Answer:
To calculate the principal repaid in Brenda's 7th payment, we need to determine the decreasing annuity payment for each year and then subtract the interest portion from it.
Given that Brenda has a 21-year loan with a decreasing annuity, the annual payments decrease by 1 each year. The first payment is 21, the second payment is 20, and so on until the 21st payment, which is 1.
To calculate the principal repaid in the 7th payment, we need to find the decreasing annuity payment for the 7th year. We can use the formula for the present value of an annuity to calculate the decreasing annuity payment:
PVA = PMT * [(1 - (1 + r)^(-n)) / r]
Where:
PVA is the present value of the annuity
PMT is the annuity payment for each period
r is the interest rate per period
n is the total number of periods
In this case, the annuity payment for each period (PMT) is the decreasing annuity payment for the first year, which is 21. The interest rate per period (r) is 4% or 0.04, and the total number of periods (n) is 21.
Using the formula, we can calculate the present value of the annuity:
PVA = 21 * [(1 - (1 + 0.04)^(-21)) / 0.04]
PVA ≈ 247.1966
The present value of the annuity is approximately 247.1966.
Now, we need to calculate the principal repaid in the 7th payment. To do this, we subtract the interest portion from the decreasing annuity payment for the 7th year.
The interest portion can be calculated using the formula:
Interest = Principal * Rate
Where:
Principal is the remaining principal balance
Rate is the interest rate
In this case, the remaining principal balance is the present value of the annuity, which is 247.1966. The interest rate is 4% or 0.04.
Interest = 247.1966 * 0.04
Interest ≈ 9.888
The interest portion is approximately 9.888.
Finally, we subtract the interest portion from the decreasing annuity payment for the 7th year:
Principal Repaid = Decreasing Annuity Payment - Interest
Principal Repaid = 7 - 9.888
Principal Repaid ≈ -2.888
The principal repaid in Brenda's 7th payment is approximately -2.888.
None of the provided answer choices match this calculation. It's possible that there was an error in the answer choices or in the calculation provided. Please double-check the question or seek clarification from your teacher or instructor.
Explanation: