Suppose you have $600 to invest in a savings plan.Suppose if, instead of depositing the $600 all at once, you deposit nothing at the beginning and you divide up the $600 into 12 envelopes each with $50.Find the balance after one year if you deposit one $50 envelope each month, all year, into an account that pays 5% APR with monthly compounding.(Round answer to the nearest cent.)
Question
Suppose you have 600 all at once, you deposit nothing at the beginning and you divide up the 50.Find the balance after one year if you deposit one $50 envelope each month, all year, into an account that pays 5% APR with monthly compounding.(Round answer to the nearest cent.)
Solution
To solve this problem, we will use the formula for the future value of a series of monthly deposits, also known as an annuity. The formula is:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
where: FV is the future value of the investment P is the amount deposited each period (in this case, $50) r is the annual interest rate (in this case, 5% or 0.05) n is the number of times the interest is compounded per year (in this case, 12 times) t is the number of years the money is invested for (in this case, 1 year)
Substituting the given values into the formula, we get:
FV = 50 * [(1 + 0.05/12)^(12*1) - 1] / (0.05/12)
Solving the equation step by step:
- Calculate the rate per period: 0.05/12 = 0.00416667
- Calculate the total number of periods: 12*1 = 12
- Calculate the factor (1 + r/n)^(nt): (1 + 0.00416667)^12 = 1.0512
- Subtract 1 from the factor: 1.0512 - 1 = 0.0512
- Divide the result by the rate per period: 0.0512 / 0.00416667 = 12.272
- Multiply the result by the amount deposited each period: 12.272 * 50 = $613.60
So, if you deposit 613.60 after one year.
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