To solve for the unknown interest rate in these scenarios, we can use the future value formula for compound interest:

$FV = PV(1 + r)^n$

Where:

• $FV$ is the future value,
• $PV$ is the present value,
• $r$ is the interest rate (expressed as a decimal),
• $n$ is the number of years.

We need to solve for $r$, the unknown interest rate. Rearranging the formula to isolate $r$, we get:

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

Let's solve each case individually.

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Case 1:

• Present value (PV) = 180
• Years (n) = 5
• Future value (FV) = 228

Using the formula for $r$:

$r = \left(\frac{228}{180}\right)^{\frac{1}{5}} - 1$

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Case 2:

• Present value (PV) = 300
• Years (n) = 19
• Future value (FV) = 859

$r = \left(\frac{859}{300}\right)^{\frac{1}{19}} - 1$

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Case 3:

• Present value (PV) = 33,000
• Years (n) = 20
• Future value (FV) = 152,407

$r = \left(\frac{152,407}{33,000}\right)^{\frac{1}{20}} - 1$

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Case 4:

• Present value (PV) = 32,261
• Years (n) = 30
• Future value (FV) = 644,965

$r = \left(\frac{644,965}{32,261}\right)^{\frac{1}{30}} - 1$

I will now compute the interest rates for each case.Here are the interest rates for each case:

1. Case 1: Present value = 180, Future value = 228, Years = 5
   Interest rate $r \approx 4.84\%$

2. Case 2: Present value = 300, Future value = 859, Years = 19
   Interest rate $r \approx 5.69\%$

3. Case 3: Present value = 33,000, Future value = 152,407, Years = 20
   Interest rate $r \approx 7.95\%$

4. Case 4: Present value = 32,261, Future value = 644,965, Years = 30
   Interest rate $r \approx 10.50\%$

Would you like more details on any of these calculations or have any questions?

Here are 5 related questions to further explore:

1. How does increasing the number of years affect the interest rate, assuming constant future and present values?
2. What happens to the interest rate if the present value is doubled but the future value remains the same?
3. Can you derive the future value if you know the interest rate, present value, and years?
4. How do simple interest and compound interest compare in these scenarios?
5. What factors can influence real-world interest rates in similar calculations?

Tip: When calculating compound interest over long periods, small changes in the rate can have a large impact on the final amount.