To determine the annual contributions Derek must make to his retirement account from his 26th birthday to his 65th birthday, we need to calculate how much he needs to contribute each year to reach his goal of $3,482,485 by the end of the year he turns 73.
Steps to Solve:
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Understand the Time Frames:
- Derek will contribute from age 26 to age 65 (for a total of 40 years).
- After age 65, he will not contribute until he turns 73.
- This means there are 8 years (from age 65 to age 73) during which the accumulated contributions will earn interest but will not be added to.
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Future Value of Contributions:
- We can use the Future Value of an Ordinary Annuity formula to calculate how much Derek needs to contribute annually.
The formula for the future value of an ordinary annuity is: \[ FV = C \times \frac{(1 + r)^n - 1}{r} \] where:
- \(FV\) = future value of the retirement account at age 73 ($3,482,485)
- \(C\) = annual contribution amount
- \(r\) = annual interest rate (0.07)
- \(n\) = number of contributions (40 years)
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Calculate Future Value at Age 72:
- First, calculate how much the amount at age 72 will be after the 8 years of interest earned until age 73.
- We need to "discount" $3,482,485 back to age 72, so we use the formula: \[ FV_{72} = \frac{FV}{(1 + r)^n} = \frac{3482485}{(1 + 0.07)^1} \]
- Calculate \(FV_{72}\): \[ FV_{72} = \frac{3482485}{1.07} \approx 3258247.89 \]
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Use the Future Value of Annuity Formula:
- Now, use the future value at age 72 to find \(C\): \[ 3258247.89 = C \times \frac{(1 + 0.07)^{40} - 1}{0.07} \]
- Calculate \((1 + 0.07)^{40}\): \[ (1 + 0.07)^{40} \approx 14.9748 \]
- Substitute in: \[ 3258247.89 = C \times \frac{14.9748 - 1}{0.07} \]
- Calculate the fraction: \[ \frac{13.9748}{0.07} \approx 199.640 \]
- Now we have: \[ 3258247.89 = C \times 199.640 \]
- Solve for \(C\): \[ C = \frac{3258247.89}{199.640} \approx 16320.12 \]
Conclusion:
Derek must make annual contributions of approximately $16,320.12 from his 26th birthday until his 65th birthday to reach his goal of $3,482,485 in his retirement account by age 73.
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