Question

Sara is planning on fully renovating her kitchen. She figures that this will cost a total of $25,750. If Sandra already has $21,400 saved up, what annual interest rate
compounded semiannually would Sara need to earn in order to have the required
funds if she invests her money for 2.5 years?

Answer 1

Answer:

She needs an annual interest rate of 7.54%.

Step-by-step explanation:

Compound interest:

The compound interest formula is given by:

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

Where 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 and t is the time in years for which the money is invested or borrowed.

She figures that this will cost a total of $25,750.

This means that $A = 25750$.

Sandra already has $21,400 saved up

This means that $P = 21400$.

Semianually compunding, 2.5 years.

This means that $n = 2, t = 2.5$

What interest rate?

We have to find r. So

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

$25750 = 21400(1 + \frac{r}{2})^{2*2.5}$

$(1 + 0.5r)^5 = \frac{25750}{21400}$

$\sqrt[5]{(1 + 0.5r)^5} = \sqrt[5]{\frac{25750}{21400}}$

$1 + 0.5r = (\frac{25750}{21400})^{\frac{1}{5}}$

$1 + 0.5r = 1.0377$

$0.5r = 0.0377$

$r = \frac{0.0377}{0.5}$

$r = 0.0754$

0.0754*100% = 7.54%

She needs an annual interest rate of 7.54%.