If you deposit $5000 into an account paying 6% annual interest compounded monthly, approximately how many years will it take until the balance in the account reaches $8000?

Introduction / Context:
This problem tests your understanding of compound interest with monthly compounding and the use of exponents to solve for time. Such questions model real savings or investment accounts where interest is added many times per year, and they require comfort with logarithms or careful estimation.

Given Data / Assumptions:

• Initial deposit (principal) P = $5000
• Final amount A = $8000
• Nominal annual interest rate r = 6% per year
• Compounding frequency: monthly (12 times per year)
• We must find the time T in years

Concept / Approach:
For nominal annual rate r, compounded monthly, the periodic rate i is:
i = r / 12The compound amount formula with monthly compounding is:
A = P * (1 + i)^(12 * T)We are given A, P, r, and need to solve for T. This involves taking logarithms or using approximate calculation and then comparing with the options given.

Step-by-Step Solution:
Step 1: Compute the monthly rate.i = 6% / 12 = 0.06 / 12 = 0.005Step 2: Write the compound interest equation.8000 = 5000 * (1.005)^(12 * T)Step 3: Divide both sides by 5000.8000 / 5000 = (1.005)^(12 * T)1.6 = (1.005)^(12 * T)Step 4: Take natural logarithms on both sides.ln(1.6) = (12 * T) * ln(1.005)T = ln(1.6) / (12 * ln(1.005))Step 5: Approximating, ln(1.6) is about 0.470, ln(1.005) is about 0.004987.T ≈ 0.470 / (12 * 0.004987) ≈ 0.470 / 0.0598 ≈ 7.85 yearsThis is closest to 7.9 years.

Verification / Alternative check:
You can verify by plugging T = 7.9 years into the formula. Compute the exponent 12 * 7.9 = 94.8 months. Then estimate (1.005)^(94.8) which is approximately 1.6. Multiplying 5000 by 1.6 yields about 8000, confirming that 7.9 years is a very good approximation consistent with the multiple-choice options.

Why Other Options Are Wrong:
6.9 years: Too small; the account would not yet grow from 5000 to 8000.8.9 and 9.9 years: These would result in an amount significantly higher than $8000.5.9 years: Much too short a time to increase the investment by 60% at only 6% nominal rate.

Common Pitfalls:
Some learners mistakenly use annual compounding instead of monthly, which leads to incorrect time estimates. Others try to avoid logarithms but then guess poorly. Remember that for such growth problems with time in the exponent, logarithms are the correct mathematical tool. Also ensure that the rate is converted to a decimal and divided by the correct compounding frequency.

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