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

She needs an annual interest rate of 7.54%.

Explanation:

Compound interest:

The compound interest formula is given by:

`A(t) = P(1 + (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 + (r)/(n))^(nt)`

`25750 = 21400(1 + (r)/(2))^(2*2.5)`

`(1 + 0.5r)^5 = (25750)/(21400)`

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

`1 + 0.5r = ((25750)/(21400))^{(1)/(5)}`

`1 + 0.5r = 1.0377`

`0.5r = 0.0377`

`r = (0.0377)/(0.5)`

`r = 0.0754`

0.0754*100% = 7.54%

She needs an annual interest rate of 7.54%.