Benjamin wants to save 54000 dollars for a down payment on a home. How much will he need to invest in an account with 8.2 percent APR, compounding daily, in order to reach his goal in five years?
This is a compound interest problem. And, to find the amount at the end of a certain period, you take 𝑃, your original investment amount. And you multiply that by one plus the rate — that’s your annual interest rate, your APR — divided by 𝑛, the number of times per year it’s compounded. And that value is taken to the power of 𝑛, the number of times it’s compounded. These two values, these two 𝑛 variables are the same number. And then you multiply that 𝑛 exponent by 𝑡, the time we’re considering. At the end of the time, Benjamin needs 54000 dollars.
We don’t know what 𝑃 is. That’s what we’re trying to find, how much he should invest in the beginning. We know that the interest rate is 8.2 percent. But we want to write that as a decimal, 0.082. This rate is compounded daily. How many times is it compounded in one year? 365 times. And that means we need 365 times for the exponent multiplied by five. Benjamin wants to reach his goal in five years.
Now, at this point, it’s important to say that the best way to solve this, the most precise way to solve this, is by keeping each step in your calculator. Starting with 0.082 divided by 365, we’ll enter that value into the calculator and then, without changing anything on the calculator, we’ll add one. On most calculators, this will mean, it will say answer plus one. And now we need to take the result here, 1.000224658, and take that value to the 365 times five power. When we do that, we see that we’re taking the answer we already got and then the 365 times five power.
This allows us to wait to round until the very last step. When we hit enter, we find out that the value is 1.5067484. 54000 has to equal the original investment 𝑃 times 1.5067484. And to find out what 𝑃 is, we need to divide both sides of the equation by the result 1.5067484. The best way to do that will be to enter 54000 divided by. And then find the button that lets you use the answer from the previous line. So it says 54000 divided by the answer. And it’s dividing 54000 by what we had in the previous line. And the result is 35838.76379. The 𝑃, the amount he needs to invest, is this.
We know we’re dealing with money. So we want to round to the hundredths place. The digit to the right of the hundredths place is less than five. So we’ll round it down. And we’ll say that Benjamin needs to invest 35838 dollars and 76 cents in order to reach his goal of 54000 dollars in five years.