Question

Use the "rule of 72" to estimate the doubling time (in years) for the interest rate, and then calculate it exactly. (Round your answers to two decimal places.) 9% compounded annually.
"rule of 72" yr

exact answer yr

Answer 1

Answer:

According to the rule of 72, the doubling time for this interest rate is 8 years.

The exact doubling time of this amount is 8.04 years.

Step-by-step explanation:

Sometimes, the compound interest formula is quite complex to be solved, so the result can be estimated by the rule of 72.

By the rule of 72, we have that the doubling time D is given by:

$D = \frac{72}{Interest Rate}$

The interest rate is in %.

In our exercise, the interest rate is 9%. So, by the rule of 72:

$D = \frac{72}{9} = 8$.

According to the rule of 72, the doubling time for this interest rate is 8 years.

Exact answer:

The exact answer is going to be found using the compound interest formula.

$A = P(1 + \frac{r}{n})^{nt}$

In which 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.

is double the initial amount, double the principal.

$A = 2P$

$r = 0.09$

The interest is compounded anually, so $n = 1$

$A = P(1 + \frac{r}{n})^{nt}$

$2P = P(1 + \frac{0.09}{1})^{t}$

$2 = (1.09)^{t}$

Now, we apply the following log propriety:

$\log_{a} a^{n} = n$

So:

$\log_{1.09}(1.09)^{t} = \log_{1.09} 2$

$t = 8.04$

The exact doubling time of this amount is 8.04 years.