Question

Account A and Account B both have a principal of $2,000 and an annual interest rate of 2%. No additional deposits or withdrawals are made. Account A earns simple interest. Account B earns interest compounded annually. Compare the amounts in the two accounts after 20 years. Which earns more interest? How much more?

After 20 years, account (A or B) will have earned $____ more.

Answer 1

Account B earns more interest.

After 20 years, account B will have earned $171.89 more.

Explanation:

Let's calculate the total for each account.

Account A:

Account A earns simple interest. We know that the principal value is $2000 and the interest rate is 2% or 0.02. We can use the simple interest formula:

`A=P(1+rt)`

Where A is the future value, P is the principal, r is the rate, and t is the time in years.

So, let's substitute 2000 for P, 0.02 for r, and 20 for t. This yields:

`A=2000(1+0.02(20))`

Multiply and add:

`A=2000(1+0.4)=2000(1.4)`

Multiply. So, the total amount of money in Account A after 20 years is:

`A=\$2800`

Since we initially deposited $2000 and our total is now $2800, this means that we earned an interest of
`2800-2000=\$ 800`

Account B:

Account B earns compound interest. Like Account A, Account B has a principal value of $2000 and the interest rate is 2% or 0.02. We also know that it's compounded annually, so once per year. We can use the compound interest formula:

`B=P(1+(r)/(n)})^(nt)`

Where B is the future value, P is the principal, r is the rate, n is the times compounded per year, and t is the time in years.

So, let's substitute 2000 for P, 0.02 for r, n for 1 (since it's compounded annually), and t for 20. This yields:

`B=2000(1+(0.02)/(1))^((1)(20))`

Simplify this to acquire:

`B=2000(1.02)^(20)`

Evaluate. Use a calculator. So, after 20 years, the amount of money in Account B is:

`B\approx\$2971.89`

Since our principal was $2000, this means that we earned an interest of approximately
`2971.89-2000=\$ 971.89`.

So, Account A earned an interest of $800 and Account B earned an approximate interest of $971.89.

So, Account B earned more interest.

And it earned
`971.89-800=\$ 171.89` more than Account A.

And we're done!