To solve for the unknown interest rates, we use the Future Value (FV) formula:

$FV = PV \cdot (1 + r)^t$

Where:

• $FV$ = Future Value
• $PV$ = Present Value
• $r$ = Interest rate (expressed as a decimal)
• $t$ = Time in years

Rearranging the formula to solve for $r$:

$r = \left( \frac{FV}{PV} \right)^{\frac{1}{t}} - 1$

We'll solve each case step by step.

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1. Given:

$PV = 48,000, \; t = 11, \; FV = 185,382$

Substitute into the formula:

$r = \left( \frac{185,382}{48,000} \right)^{\frac{1}{11}} - 1$

$r = \left( 3.862125 \right)^{\frac{1}{11}} - 1$

$r = 1.132 - 1 = 0.132 \; \text{or} \; 13.20\%$

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2. Given:

$PV = 40,353, \; t = 25, \; FV = 531,618$

Substitute into the formula:

$r = \left( \frac{531,618}{40,353} \right)^{\frac{1}{25}} - 1$

$r = \left( 13.178 \right)^{\frac{1}{25}} - 1$

$r = 1.127 - 1 = 0.127 \; \text{or} \; 12.70\%$

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Final Answers:

1. Interest rate: 13.20%
2. Interest rate: 12.70%

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Let me know if you'd like detailed steps or further clarifications! Here are 5 questions to expand your knowledge:

1. How does the compounding frequency (e.g., annual, semi-annual) affect the interest rate calculation?
2. Can the Future Value formula be rearranged to solve for time ($t$) instead of $r$?
3. How would the process differ if interest was compounded continuously?
4. What does a negative interest rate mean in a financial context?
5. Why is it essential to use the same time units for both $r$ and $t$ (e.g., annual interest rate with years)?

Tip: Always keep track of the units and compounding frequency when working with interest rate formulas!