Monthly withdrawals of $220 are made from an account that has an opening balance of $35 300, invested at 7% per annum, compounding monthly. The balance of the account after 1 year is closest to:单项选择题
A
a. $32 660
B
b. $33 500
C
c. $40 578
D
d. $35 125

Question

Monthly withdrawals of $220 are made from an account that has an opening balance of $35 300, invested at 7% per annum, compounding monthly. The balance of the account after 1 year is closest to:

BlackTom AI · Accepted Answer

Use monthly compounding: the monthly interest rate is i = 0.07/12 = 0.005833333333333333. Each month interest is credited and then $220 is withdrawn (end-of-month withdrawals). After 12 months the balance equals the initial balance grown for 12 months minus the accumulated effect of the 12 withdrawals (an ordinary annuity).

1) Relevant formulas and concepts
- Compound growth: A(0)*(1+i)^n for n periods.
- Future value of an ordinary annuity (withdrawals at end of each period): payment * [((1+i)^n - 1)/i].
- Net balance after n periods = initial*(1+i)^n - payment * [((1+i)^n - 1)/i].

2) Step-by-step reasoning and calculations
- Monthly rate: i = 0.07/12 = 0.0058333333.
- Growth factor for 12 months: (1 + i)^12 ≈ (1.0058333333)^12 ≈ 1.072292.
- Initial balance grown: 35,300 * 1.072292 ≈ 35,300 + 35,300*0.072292 = 35,300 + 2,551.9076 = 37,851.9076.
- Annuity factor for the 12 withdrawals: ((1+i)^12 - 1)/i.
  Compute numerator: (1+i)^12 - 1 ≈ 0.072292.
  Divide by i: 0.072292 / 0.0058333333 ≈ 12.3929143.
- Total of accumulated withdrawals (future value at month 12): 220 * 12.3929143 ≈ 2,726.4411.
- Net balance after 12 months: 37,851.9076 − 2,726.4411 = 35,125.4665 ≈ $35,125.

3) Why the correct answer is correct
The formula correctly accounts for monthly compounding of the opening balance and the time value impact of twelve end-of-month withdrawals. The computed balance rounds to $35,125, which matches option d.

4) Why the other choices are incorrect
- a. $32,660 and b. $33,500 are too low — they would imply either larger withdrawals or ignoring the earned interest.
- c. $40,578 is too high — it would correspond to treating withdrawals as deposits or failing to subtract the withdrawals’ accumulated effect.

Conclusion: The balance after one year is approximately $35,125, so the correct choice is d.

Monthly withdrawals of $220 are made from an account that has an opening balance of $35 300, invested at 7% per annum, compounding monthly. The balance of the account after 1 year is closest to:单项选择题

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