Suppose you wish to retire at the age of 60 with \( \$ 70,000 \) 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. Formulas

To solve this savings goal, you can use the future value of an annuity formula, which is: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Here, \( FV \) is the future value of the annuity, \( P \) is the monthly payment, \( r \) is the monthly interest rate, and \( n \) is the total number of payments. If you want to retire with \( \$ 70,000 \) at \( 60 \) years old and start saving at \( 25 \), you'll be saving for \( 35 \) years, which is \( 35 \times 12 = 420 \) months. The annual interest rate of \( 6.5\% \) translates to a monthly rate of: \[ r = \frac{6.5\%}{12} = 0.00541667 \] Now plug in the values: \[ 70,000 = P \times \frac{(1 + 0.00541667)^{420} - 1}{0.00541667} \] Calculate \( (1 + 0.00541667)^{420} \), which is approximately \( 6.96006 \). Thus: \[ 70,000 = P \times \frac{6.96006 - 1}{0.00541667} \] \[ 70,000 = P \times \frac{5.96006}{0.00541667} \] \[ 70,000 = P \times 109.31875 \] \[ P \approx \frac{70,000}{109.31875} \approx 639.62 \] So your monthly payment into the IRA would be approximately \( \$ 639.62 \). --- Knowing that the concept of compound interest can feel like magic, let’s appreciate its historical roots. Compounding interest has been employed since ancient civilizations, with the Babylonians, Greeks, and even Roman times utilizing early forms of interest calculation. They understood this financial principle long before it became a fixture of modern finance—essentially creating the building blocks of economics as we know it! In today's world, putting money into an IRA allows your investments to grow tax-deferred! This means you won’t pay taxes on your earnings until you withdraw them in retirement, giving your money the chance to grow significantly over time. This is particularly beneficial for long-term savers, as it compounds more over the years—cumulatively leading to wealth accumulation. So, start early, save consistently, and enjoy watching your financial garden flourish!