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To solve this problem, we’ll use the compound interest formula:

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

where:
- $ A $ is the future value of the investment,
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times interest is compounded per year, and
- $ t $ is the number of years the money is invested.

Given values:
- $ P = 42200 $ dollars,
- $ r = 5\% = 0.05 $,
- $ t = 5 $ years.

Let's calculate for each compounding frequency:

### (a) Annual Compounding
- $ n = 1 $

\[
A = 42200 \left(1 + \frac{0.05}{1}\right)^{1 \times 5}
\]

\[
A = 42200 \left(1 + 0.05\right)^5
\]

### (b) Semiannual Compounding
- $ n = 2 $

\[
A = 42200 \left(1 + \frac{0.05}{2}\right)^{2 \times 5}
\]

\[
A = 42200 \left(1 + 0.025\right)^{10}
\]

### (c) Monthly Compounding
- $ n = 12 $

\[
A = 42200 \left(1 + \frac{0.05}{12}\right)^{12 \times 5}
\]

\[
A = 42200 \left(1 + 0.0041667\right)^{60}
\]

### (d) Daily Compounding
- $ n = 365 $

\[
A = 42200 \left(1 + \frac{0.05}{365}\right)^{365 \times 5}
\]

\[
A = 42200 \left(1 + 0.000136986\right)^{1825}
\]

Now I will calculate each of these values to find the final amounts.The value of the investment at the end of 5 years, for each compounding method, is as follows (rounded to the nearest cent):

- (a) **Annual Compounding**: $53,859.08
- (b) **Semiannual Compounding**: $54,019.57
- (c) **Monthly Compounding**: $54,157.74
- (d) **Daily Compounding**: $54,184.94

Would you like a detailed breakdown of each step or have any further questions?

### Related Questions:
1. How would the results change if the interest rate were 6% instead of 5%?
2. What is the difference in final amounts between monthly and daily compounding for this investment?
3. How does increasing the time period to 10 years affect the investment values for each compounding method?
4. Can we use continuous compounding to calculate a similar investment return? If so, what would it be?
5. What would happen if the investment were compounded quarterly instead?

### Tip:
In compound interest calculations, the frequency of compounding significantly affects the final amount. More frequent compounding yields higher returns.

If 42200 dollars is invested at an interest rate of 5 percent per year, find the value of the investment at the end of 5 years for the following compounding methods, to the nearest cent.

Learn how to calculate the future value of a $42,200 investment at a 5% interest rate over 5 years. This solution explains the compound interest formula with various compounding methods, including annual, semiannual, monthly, and daily. Ideal for students in Grades 10-12.

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