Respuesta :
Answer:
Using the rule of 72, the doubling time is 9.35 years.
The exact answer is that the doubling time is 8.89 years.
Step-by-step explanation:
By the rule of 72, we have that the doubling time D is given by:
[tex]D = \frac{72}{Interest Rate}[/tex]
The interest rate is in %.
In our exercise, the interest rate is 7.7%. So, by the rule of 72:
[tex]D = \frac{72}{7.7} = 9.35[/tex].
Exact answer:
The exact answer is going to be found using the compound interest formula(since the rule of 72 is a simplification of this formula).
The compound interest formula is given by:
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.
So, for this exercise, we have:
We want to find the doubling time, that is, the time in which the amount is double the initial amount, double the principal.
[tex]A = 2P[/tex]
[tex]r = 0.077[/tex]
There are 52 weeks in a year, so [tex]n = 52[/tex]
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
[tex]2P = P(1 + \frac{0.077}{52})^{52t}[/tex]
[tex]2 = (1.0015)^{52t}[/tex]
Now, we apply the following log propriety:
[tex]\log_{a} a^{n} = n[/tex]
So:
[tex]\log_{1.0015}(1.0015)^{52t} = \log_{1.0015} 2[/tex]
[tex]52t = 462.44[/tex]
[tex]t = \frac{462.44}{52}[/tex]
[tex]t = 8.89[/tex]
The exact answer is that the doubling time is 8.89 years.
