A loan of $7000 at 6.7% interest per annum compounding monthly on the reducing balance is repaid over five years with equal monthly payments. The amount of the repayment is closest to:单项选择题
A
a. $141.13
B
b. $137.60
C
c. $1423.54
D
d. $155.75

Question

A loan of $7000 at 6.7% interest per annum compounding monthly on the reducing balance is repaid over five years with equal monthly payments. The amount of the repayment is closest to:

BlackTom AI · Accepted Answer

Monthly payment for a fully amortizing loan uses the annuity formula: R = P * i / (1 - (1 + i)^(-n)).

1) Relevant knowledge points:
- Convert the annual nominal rate (compounded monthly) to a monthly interest rate: i = r/12.
- Total number of monthly payments n = years × 12.
- Use the standard loan/annuity formula above to find the equal monthly payment R.

2) Step-by-step reasoning and calculations:
- Principal P = $7000.
- Annual nominal rate r = 6.7% = 0.067, so monthly rate i = 0.067 / 12 = 0.005583333333333333 (about 0.558333% per month).
- Term = 5 years → n = 5 × 12 = 60 monthly payments.
- Compute (1 + i)^n = (1.0055833333)^60.
  Use logarithms/expansion: ln(1.0055833333) ≈ 0.00556775, so 60·ln(1+i) ≈ 0.33406498 and (1+i)^60 = e^{0.33406498} ≈ 1.396598.
- Compute the denominator: 1 - (1 + i)^{-n} = 1 - 1/1.396598 ≈ 1 - 0.7160006 = 0.2839994.
- Compute numerator: P · i = 7000 × 0.005583333333 = 39.083333333.
- Monthly payment R = 39.083333333 / 0.2839994 ≈ 137.605 ≈ $137.60.

3) Why this answer is correct:
- The calculation follows the exact amortization formula for monthly-compounded interest with equal payments. The intermediate computation of the compound factor (1+i)^60 and its reciprocal lead directly to the denominator used; dividing the monthly interest portion of the principal by that denominator yields the correct monthly payment.

4) Why the other options are incorrect:
- a. $141.13 and d. $155.75 are larger than the correct amortized payment — they would pay off the loan faster (or imply a higher interest rate). 
- c. $1423.54 is an order-of-magnitude larger and clearly inconsistent with standard monthly payments on a $7,000 loan over five years.

Conclusion: The monthly repayment is approximately $137.60, matching option b.

A loan of $7000 at 6.7% interest per annum compounding monthly on the reducing balance is repaid over five years with equal monthly payments. The amount of the repayment is closest to:单项选择题

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