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Suppose you want to have $300,000\$300,000 for retirement in 2525 years. Your account earns 6%6\% interest. a) How much would you need to deposit in the account each month? $\$ b) How much interest will you earn? $\$ Enter an integer or decimal number.

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Full solution

Q. Suppose you want to have $300,000\$300,000 for retirement in 2525 years. Your account earns 6%6\% interest. a) How much would you need to deposit in the account each month? $\$ b) How much interest will you earn? $\$ Enter an integer or decimal number.
  1. Use Annuity Formula: To solve part a) of the problem, we need to use the formula for the future value of an annuity, which is used to calculate the amount of regular deposits needed to reach a certain future value with a given interest rate and time period. The formula is:\newlinePMT=FV((1+rn)nt−1)/(rn)PMT = \frac{FV}{\left(\left(1 + \frac{r}{n}\right)^{nt} - 1\right) / \left(\frac{r}{n}\right)}\newlineWhere:\newlinePMTPMT = monthly payment\newlineFVFV = future value ($300,000\$300,000)\newlinerr = annual interest rate (6%6\% or 0.060.06)\newlinenn = number of times the interest is compounded per year (1212 for monthly)\newlinett = number of years (PMTPMT00)\newlineFirst, we need to convert the annual interest rate to a monthly rate by dividing by 1212.
  2. Calculate Monthly Rate: Calculate the monthly interest rate: r/n=0.06/12=0.005r/n = 0.06 / 12 = 0.005
  3. Calculate Total Periods: Now, we calculate the total number of periods (months) over 2525 years:\newlinent=12×25=300nt = 12 \times 25 = 300
  4. Calculate Monthly Payment: Next, we plug these values into the formula to calculate the monthly payment (PMT): PMT=300,000((1+0.005)300−1)/0.005\text{PMT} = \frac{300,000}{\left(\left(1 + 0.005\right)^{300} - 1\right) / 0.005}
  5. Calculate Numerator: Calculate the numerator of the formula: (1+0.005)300−1(1 + 0.005)^{300} - 1
  6. Evaluate Expression: Evaluate the expression:\newline(1+0.005)300≈4.291870(1 + 0.005)^{300} \approx 4.291870\newline4.291870−1≈3.2918704.291870 - 1 \approx 3.291870
  7. Calculate Denominator: Now, calculate the denominator of the formula: 0.0050.005
  8. Divide to Find PMT: Divide the numerator by the denominator to find the monthly payment: PMT=300,000(3.291870/0.005)PMT = \frac{300,000}{(3.291870 / 0.005)}
  9. Calculate Total Payments: Complete the division to find the PMT: PMT≈300,000658.374\text{PMT} \approx \frac{300,000}{658.374} PMT≈455.63\text{PMT} \approx 455.63
  10. Find Total Interest: For part b) of the problem, we need to calculate the total amount paid over the 2525 years and then subtract the future value to find the total interest earned.\newlineFirst, calculate the total amount paid:\newlineTotal payments = PMT×n×t\text{PMT} \times n \times t
  11. Correct Total Interest: Plug in the values to find the total payments:\newlineTotal payments = 455.63×300455.63 \times 300
  12. Subtract to Find Interest: Calculate the total payments: Total payments ≈136,689\approx 136,689
  13. Subtract to Find Interest: Calculate the total payments:\newlineTotal payments ≈136,689\approx 136,689Now, subtract the future value from the total payments to find the total interest earned:\newlineTotal interest = Total payments - FV
  14. Subtract to Find Interest: Calculate the total payments:\newlineTotal payments ≈136,689\approx 136,689Now, subtract the future value from the total payments to find the total interest earned:\newlineTotal interest = Total payments - FVCalculate the total interest:\newlineTotal interest = 136,689−300,000136,689 - 300,000
  15. Subtract to Find Interest: Calculate the total payments:\newlineTotal payments ≈136,689\approx 136,689Now, subtract the future value from the total payments to find the total interest earned:\newlineTotal interest = Total payments - FVCalculate the total interest:\newlineTotal interest = 136,689−300,000136,689 - 300,000Correct the calculation for the total interest:\newlineTotal interest = 300,000−136,689300,000 - 136,689
  16. Subtract to Find Interest: Calculate the total payments:\newlineTotal payments ≈136,689\approx 136,689Now, subtract the future value from the total payments to find the total interest earned:\newlineTotal interest = Total payments - FVCalculate the total interest:\newlineTotal interest = 136,689−300,000136,689 - 300,000Correct the calculation for the total interest:\newlineTotal interest = 300,000−136,689300,000 - 136,689Complete the subtraction to find the total interest:\newlineTotal interest ≈163,311\approx 163,311

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