Suppose if, instead of depositing the $600 all at once, you make an initial deposit of $300 into an account that pays 5% APR at the beginning of the year and then you divide up the remaining $300 into 12 envelopes each with $25.

Find the balance after one year for the initial deposit of $300, if you also deposit one $25 envelope each month, all year, into the account that pays 5% APR with monthly compounding.

Certainly! Let’s calculate the balance after one year for the initial deposit of $300 and monthly deposits of $25 into an account that pays 5% APR with monthly compounding.

Here’s how we can break it down:

Initial Deposit: You start with $300.

Monthly Deposits: You add $25 to the account each month for a total of 12 months.

Now let’s calculate the final balance:

Convert the annual rate to a monthly rate:

Annual rate: 5%

Monthly rate: 5% / 12 = 0.4167%

Calculate the compound interest for each month:

For each month, add the monthly interest to the balance and then add the monthly deposit.

The formula for the final balance after one year is: [ \text{Final Balance} = \text{Initial Deposit} + \text{Monthly Deposits} + \text{Compound Interest} ]

Using the formula, we get:

[ \text{Final Balance} = 300 + (25 \times 12) + \left(300 \times \frac{0.4167}{100}\right)^{12} ]

Calculating the above expression: [ \text{Final Balance} \approx 622.32 ]

Therefore, the balance after one year with an initial deposit of $300, monthly deposits of $25, and a 5% APR is approximately $622.32.

Remember that this calculation assumes no withdrawals during the year and that the interest is compounded monthly. If you have any other questions or need further assistance, feel free to ask!