Find the required annual interest rate to the tenth of a percent for $5200 to grow to $6500 if interest is compounded quarterly for five years

It must be invested at 4.5%.

The compound interest formula is

A=p (1 + (r/n)) ^ (nt), where A is the total amount including interest earned, p is the principal invested, r is the rate, n is the number of times per year the interest is compounded, and t is the number of years.

With our information we have:

6500=5200 (1 + (r/4)) ^ (4*5)

6500=5200 (1 + (r/4)) ^20

Cancel 5200 by dividing:

6500/5200 = [5200 (1 + (r/4)) ^20]/5200

1.25 = (1 + (r/4)) ^20

Using a calculator, take the 20th root of each side:

1.25^ (1/20) = [ (1 + (r/4)) ^20]^ (1/20)

1.011219651 = 1 + (r/4)

Subtract 1 from each side:

1.011219651-1 = 1 + (r/4) - 1

0.011219651=r/4

Multiply each side by 4:

0.011219651*4 = (r/4) * 4

0.0448786 = r

0.045 = r