Question

Carter invested $540 in an account paying an interest rate of 4. 875 % compounded monthly. Jack invested $540 in an account paying an interest rate of 5. 5% compounded continuously. To the nearest hundredth of a year, how much longer would it take for Carter's money to triple than for Jack's money to triple?

Answer 1

Using compound interest and continuous compounding, it is found that it would take 2.61 years longer for Carter's money to triple.

What is compound interest?

The amount of money earned, in compound interest, after t years, is given by:

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

In which:

• A(t) is the amount of money after t years.

• P is the principal(the initial sum of money).

• r is the interest rate(as a decimal value).

• n is the number of times that interest is compounded per year.

For Carter, we have that the rate of interest and the number of compoundings are, respectively, r = 0.04875 and n = 12. The time to triple is t for which A(t) = 3P, hence:

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

$3P = P\left(1 + \frac{0.04875}{12}\right)^{12t}$

$(1.0040625)^{12t} = 3$

$\log{(1.0040625)^{12t}} = \log{3}$

$12t\log{(1.0040625)} = \log{3}$

$t = \frac{\log{3}}{12\log{(1.0040625)}}$

t = 22.58.

What is continuous compounding?

The amount of money is given by:

$A(t) = Pe^{kt}$

For Jack, we have that the rate of interest is of k = 0.055, hence:

$A(t) = Pe^{kt}$

$3P = Pe^{0.055t}$

$e^{0.055t} = 3$

$\ln{e^{0.055t}} = \ln{3}$

$0.055t = \ln{3}$

$t = \frac{\ln{3}}{0.055}$

t = 19.97.

The difference is given by:

22.58 - 19.97 = 2.61

It would take 2.61 years longer for Carter's money to triple.

More can be learned about compound interest at https://brainly.com/question/25781328