Question18 You feel that you will need $1.85 Million in your retirement account, and when you reach that amount, you plan to retire. You believe you can earn an APR of 9.2% compounded monthly and plan to save $420 per month until you reach your goal. How many years will it be until you reach your goal and retire? 34.18 years 38.72 years 41.56 years 45.03 years ResetMaximum marks: 2 Flag question undefined单项选择题
A
34.18 years
B
38.72 years
C
41.56 years
D
45.03 years

Question

Question18 You feel that you will need $1.85 Million in your retirement account, and when you reach that amount, you plan to retire. You believe you can earn an APR of 9.2% compounded monthly and plan to save $420 per month until you reach your goal. How many years will it be until you reach your goal and retire? 34.18 years 38.72 years 41.56 years 45.03 years ResetMaximum marks: 2 Flag question undefined单项选择题

A

34.18 years

B

38.72 years

C

41.56 years

D

45.03 years

BlackTom AI · Accepted Answer

The future value of an ordinary monthly annuity is FV = PMT * [((1 + r)^n − 1) / r], with r = APR/12. Here PMT = $420, FV = $1,850,000, APR = 9.2% so r = 0.092/12 = 0.007666666666666667.

1) Set up the equation and isolate (1 + r)^n:
FV = PMT * [((1 + r)^n − 1) / r]
=> (1 + r)^n − 1 = FV * r / PMT
=> (1 + r)^n = 1 + (FV * r / PMT)

2) Compute intermediate values:
FV * r = 1,850,000 * 0.007666666666666667 = 14,183.333333333334
(FV * r) / PMT = 14,183.333333333334 / 420 = 33.76984126984127
1 + (FV * r / PMT) = 34.76984126984127

3) Solve for n using natural logs:
(1 + r)^n = 34.76984126984127
n = ln(34.76984126984127) / ln(1.0076666666666667)

Compute logs:
ln(34.76984126984127) ≈ 3.54872
ln(1.0076666666666667) ≈ 0.00763743

n ≈ 3.54872 / 0.00763743 ≈ 464.65 months

4) Convert months to years:
464.65 months / 12 ≈ 38.72 years

Why this is correct: the formula used is the standard future-value formula for an ordinary annuity with monthly deposits and monthly compounding; algebra and logarithms yield the number of monthly deposits required. Each arithmetic step above is shown, leading to 464.65 months (≈38.72 years).

Why the other options are incorrect:
- 34.18 years (≈410.2 months) is too short — it underestimates the number of deposits needed.
- 41.56 years (≈498.7 months) and 45.03 years (≈540.4 months) are too long — they overestimate required deposits or imply a smaller growth rate/PMT than given.

Final answer: 38.72 years (matches the provided correct answer).

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