if I am missing something.
This doesn't make much sense, if the interest is evaluated only once a year. Clearly, after 14 years you would only have $1.05^{14}=1.9799\ldots$ times the initial amount and only after 15 years you will have twice the initial amount (and a little bit more).
I think there is an error in the book, unless it is understood that the interest is evaluated continuously, but then anyway none of the options would be correct.
The $\ln 2/\ln (1.05)\simeq 14.206\ldots$ is just the "average" in the following sense: in the long run, the initial amount will be at least $2^n$ times the initial amount after $\lceil n\times (\ln2/\ln (1.05))\rceil$ years, where $\lceil x \rceil$ is the smallest integer larger than $x$.