Question

Sung Lee invests $5,000 at age 18. He hopes the investment will be worth $20,000 when he turns 25. 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

Answer:

19.8%

Step-by-step explanation:

We have the following formula for continuous compound interest:

A = P * e ^ (i * t)

Where:

A is the final value

P is the initial investment

i is the interest rate in decimal

t is time.

The time can be calculated as follows:

25 - 18 = 7

That is, the time corresponds to 7 years. In addition, A is 20,000 for A and P would be 5,000, we replace:

20000 = 5000 * e ^ (7 * i)

20000/5000 = e ^ (7 * i)

e ^ (7 * i) = 4

ln e ^ (7 * i) = ln 4

7 * i = ln 4

i = (ln 4) / 7

i = 0.198

Which means that the rounded percentage will be 19.8% per year

Answer 2

The rate of growth needed to achieve this goal is 22%

A compound interest is given by the formula:

$A=P(1+\frac{r}{n} )^{nt}$

Where A is the final amount, r is the rate, t is the period, P is the principal and n is the number of times compounded in a period.

Given that A = 20000, P = 5000, t = 7 (25 - 18). hence:

$20000=5000(1+\frac{r}{1} )^{7*1}\\\\4=(1+r )^{7}\\\\\\ln(4)=7\ ln(1+r)\\\\r=0.22$

Therefore the rate of growth needed to achieve this goal is 22%

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