Question

If $30,000 is invested at 5% annual interest compounded monthly, how long would it take for the account balance to reach $90,000? Round your answer to the nearest tenth.

1) 20.5 years
2) 21.7 years
3) 19.4 years
4) 22.0 years

Answer 1

To determine how long it takes for $30,000 to grow to $90,000 at 5% interest compounded monthly, we use the compound interest formula A = P(1 + r/n)^(nt). After solving, it takes approximately 21.7 years for the investment to reach $90,000.

Step-by-step explanation:

To solve for the time it takes for an investment to grow to a certain amount with compound interest, you can use the formula for compound growth, which is:

A = P(1 + r/n)nt

Where:

A is the future value of the investment/loan, including interest

• P is the principal investment amount (initial deposit or loan amount)
• r is the annual interest rate (decimal)
• n is the number of times that interest is compounded per year
• t is the time in years

We know that A=$90,000, P=$30,000, r=5% or 0.05, and n=12 (since interest is compounded monthly). Plugging these values into the formula, we get:

90000 = 30000(1 + 0.05/12)12t

You'll then solve for t, which represents the time in years. After calculations, we find that t ≈ 21.7 years, so the correct answer is (2) 21.7 years.

To solve for t, follow these steps:

Divide both sides by the principal to get 3 = (1 + 0.05/12) 12t

1. Take the natural logarithm of both sides to get rid of the exponent, which gives you ln(3) = 12t * ln(1 + 0.05/12)
2. Solve for t, ensuring to round the answer to the nearest tenth, and you get t ≈ 21.7 years