### Instruction: Angela has deposited $\$8,\!000$ into an account that pays $6\%$ interest compounded annually.

Bob has deposited $\$10,\!000$ into an account that pays $7\%$ simple annual interest.

In $20$ years Angela and Bob compare their respective balances. To the nearest dollar, what is the positive difference between their balances?### Response: I know that the formula for the future value of an account that pays interest compounded annually is $FV = PV(1 + r)^n$, where $FV$ is the future value, $PV$ is the present value, $r$ is the annual interest rate, and $n$ is the number of years.
So, for Angela's account, I can plug in the given values and get $FV = 8,\!000(1 + 0.06)^{20}$.
To simplify this expression, I can use the exponent rules and write $(1 + 0.06)^{20}$ as $1.06^{20}$.
Then, I can use a calculator to evaluate this power and get $1.06^{20} \approx 3.207$.
Multiplying this by $8,\!000$, I get $FV \approx 25,\!656$.
This means that in $20$ years, Angela's balance will be about $\$25,\!656$.
Similarly, I know that the formula for the future value of an account that pays simple annual interest is $FV = PV(1 + rn)$, where the variables are the same as before, except that $r$ is multiplied by $n$ instead of being raised to the power of $n$.
So, for Bob's account, I can plug in the given values and get $FV = 10,\!000(1 + 0.07 \times 20)$.
To simplify this expression, I can use the distributive property and write $1 + 0.07 \times 20$ as $1 + 1.4$.
Adding these two terms, I get $1 + 1.4 = 2.4$.
Then, multiplying this by $10,\!000$, I get $FV = 24,\!000$.
This means that in $20$ years, Bob's balance will be $\$24,\!000$.
To find the positive difference between their balances, I need to subtract the smaller balance from the larger balance.
In this case, the smaller balance is Bob's and the larger balance is Angela's.
So, the positive difference is $25,\!656 - 24,\!000 = 1,\!656$.
Rounding this to the nearest dollar, I get $\$1,\!656$.