Question

Nicole invests $2000 in an account. The account pays compound interest at a rate of K% per year. At the end of the first year, the money in the account is $2036. (i) Show that K = 1.8. (ii) Find the number of complete years before Nicole has at least $2150 in the account. Show full working​

Answer 1

Answer:

i) See proof below.

ii) 5 years

Step-by-step explanation:

To solve this problem, we can use the formula for annual compound interest:

$\boxed{\begin{minipage}{7 cm}\underline{Annual Compound Interest Formula}\\\\ A=P\left(1+r\right)^{t} \\\\where:\\\\ \phantom{ww} \bullet A = final amount \\ \phantom{ww} \bullet P = principal amount \\ \phantom{ww} \bullet r = interest rate (in decimal form) \\ \phantom{ww} \bullet t = time (in years) \\ \end{minipage}}$

Given values:

• A = $2036
• P = $2000
• t = 1 year

Substitute the values into the formula and solve the equation for r:

$2036=2000(1+r)^1$

$2036=2000(1+r)$

$\dfrac{2036}{2000}=1+r$

$1.018=1+r$

$r=1.018-1$

$r=0.018$

Multiply by 100 to convert to a percentage:

$r = 0.018 \times 100=1.8\%$

Therefore, the interest rate per year, K, is 1.8%.

To find the number of complete years before Nicole has at least $2,150 in the account, substitute A = 2150, P = 2000, and r = 0.018 into the formula, and solve for t.

$2150=2000(1+0.018)^t$

$2150=2000(1.018)^t$

$\dfrac{2150}{2000}=(1.018)^t$

$1.075=(1.018)^t$

Take natural logs of both sides of the equation:

$\ln 1.075= \ln (1.018)^t$

$\textsf{Apply the log power law:} \quad \ln x^n=n \ln x$

$\ln 1.075= t \ln 1.018$

Divide both sides of the equation by ln(1.018):

$\dfrac{\ln 1.075}{\ln 1.018}= \dfrac{t \ln 1.018}{\ln 1.018}$

$t=4.05386734...$

As we need to find the number of complete years before Nicole has at least $2,150 in the account, we need to round up the found value of t to the nearest complete year.  

From our calculations, t ≈ 4.05 years, so the balance of the account reaches $2,150 after 4 years and during the 5th year. (After 4 years, the account balance is $2,147.93).

Therefore, the number of complete years before Nicole has at least $2,150 in the account is 5 years.