QuestionJuly 18, 2025

Suppose you wish to retire at the age of 60 with 80,000 in savings. Determine your monthly payment into an IRA if the APR is 6.5% compounded monthly and you begin making payments at 25 years old. Round your answer to the nearest cent, if necessary.

Suppose you wish to retire at the age of 60 with 80,000 in savings. Determine your monthly payment into an IRA if the APR is 6.5% compounded monthly and you begin making payments at 25 years old. Round your answer to the nearest cent, if necessary.
Suppose you wish to retire at the age of 60 with 80,000 in savings. Determine your monthly payment into an IRA if the APR is 6.5%  compounded monthly and you
begin making payments at 25 years old. Round your answer to the nearest cent, if necessary.

Solution
4.3(212 votes)

Answer

387.53 Explanation 1. Determine the number of payments Calculate the total number of monthly payments from age 25 to 60. This is 35 years, so 35 \times 12 = 420 payments. 2. Identify the future value formula for annuities Use the future value of an ordinary annuity formula: **FV = P \frac{(1 + r)^n - 1}{r}**, where FV is the future value, P is the monthly payment, r is the monthly interest rate, and n is the number of payments. 3. Calculate the monthly interest rate Convert the annual interest rate to a monthly rate: r = \frac{6.5\%}{12} = 0.0054167. 4. Solve for the monthly payment Rearrange the formula to solve for P: **P = \frac{FV \cdot r}{(1 + r)^n - 1}**. Substitute FV = 80000, r = 0.0054167, and n = 420 into the formula: \[ P = \frac{80000 \cdot 0.0054167}{(1 + 0.0054167)^{420} - 1} \] 5. Compute the monthly payment Calculate the value using the formula: \[ P = \frac{80000 \cdot 0.0054167}{(1.0054167)^{420} - 1} \approx 387.53 \]

Explanation

1. Determine the number of payments<br /> Calculate the total number of monthly payments from age 25 to 60. This is $35$ years, so $35 \times 12 = 420$ payments.<br /><br />2. Identify the future value formula for annuities<br /> Use the future value of an ordinary annuity formula: **$FV = P \frac{(1 + r)^n - 1}{r}$**, where $FV$ is the future value, $P$ is the monthly payment, $r$ is the monthly interest rate, and $n$ is the number of payments.<br /><br />3. Calculate the monthly interest rate<br /> Convert the annual interest rate to a monthly rate: $r = \frac{6.5\%}{12} = 0.0054167$.<br /><br />4. Solve for the monthly payment<br /> Rearrange the formula to solve for $P$: **$P = \frac{FV \cdot r}{(1 + r)^n - 1}$**.<br /> Substitute $FV = 80000$, $r = 0.0054167$, and $n = 420$ into the formula:<br />\[ <br />P = \frac{80000 \cdot 0.0054167}{(1 + 0.0054167)^{420} - 1}<br />\]<br /><br />5. Compute the monthly payment<br /> Calculate the value using the formula:<br />\[ <br />P = \frac{80000 \cdot 0.0054167}{(1.0054167)^{420} - 1} \approx 387.53<br />\]
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