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To pay for a home improvement project that totals $20,000, a homeowner is choosing between two different credit card loans with an interest rate of 9%. The first credit card compounds interest quarterly, while the second credit card compounds monthly. The homeowner plans to pay off the loan in 10 years.

Part A: Determine the total value of the loan with the quarterly compounded interest. Show all work and round your answer to the nearest hundredth. (4 points)

Part B: Determine the total value of the loan with the monthly compounded interest. Show all work and round your answer to the nearest hundredth. (4 points)

Part C: What is the difference between the total interest accrued on each loan? Explain your answer in complete sentences. (2 points)

1 Answer

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Part A: To determine the total value of the loan with quarterly compounded interest, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where:

A is the total value of the loan,

P is the principal amount (the initial loan amount),

r is the annual interest rate (as a decimal),

n is the number of times interest is compounded per year,

t is the number of years.

In this case, the principal amount (P) is $20,000, the annual interest rate (r) is 9% or 0.09, the number of times interest is compounded per year (n) is 4 (quarterly compounded), and the number of years (t) is 10.

Using the formula, we can calculate the total value of the loan with quarterly compounded interest:

A = 20000(1 + 0.09/4)^(4*10)

A ≈ $53,262.47 (rounded to the nearest hundredth)

Therefore, the total value of the loan with quarterly compounded interest is approximately $53,262.47.

Part B: To determine the total value of the loan with monthly compounded interest, we use the same formula but with a different value for n.

In this case, the number of times interest is compounded per year (n) is 12 (monthly compounded).

Using the formula, we can calculate the total value of the loan with monthly compounded interest:

A = 20000(1 + 0.09/12)^(12*10)

A ≈ $53,758.68 (rounded to the nearest hundredth)

Therefore, the total value of the loan with monthly compounded interest is approximately $53,758.68.

Part C: To find the difference between the total interest accrued on each loan, we subtract the principal amount from the total value of each loan.

For the loan with quarterly compounded interest:

Total interest accrued = Total value - Principal

Total interest accrued = $53,262.47 - $20,000

Total interest accrued ≈ $33,262.47

For the loan with monthly compounded interest:

Total interest accrued = Total value - Principal

Total interest accrued = $53,758.68 - $20,000

Total interest accrued ≈ $33,758.68

The difference between the total interest accrued on each loan is approximately $33,758.68 - $33,262.47 = $496.21.

Therefore, the difference between the total interest accrued on each loan is approximately $496.21. This means that the loan with monthly compounded interest will accrue approximately $496.21 more in interest compared to the loan with quarterly compounded interest over the 10-year period.

User Kevin Chandra
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