To calculate the future value of an investment with compound interest, we use the formula:

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

where:

• $A$ is the future value of the investment,
• $P$ is the principal (initial investment),
• $r$ is the annual interest rate (as a decimal),
• $n$ is the number of compounding periods per year,
• $t$ is the time in years.

Given:

• $P = 33800$,
• $r = 0.05$ (5% annual interest rate),
• $t = 5$ years.

Calculations:

(a) Annual Compounding ($n = 1$):

$A = 33800 \left(1 + \frac{0.05}{1}\right)^{1 \times 5}$ $A = 33800 \times (1.05)^5$

(b) Semiannual Compounding ($n = 2$):

$A = 33800 \left(1 + \frac{0.05}{2}\right)^{2 \times 5}$ $A = 33800 \times \left(1.025\right)^{10}$

(c) Monthly Compounding ($n = 12$):

$A = 33800 \left(1 + \frac{0.05}{12}\right)^{12 \times 5}$ $A = 33800 \times \left(1.0041667\right)^{60}$

(d) Daily Compounding ($n = 365$):

$A = 33800 \left(1 + \frac{0.05}{365}\right)^{365 \times 5}$ $A = 33800 \times \left(1.000136986\right)^{1825}$

Let’s calculate these values to the nearest cent.The future values of the investment, to the nearest cent, are:

• (a) Annual Compounding: $43,138.32
• (b) Semiannual Compounding: $43,266.86
• (c) Monthly Compounding: $43,377.52
• (d) Daily Compounding: $43,399.32

Would you like further details on any of these calculations or explanations of the formulas?

Here are some related questions to explore:

1. How would the investment change with a 10% interest rate under each compounding method?
2. What would the future value be if compounded continuously?
3. How does the effect of compounding frequency vary with different time periods, like 10 years?
4. What if the principal amount was doubled? How would each result change?
5. What if the compounding frequency is changed to quarterly instead of semiannual?

Tip: The more frequently interest is compounded, the greater the final amount due to interest on previously earned interest.