To calculate the nominal rate of interest, we will use the compound interest formula:

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

Where:

• $A$ = Maturity Amount
• $P$ = Principal
• $r$ = Nominal interest rate (annual rate we need to find)
• $n$ = Number of times the interest is compounded per year
• $t$ = Time in years

We need to isolate $r$ from the formula, so we will rearrange the equation:

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

Let's calculate the nominal rate for each scenario:

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1. First Scenario:

Principal = $44,000
Maturity Amount = $102,827.19
Compounding Frequency = Semi-annually ($n = 2$)
Term = 10 years, 6 months = 10.5 years

$r = 2 \left( \left(\frac{102,827.19}{44,000}\right)^{\frac{1}{2 \times 10.5}} - 1 \right)$

2. Second Scenario:

Principal = $63,000
Maturity Amount = $96,375.11
Compounding Frequency = Quarterly ($n = 4$)
Term = 9 years, 6 months = 9.5 years

$r = 4 \left( \left(\frac{96,375.11}{63,000}\right)^{\frac{1}{4 \times 9.5}} - 1 \right)$

3. Third Scenario:

Principal = $80,000
Maturity Amount = $165,704.77
Compounding Frequency = Monthly ($n = 12$)
Term = 12 years, 2 months = 12.17 years

$r = 12 \left( \left(\frac{165,704.77}{80,000}\right)^{\frac{1}{12 \times 12.17}} - 1 \right)$

Let me now calculate these values. Let's calculate the nominal interest rate for each scenario

Scenario 1

P1 = 44000 # Principal A1 = 102827.19 # Maturity Amount n1 = 2 # Semi-annually t1 = 10.5 # Years

r1 = n1 * ((A1 / P1) ** (1 / (n1 * t1)) - 1)

Scenario 2

P2 = 63000 # Principal A2 = 96375.11 # Maturity Amount n2 = 4 # Quarterly t2 = 9.5 # Years

r2 = n2 * ((A2 / P2) ** (1 / (n2 * t2)) - 1)

Scenario 3

P3 = 80000 # Principal A3 = 165704.77 # Maturity Amount n3 = 12 # Monthly t3 = 12.17 # Years

r3 = n3 * ((A3 / P3) ** (1 / (n3 * t3)) - 1)

Convert rates to percentage

r1_percent = r1 * 100 r2_percent = r2 * 100 r3_percent = r3 * 100

(r1_percent, r2_percent, r3_percent)Here are the nominal interest rates for each scenario:

1. First Scenario: 8.25% (semi-annually, 10 years, 6 months)
2. Second Scenario: 4.50% (quarterly, 9 years, 6 months)
3. Third Scenario: 6.00% (monthly, 12 years, 2 months)

Would you like further details or have any questions on these calculations? Here are some additional questions to explore further:

1. How does the compounding frequency affect the nominal interest rate?
2. What happens if the time period is shortened or lengthened?
3. Can we calculate the effective annual rate (EAR) from these nominal rates?
4. How do inflation and tax rates impact the real return on these investments?
5. What other methods can be used to compare investment returns over different terms?

Tip: When dealing with compound interest, a higher compounding frequency usually leads to a slightly higher effective return, even if the nominal rate is the same!