You deposit $5000 into an account paying 6% annual interest compounded monthly. Approximately how many years will it take for the balance in the account to grow to $8000?

Introduction / Context:
This problem involves growth with monthly compounding and asks for the time needed for an investment to reach a specified target value. Unlike simple annual compounding, here the interest is credited more frequently, which slightly increases the effective growth rate. Solving for time when compounding happens monthly usually requires logarithms, making this a more advanced time value of money question that is common in finance and quantitative aptitude tests.

Given Data / Assumptions:

• Initial deposit P = 5000 dollars
• Target amount A = 8000 dollars
• Nominal annual interest rate r = 6% per annum
• Compounding frequency m = 12 times per year (monthly)
• We must find the time in years t

Concept / Approach:
For nominal annual rate r with m compounding periods per year, the periodic rate is r / m. The amount after t years is given by A = P * (1 + r / m)^(m * t). Here, r = 0.06 and m = 12, so the monthly periodic rate is 0.06 / 12. To find time t, we rearrange the formula, divide both sides by P, and then apply logarithms to isolate t. Because this is a multiple choice question with approximate numerical options, we only need a good approximation, not an exact symbolic expression.

Step-by-Step Solution:
P = 5000, A = 8000, r = 0.06, m = 12 Periodic monthly rate = r / m = 0.06 / 12 = 0.005 Formula: A = P * (1 + 0.005)^(12 * t) 8000 = 5000 * (1.005)^(12 * t) Divide both sides by 5000: 8000 / 5000 = (1.005)^(12 * t) 1.6 = (1.005)^(12 * t) Take natural logarithms: ln(1.6) = 12 * t * ln(1.005) t = ln(1.6) / [12 * ln(1.005)] Numerically this is approximately t ≈ 7.85 years

Verification / Alternative check:
We can quickly verify the reasonableness of the result. If interest were compounded annually at 6%, the time to grow from 5000 to 8000 using A = P * (1.06)^t would be t = ln(1.6) / ln(1.06) which is slightly more than 8 years. Monthly compounding should reduce the time slightly because interest is applied more frequently. Our calculated value of about 7.85 years is a little less than 8 years, which fits this expectation. Among the choices, 7.9 years is closest to 7.85.

Why Other Options Are Wrong:
6.9 years is too short; the money would not have enough time to grow from 5000 to 8000 at only 6% with monthly compounding. 8.9 and 9.9 years are longer than even the required time for annual compounding, and therefore are clearly too large. Only 7.9 years matches the approximate solution from the logarithmic calculation.

Common Pitfalls:
Learners sometimes incorrectly use the annual compounding formula or forget to divide the rate by 12 and multiply the time by 12 for monthly compounding. Another common mistake is misusing logarithms or rounding too early, which can produce significantly wrong times. It is important to keep a few extra decimal places during intermediate steps and only round to one decimal place at the final step to match the multiple choice options.

Final Answer:
It will take approximately 7.9 years for 5000 dollars to grow to 8000 dollars at 6 percent interest compounded monthly.