(4 marks, difficulty level: Easy) Suppose that you invest $28,000 in an account paying 8% interest. You plan to withdraw $2300 at the end of each year for 20 years. How much money will be left in the account after 20 years?

Question

(4 marks, difficulty level: Easy) Suppose that you invest $28,000 in an account paying 8% interest. You plan to withdraw $2300 at the end of each year for 20 years. How much money will be left in the account after 20 years?

BlackTom AI · Accepted Answer

Question restatement: An initial deposit of $28,000 earns 8% interest per year, and you withdraw $2,300 at the end of each year for 20 years. After 20 years, how much remains in the account?

Option-by-option analysis:
- Option A: $22,361
  This value is substantially lower than the expected remaining balance. If you apply the standard future value-with-withdrawals formula, you would expect the balance to stay above the initial amount after many years because the interest (8%) compounds on the existing balance, and withdrawals are fixed. A balance this low suggests either a much larger withdrawal or a much smaller interest effect, neither of which matches the given data.
- Option B: $25,254
  This aligns closely with the correct calculation. Using the closed-form expression for the balance after n years with a starting principal P0, interest rate i, and annual withdrawal W is B_n = P0*(1+i)^n − W * [((1+i)^n − 1)/i]. Substituting P0 = 28,000, i = 0.08, W = 2,300, n = 20 gives B_20 ≈ 28,000*(1.08)^20 − 2,300 * [((1.08)^20 − 1)/0.08], which evaluates to about $25,254. The calculation uses (1.08)^20 ≈ 4.663, leading to the approximate result B_20 ≈ 130,564 − 105,310 ≈ 25,254.
- Option C: $26,371
  This is somewhat higher than the correct value. It would occur if either the future value factor (1.08)^20 were underestimated or the accumulated value of withdrawals were underestimated. Since the standard formula yields around 25,254, this option overstates the remaining balance by a modest amount, indicating a miscalculation in the substitution or rounding.
- Option D: $11,424
  This is far too small relative to the starting amount and the interest earned. Maintaining even modest compounding at 8% for 20 years without withdrawals would already exceed 11,424 by a large margin, and with withdrawals of 2,300 per year, the balance would not drop to such a low figure in this scenario.

Summary: The problem uses the standard balance-after-n-years with annual withdrawals model: B_n = P0*(1+i)^n − W * [((1+i)^n − 1)/i]. Plugging in P0 = 28,000, i = 0.08, W = 2,300, n = 20 yields approximately 25,254, which matches Option B. The other options either underestimate, overestimate, or are far from the computed value due to incorrect application or rounding of the formula.

(4 marks, difficulty level: Easy) Suppose that you invest $28,000 in an account paying 8% interest. You plan to withdraw $2300 at the end of each year for 20 years. How much money will be left in the account after 20 years?单项选择题

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