Question

Suppose that $4000 is placed in a savings account at an annual rate of 5.1% , compounded semiannually. Assuming that no withdrawals are made, how long will it take for the account to grow to $6000?

Answer 1

9.8 years (if rounded, 10 years)

Explanation:

To determine how long it will take for the savings account balance to grow to $6000, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the future value of the investment ($6000 in this case)

P = the initial deposit ($4000)

r = the annual interest rate (5.1% or 0.051)

n = the number of times interest is compounded per year (semiannually, so n = 2)

t = the number of years

Substituting the given values into the formula, we have:

6000 = 4000(1 + 0.051/2)^(2t)

Now, we can solve for t by isolating it in the equation:

(1 + 0.051/2)^(2t) = 6000/4000

(1 + 0.0255)^(2t) = 1.5

Taking the natural logarithm (ln) of both sides, we get:

ln[(1 + 0.0255)^(2t)] = ln(1.5)

2t * ln(1.0255) = ln(1.5)

t = ln(1.5) / (2 * ln(1.0255))

Using a calculator, we find:

t ≈ 9.8

Therefore, it will take approximately 9.8 years for the account balance to grow to $6000 when compounded semiannually at an annual interest rate of 5.1%.

Hope this helped!