Suppose you have $600 to invest in a savings plan.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.(Round answer to the nearest cent.)
Question
Suppose you have 600 all at once, you make an initial deposit of 300 into 12 envelopes each with 300, if you also deposit one $25 envelope each month, all year, into the account that pays 5% APR with monthly compounding.(Round answer to the nearest cent.)
Solution
To solve this problem, we need to calculate the future value of two different types of investments: a lump sum and a series of monthly deposits.
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First, calculate the future value of the initial $300 deposit. This is a lump sum investment, so we use the formula for the future value of a lump sum:
FV = PV * (1 + r/n)^(nt)
where: FV = future value PV = present value (the initial deposit) r = annual interest rate (in decimal form) n = number of times interest is compounded per year t = number of years
Plugging in the values:
FV = 315.79
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Next, calculate the future value of the $25 monthly deposits. This is a series of equal payments, so we use the formula for the future value of an annuity:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
where: FV = future value P = payment amount per period r = annual interest rate (in decimal form) n = number of times interest is compounded per year t = number of years
Plugging in the values:
FV = 310.08
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Finally, add the future values of the initial deposit and the monthly deposits to find the total balance after one year:
Total balance = 310.08 = $625.87
So, the balance after one year for the initial deposit of 25 envelope each month, all year, into the account that pays 5% APR with monthly compounding, is approximately $625.87.
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