Respuesta :
Using compound interest, it is found that:
a) Pietro will have $1,719.49 in his account after 5 years.
b) The interest rate is of 8.19%.
What is compound interest?
The amount of money earned, in compound interest, after t years, is given by:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
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.
- t is the time in years for which the money is invested or borrowed.
Item a:
- Investment of 1500 euros, hence [tex]P = 1500[/tex].
- Interest rate of 2.75%, compounded half-yearly, hence [tex]r = 0.0275, n = 2[/tex].
- 5 years, hence [tex]t = 5[/tex].
Then:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
[tex]A(5) = 1500\left(1 + \frac{0.0275}{2}\right)^{2(5)}[/tex]
[tex]A(5) = 1719.49[/tex]
Pietro will have $1,719.49 in his account after 5 years.
Item b:
- Amount increases by 1.5 times, hence [tex]A(t) = 1.5(1500)[/tex]
- Investment of 1500 euros, hence [tex]P = 1500[/tex].
- Compounded quarterly, hence [tex]n = 8[/tex].
- 5 years, hence [tex]t = 5[/tex].
Then:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
[tex]1.5(1500) = 1500\left(1 + \frac{r}{4}\right)^{20}[/tex]
[tex]1.5 = (1 + \frac{r}{4}\right)^{20}[/tex]
[tex]\sqrt[20]^{(1 + \frac{r}{4}\right)^{20}} = \sqrt[20]{1.5}[/tex]
[tex]1 + 0.25r = 1.02048015365[/tex]
[tex]0.25r = 0.02048015365[/tex]
[tex]r = \frac{0.02048015365}{0.25}[/tex]
[tex]r = 0.0819[/tex]
The interest rate is of 8.19%.
To learn more about compound interest, you can take a look at https://brainly.com/question/25781328
