Answer:
Step-by-step explanation:
This can be derived using the formula to calculate future value of an amount which is given as:
FV=PV(1+i)^{n}
Where
FV is the future value
PV is the present value
i is the compound interest rate
n is the time period
To calculate how much time it takes to triple an amount, let's substitute $1 in place of PV and $3 in place of FV in the above equation.
3=(1+i)^{n}
Taking natural log on both sides, we get
log_{e} 3= n log_{e}(1+i)
n={\log_{e}3}/{log_{e}(1+i)}....................(1)
Using some log magic, this formula can also be written as
n=log_{(1+i)}3................(2)
Both equations (1) and (2) work fine.
(b)
In order to derive a rule 72 style formula start from equation (1) above. In that formula, we can say that for small values of interest rate,
log_{e}(1+i) ~ i ( This is a property of log)
Substituting this is equation (1), we get
n=log_{e}3/{i}
n={1.098}/{i}
Since, we want to use i as an integer and not as a percentage (so, 3% will be used as 3 and not 0.03), we multiply the RHS by 100
n={109.8}/{i}
and then to make it cleaner, we round off 109.8 to 110.
n={110}/{i}
Now, many times, people round this 109.8 off to 114. This is to make the calculations a little bit easier as 114 is number that has a lot of factors. So, if the numerator is rounded off to 114, the formula becomes
n={114}/{i}
