Let's break down each problem using the formulas provided.

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Problem 1: Finding the Interest Rate

The formula for compound interest, as given, is:

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

Where:

• $A = 68,211.38$ (final amount)
• $P = 55,490$ (initial investment)
• $n = 2$ (since the compounding is semiannual, so twice per year)
• $t = 36$ months = 3 years (time in years)

Now, substitute the values into the formula:

$r = 2 \left[ \left( \frac{68,211.38}{55,490} \right)^{\frac{1}{2 \times 3}} - 1 \right]$

First, calculate $\frac{A}{P}$:

$\frac{68,211.38}{55,490} \approx 1.2296$

Now, raise this value to the power of $\frac{1}{6}$ (since $n \times t = 6$):

$1.2296^{\frac{1}{6}} \approx 1.0355$

Now, subtract 1:

$1.0355 - 1 = 0.0355$

Finally, multiply by 2 (since $n = 2$):

$r = 2 \times 0.0355 \approx 0.0710$

So, the interest rate is approximately 7.10% per year.

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Problem 2: Finding Time for Investment to Grow

The formula for compound interest time is:

$t = \frac{\log(A / P)}{n \log(1 + \frac{r}{n})}$

Where:

• $A = 50,692.17$ (final amount)
• $P = 45,000$ (borrowed amount)
• $r = 6\% = 0.06$ (annual interest rate)
• $n = 4$ (since the interest is compounded quarterly)

Now, substitute the values into the formula:

$t = \frac{\log(50,692.17 / 45,000)}{4 \log(1 + \frac{0.06}{4})}$

First, calculate $\frac{A}{P}$:

$\frac{50,692.17}{45,000} \approx 1.1276$

Now, take the logarithm of this value:

$\log(1.1276) \approx 0.0512$

Next, calculate $\left(1 + \frac{0.06}{4}\right) = 1 + 0.015 = 1.015$, and then take the logarithm:

$\log(1.015) \approx 0.00647$

Now substitute these into the equation:

$t = \frac{0.0512}{4 \times 0.00647} = \frac{0.0512}{0.02588} \approx 1.98 \text{ years}$

So, it will take approximately 1.98 years or about 2 years for the amount to grow.

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Summary of Answers:

1. Interest Rate = 7.10% per year.
2. Time to Reach Final Amount = 2 years.

Would you like a detailed explanation of any step or additional assistance with similar problems?

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Follow-up Questions:

1. How do you calculate compound interest when interest is compounded monthly?
2. What is the difference between compound interest and simple interest?
3. Can you explain how to adjust the formula when compounding occurs annually?
4. How can you calculate the future value of an investment using continuous compounding?
5. What happens to the interest rate if the compounding frequency increases?

Tip: When solving compound interest problems, always ensure that the time and interest rate are in the correct units relative to the compounding frequency.