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If you save $8,152\$8,152 now and the account pays 3.3%3.3\% per annum, compounding monthly, how much is the outstanding balance at the end of year 99?

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Full solution

Q. If you save $8,152\$8,152 now and the account pays 3.3%3.3\% per annum, compounding monthly, how much is the outstanding balance at the end of year 99?
  1. Identify variables needed: Identify the variables needed to calculate the future value of the investment.\newlinePrincipal amount PP = $8,152\$8,152\newlineAnnual interest rate rr = 3.3%3.3\% or 0.0330.033 (as a decimal)\newlineNumber of times the interest is compounded per year nn = 1212 (monthly compounding)\newlineNumber of years tt = 99\newlineWe will use the formula for compound interest: A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}
  2. Convert interest rate: Convert the annual interest rate from a percentage to a decimal by dividing by 100100.\newliner=3.3%100=0.033r = \frac{3.3\%}{100} = 0.033
  3. Substitute values into formula: Substitute the values into the compound interest formula. A=8152(1+0.03312)(12×9)A = 8152(1 + \frac{0.033}{12})^{(12\times9)}
  4. Calculate value inside parentheses: Calculate the value inside the parentheses (1+r/n)(1 + r/n). \newline1+0.033/12=1+0.00275=1.002751 + 0.033/12 = 1 + 0.00275 = 1.00275
  5. Calculate exponent: Calculate the exponent (nt)(n^t). \newline12×9=10812 \times 9 = 108
  6. Raise value to power: Raise the value inside the parentheses to the power of the exponent. \newline(1.00275)108(1.00275)^{108}\newlineTo calculate this, we can use a calculator.\newline(1.00275)108≈1.349858807576003(1.00275)^{108} \approx 1.349858807576003
  7. Multiply principal amount: Multiply the principal amount by the result from the previous step to find the future value. \newlineA=8152×1.349858807576003A = 8152 \times 1.349858807576003\newlineAgain, using a calculator for this multiplication.\newlineA≈8152×1.349858807576003≈11001.97A \approx 8152 \times 1.349858807576003 \approx 11001.97

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