Question

dajon invested $41,300 in a mutual fund at 4.1% compounded semi-annually. after 3 years, the interest rate was changed to 7.2% compounded monthly.
a) how much was the value of the fund 7 years after the rate change? $
b) how much was the total compound interest earned during the whole term? $

Explanation:

Step1: Calculate value after first 3 years (semi - annual compounding)

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of times interest is compounded per year, and $t$ is the number of years.
For the first part, $P=\$41300$, $r = 0.041$, $n = 2$ (semi - annually), and $t = 3$ years.
Substitute the values into the formula:
$A_1=41300(1 +\frac{0.041}{2})^{2\times3}=41300(1 + 0.0205)^{6}$
First, calculate $(1 + 0.0205)^{6}\approx1.0205^{6}\approx1.1297$
Then, $A_1 = 41300\times1.1297\approx\$46656.61$

Step2: Calculate value 7 years after rate change (monthly compounding)

Now, the principal for the second part is $A_1\approx\$46656.61$, $r = 0.072$, $n = 12$ (monthly), and $t = 7$ years.
Using the compound - interest formula again: $A_2=A_1(1+\frac{r}{n})^{nt}$
Substitute the values: $A_2 = 46656.61(1+\frac{0.072}{12})^{12\times7}=46656.61(1 + 0.006)^{84}$
Calculate $(1 + 0.006)^{84}\approx1.006^{84}\approx1.6312$
Then, $A_2=46656.61\times1.6312\approx\$76106.66$ (This is the answer for part a)

Step3: Calculate total compound interest (part b)

The total amount after the whole term is $A_2\approx\$76106.66$, and the initial principal $P = \$41300$
The compound interest $I=A_2 - P$
$I=76106.66 - 41300=\$34806.66$

Answer:

a) $\$76106.66$ (approximate)
b) $\$34806.66$ (approximate)