Question14 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 undefinedSingle choice
A
34.18 years
B
38.72 years
C
41.56 years
D
45.03 years

Question

Question14 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 undefinedSingle choice

A

34.18 years

B

38.72 years

C

41.56 years

D

45.03 years

BlackTom AI · Accepted Answer

Monthly savings accumulate according to the future value of an ordinary annuity:
FV = PMT * [ (1 + r)^n - 1 ] / r. Here PMT = $420, target FV = $1,850,000, and APR = 9.2% compounded monthly so the monthly rate r = 0.092/12.

Step-by-step solution:
1) Monthly interest rate: r = 0.092 / 12 = 0.007666666666666667.
2) Plug into the annuity formula and solve for the number of periods n:
   1,850,000 = 420 * [ (1 + r)^n - 1 ] / r
   Rearranged: (1 + r)^n = 1 + (FV * r) / PMT.
3) Compute FV * r: 1,850,000 * 0.007666666666666667 = 170,200 / 12 = 14,183.333333333334.
4) Compute (FV * r) / PMT: 14,183.333333333334 / 420 = 33.76984126984032.
   So (1 + r)^n = 1 + 33.76984126984032 = 34.76984126984032.
5) Take natural logs to solve for n:
   n = ln(34.76984126984032) / ln(1.0076666666666667).
   ln(34.76984126984032) ≈ 3.548606; ln(1.0076666666666667) ≈ 0.007637428.
   Therefore n ≈ 3.548606 / 0.007637428 ≈ 464.711 months.
6) Convert months to years: 464.711 / 12 ≈ 38.7259 years ≈ 38.72 years.

Why this is correct: the algebra and numeric evaluation directly follow the future-value formula for level end-of-period contributions with monthly compounding. The computed n gives the time required for $420 monthly deposits at 9.2% APR (monthly compounding) to accumulate to $1.85 million.

Why the other choices are incorrect: 34.18, 41.56, and 45.03 years do not satisfy the annuity equation with the given r and PMT; they are respectively too short or too long compared with the precise solution (≈38.72 years).

Conclusion: The correct answer is 38.72 years, matching the provided answer.

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