Answer:

To find the interest rate when compounding quarterly, you can use the formula for compound interest:

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

Where:

A = the future balance (€800)

P = the principal amount (€520)

r = the annual interest rate (which we want to find)

n = the number of times interest is compounded per year (quarterly, so 4 times)

t = the number of years (2 years)

Let's solve for r:

800 = 520(1 + r/4)^(4*2)

First, divide both sides by 520:

800/520 = (1 + r/4)^8

Now, take the eighth root of both sides:

(800/520)^(1/8) = 1 + r/4

(1.53846)^(1/8) = 1 + r/4

Now, subtract 1 from both sides:

(1.53846)^(1/8) - 1 = r/4

Now, multiply both sides by 4 to isolate r:

4 * [(1.53846)^(1/8) - 1] = r

Let's calculate the expression:

4 * [(1.53846)^(1/8) - 1]

Approximately, this expression equals:

4 * [0.0558]

Which simplifies to:

0.2232

Now, to convert it to a percentage, multiply by 100:

0.2232 * 100 ≈ 22.32%

So, the annual interest rate, when compounded quarterly, is approximately 22.32%.