A sum doubles itself in one year at a certain rate of interest, compounded annually. In how many years will a sum become six times itself under the same investment scheme?
I got confused in this simple problem because of two approaches I can think of -
First - Clearly rate of interest = 100%, So $$6P = P(1+1)^n$$ solving We get n = log6/log2, n = 2.58 Years
Second - At the end of 2 years Amount = 4P, now to convert 4P to 6P we need say x months, then $$2P=(4P*1*x)/12$$; x = 6 months, hence answer = 2.5 years
Why the difference?
The amount of money after $x$ year ($x$ a real number) is not multiplied by $(1+r)^x$. Because the compound is not continuous. The amount invested after $k\in \Bbb N$ years is multiplied by $(1+r)^x$ and this fixed amount is invested until the end of the year, and only then the interests are reinvested.
This leads to the multiplier (with $E(x)$ the integer part of $x$): $$ (1+r)^{E(x)}\times (1 + r(x-E(x))) $$ (the last factor is for the year not yet finished).
In your case this gives $$ 2^{E(x)}\times (1 + {x-E(x)}) = 6 $$
which gives $E(x) = 2$ and $$ 1 + {x-2} = \frac 64 = 1.5\implies x = 2.5 $$