investment b: a bond earns 9% per year, on average. how long will it take for an investment at this rate to…

Explanation:

Step1: Use the compound - interest formula for doubling

The rule of 72 is a quick way to estimate the number of years it takes for an investment to double at a given interest rate $r$. The formula is $t=\frac{72}{r}$, where $t$ is the time in years and $r$ is the interest rate percentage. Here $r = 9$, so $t=\frac{72}{9}=8$ years for the investment to double.

Step2: Calculate the number of doublings in a given time

For 5 years: Since it takes 8 years to double, the number of doublings $n_1=\lfloor\frac{5}{8}\rfloor = 0$ (using the floor function because we are looking for whole - number of doublings). For 10 years: $n_2=\lfloor\frac{10}{8}\rfloor = 1$. For 15 years: $n_3=\lfloor\frac{15}{8}\rfloor = 1$. For 20 years: $n_4=\lfloor\frac{20}{8}\rfloor = 2$. For 25 years: $n_5=\lfloor\frac{25}{8}\rfloor = 3$.

Step3: Calculate the final value of $1000$ after 25 years

If the investment doubles $n$ times, the final value $V$ of an initial investment $P$ is $V = P\times2^n$. Here $P = 1000$ and $n = 3$ (from 25 years), so $V=1000\times2^3=8000$.

Answer:

5 years: 0 times; 10 years: 1 time; 15 years: 1 time; 20 years: 2 times; 25 years: 3 times. The value of $1000$ after 25 years is $8000$.