Let's solve each problem step by step using the formulas provided.

Problem 1: Suzanne's Personal Loan

Formulas:

• Interest Accrued: $I = P \times \left(1 + \frac{R}{N}\right)^{NT} - P$
• Accrued Amount: $A = P \times \left(1 + \frac{R}{N}\right)^{NT}$

Where:

• $P$ = Principal amount (initial loan amount)
• $R$ = Annual interest rate (as a decimal)
• $N$ = Number of compounding periods per year
• $T$ = Time the money is invested or borrowed for (in years)

Opportunity Loans:

• $P = 1600$
• $R = 3.45\% = 0.0345$
• $N = 12$ (monthly compounding)
• $T = 1$ year

Using the interest formula: $I = 1600 \times \left(1 + \frac{0.0345}{12}\right)^{12 \times 1} - 1600$

Let's calculate this:

$I = 1600 \times \left(1 + 0.002875\right)^{12} - 1600$ $I = 1600 \times \left(1.035106\right) - 1600$ $I = 1600 \times 1.035106 - 1600$ $I = 1656.17 - 1600$ $I = 56.17$

Opportunity Loans Interest: $56.17

General Loans:

• $P = 1600$
• $R = 4.2\% = 0.042$
• $N = 4$ (quarterly compounding)
• $T = 1$ year

Using the interest formula: $I = 1600 \times \left(1 + \frac{0.042}{4}\right)^{4 \times 1} - 1600$

Let's calculate this:

$I = 1600 \times \left(1 + 0.0105\right)^{4} - 1600$ $I = 1600 \times \left(1.042832\right) - 1600$ $I = 1668.53 - 1600$ $I = 68.53$

General Loans Interest: $68.53

Comparison:

• Least Interest: Suzanne will earn the least interest with Opportunity Loans ($56.17).

Problem 2: Gavin's Savings Account

Community Bank:

• $P = 3475$
• $R = 1.90\% = 0.019$
• $N = 365$ (daily compounding)
• $T = 5$ years

Using the accrued amount formula: $A = 3475 \times \left(1 + \frac{0.019}{365}\right)^{365 \times 5}$

Let's calculate this:

$A = 3475 \times \left(1 + 0.00005205\right)^{1825}$ $A = 3475 \times \left(1.098388\right)$ $A = 3818.15$

Community Bank Accrued Value: $3818.15

Capital Bank:

• $P = 3475$
• $R = 2.05\% = 0.0205$
• $N = 12$ (monthly compounding)
• $T = 5$ years

Using the accrued amount formula: $A = 3475 \times \left(1 + \frac{0.0205}{12}\right)^{12 \times 5}$

Let's calculate this:

$A = 3475 \times \left(1 + 0.00170833\right)^{60}$ $A = 3475 \times \left(1.106507\right)$ $A = 3844.63$

Capital Bank Accrued Value: $3844.63

Comparison:

• Higher Accrued Value: The Capital Bank account will have the higher accrued value after 5 years ($3844.63).

Summary:

1. Opportunity Loans Interest Accrued: $56.17
2. General Loans Interest Accrued: $68.53
3. Least Interest Loan: Opportunity Loans
4. Community Bank Accrued Value: $3818.15
5. Capital Bank Accrued Value: $3844.63
6. Higher Accrued Value: Capital Bank

Would you like more details or have any questions?

Related Questions:

1. How does increasing the compounding frequency affect the interest accrued on a loan?
2. What is the impact of a longer loan term on the total interest accrued?
3. How do different interest rates affect the decision to choose between two savings accounts?
4. How would semi-annual compounding compare to monthly compounding for these loans?
5. What happens to the accrued interest if Suzanne decides to pay off the loan early?

Tip: When comparing financial options, always consider both the interest rate and the compounding frequency, as these can significantly affect the final amount of interest or accrued value.