Question

Johan invests $4,000 at age 22 from the signing bonus of his new job. He hopes the investment will be worth $240,000 when he turns 66. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Round to the nearest tenth of a percent.

Answer 1

The interest rate needs to be 9.8% for the sum of $4000 to compound to $240,000

Explanation:

Since the interest compounds itself, hence the question concerns the compound interest.

Details provided-

Principal (Initial contribution)- $ 4000

Amount (Expected amount)- $ 240,000

Time period- investment started at 22 years and would continue until 66 years.

Time period= 66-22 years= 44 years

Rate of interest=

We know that for compound interest-

⇒Amount= principal (1+rate/100)ⁿ

Substituting the values if Amount, principal and time ("n") in the above equation

240,000= 4000 (1+rate/100) ⁴⁴

240,000/4000= (1+rate/100) ⁴⁴

Solving the above equation would yield us with the rate as 9.75% ≈ 9.8% (rounded off to tenth place after decimal)

Hence the Interest rate required by John would be 9.8%

Answer 2

Answer: 9.3%

Explanation:

Identify the variables of the formula:

A= $240,000

P= $4,000

r= ?

t= 44 years (66-22=44)

A= Pe^rt

Substitute the values into the formula

240,000= 4,000e^r•44

Solve for r. Divide each side by 4,000

60= e^44r

Take the natural log of each side

ln60= ln e^44r

Use the power property and then simplify

ln60= 44r ln e

ln60= 44r

Divide each side by 44

ln60/44=r

Approximate the answer

r= 0.09305 ——> r= 9.3%