Let's solve each part of the problem using the relevant formulas for interest calculations.

(a) 7% Simple Interest:

The formula for simple interest is:

$A = P(1 + rt)$

Where:

• $A$ is the amount after interest,
• $P$ is the principal (initial investment),
• $r$ is the annual interest rate (as a decimal),
• $t$ is the time in years.

Given:

• $P = 19,000$,
• $r = 0.07$,
• $t = 25$.

Now substitute the values:

$A = 19,000(1 + (0.07 \times 25)) = 19,000(1 + 1.75) = 19,000 \times 2.75 = 52,250.$

So, the value of the investment after 25 years with simple interest is $52,250.00.

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(b) 7% Compounded Monthly:

The formula for compound interest is:

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

Where:

• $A$ is the amount after interest,
• $P$ is the principal (initial investment),
• $r$ is the annual interest rate (as a decimal),
• $n$ is the number of times the interest is compounded per year,
• $t$ is the time in years.

Given:

• $P = 19,000$,
• $r = 0.07$,
• $n = 12$ (since interest is compounded monthly),
• $t = 25$.

Substitute the values:

$A = 19,000 \left( 1 + \frac{0.07}{12} \right)^{12 \times 25} = 19,000 \left( 1 + 0.0058333 \right)^{300}$

Now let's calculate this value.

$A = 19,000 \times (1.0058333)^{300}$

Let me calculate this final value.The value of the investment after 25 years with 7% interest compounded monthly is approximately $108,782.95.

Summary:

• (a) With 7% simple interest: $52,250.00.
• (b) With 7% compounded monthly: $108,782.95.

Would you like further details or explanations on any part of this?

Here are 5 related questions you might find useful:

1. How does compounding frequency affect the final amount?
2. What is the difference between annual and monthly compounding in terms of interest growth?
3. How can you calculate the time needed for an investment to double with compound interest?
4. How would the final amount change if the interest rate was 5% instead of 7%?
5. How can inflation affect the real value of your investment after 25 years?

Tip: When interest is compounded more frequently, your money grows faster due to interest being added to the principal more often.