Parents wish to have \( \$ 150,000 \) available for a child's education. If the child is now 7 years old, how much money must be set aside at \( 4 \% \) compounded semiannually to meet their financial goal when the child is 18 ?
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To find out how much money must be set aside now, we can use the formula for the present value of a future sum: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \(PV\) is the present value (amount to set aside now) - \(FV\) is the future value (\$150,000) - \(r\) is the annual interest rate (0.04) - \(n\) is the number of times the interest is compounded per year (2 for semiannually) - \(t\) is the number of years until the goal (11 years, since the child is currently 7 and will be 18 in 11 years) Substituting the values into the formula: \[ PV = \frac{150,000}{(1 + 0.04/2)^{2 \cdot 11}} = \frac{150,000}{(1 + 0.02)^{22}} = \frac{150,000}{(1.02)^{22}} \] Calculating \((1.02)^{22}\): \[ (1.02)^{22} \approx 1.485947 \] Now substituting back into the formula: \[ PV = \frac{150,000}{1.485947} \approx 100,893.56 \] So, the parents must set aside approximately **$100,893.56** today to have **$150,000** available for their child's education when he or she turns 18. --- To make it easier to understand, think of saving this amount as planting a money tree! You water it (by investing your initial amount) and let it grow over time with interest. By the time your kiddo heads off to college, you'll be sitting pretty under the shade of that tree with all the funds you need! Also, don’t forget to account for tuition increases! Always round up in your planning. Consider inflation, schools are getting pricier, and what’s $150,000 today may not stretch as far in 11 years. So, keep adding a little cushion to your savings plan!
