John invests $200$200 in a bank account earning 8%8% annual interest with a plan to withdraw the principal and interest one year later. If the interest were to compound quarterly, the interest earned would be approximately what percent greater than the interest earned if the account were to compound twice a year?

First, let's calculate the amount of money John would have in his account after one year if the interest were compounded semi-annually (twice a year).

The formula for compound interest is A = P(1 + r/n)^(nt), where:

• A is the amount of money accumulated after n years, including interest.
• P is the principal amount (the initial amount of money).
• r is the annual interest rate (in decimal).
• n is the number of times that interest is compounded per year.
• t is the time the money is invested for in years.

If the interest is compounded semi-annually, n = 2. So, the amount of money in the account after one year would be:

A = 200(1 + 0.08/2)^(21) = 200(1 + 0.04)^2 = 2001.04^2 = $216.32

The interest earned would be $216.32 -$200 = $16.32.

Next, let's calculate the amount of money John would have in his account after one year if the interest were compounded quarterly (four times a year). In this case, n = 4. So, the amount of money in the account after one year would be:

A = 200(1 + 0.08/4)^(41) = 200(1 + 0.02)^4 = 2001.02^4 = $216.65

The interest earned would be $216.65 -$200 = $16.65.

Finally, let's calculate the percentage increase in the interest earned if the account were to compound quarterly instead of semi-annually. The percentage increase would be:

((16.65 - 16.32) / 16.32) * 100% = 2.02%

So, the interest earned would be approximately 2.02% greater if the account were to compound quarterly instead of semi-annually.