How many annual deposits of $1,000, beginning next year, would you need to make before you had accumulated $30,000, if the money earns 8% per year, compounded annually? Assume the account begins with a $0 balance.
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A. B. C. D. E.A
On the BAII Plus, press 8 I/Y, 0 PV, 1000 PMT, 30000 +/- FV, CPT N. On the HP12C, press 8 i, 0 PV, 1000 PMT, 30000 CHS FV, n. Note that the HP12C will display 16 as the answer.
To calculate the number of annual deposits required to accumulate $30,000 with an 8% annual interest rate, compounded annually, we can use the future value of an ordinary annuity formula:
Future Value = Payment × [(1 + Interest Rate)^(Number of Periods) - 1] / Interest Rate
In this formula:
We want to solve for the Number of Periods, so we rearrange the formula:
Number of Periods = log(1 + (Future Value × Interest Rate) / Payment) / log(1 + Interest Rate)
Let's plug in the given values into the formula and calculate:
Future Value = $30,000 Payment = $1,000 Interest Rate = 8% or 0.08
Number of Periods = log(1 + ($30,000 × 0.08) / $1,000) / log(1 + 0.08)
Calculating the numerator: Numerator = ($30,000 × 0.08) / $1,000 = $2,400 / $1,000 = 2.4
Calculating the denominator: Denominator = 1 + 0.08 = 1.08
Now, we can calculate the Number of Periods using logarithms:
Number of Periods = log(1 + 2.4) / log(1.08)
Using a calculator, the value of log(1 + 2.4) is approximately 0.87547, and log(1.08) is approximately 0.03342.
Number of Periods = 0.87547 / 0.03342 ≈ 26.150
Since we're looking for the number of annual deposits, we round the result up to the nearest whole number, giving us 27.
Therefore, the correct answer is not provided among the given choices.