Ex 1:  Compounded Interest Formula - Quarterly

A bank features a savings account that has an annual percentage rate of three point four percent with interest compounded quarterly. Ashley deposits two thousand dollars into the account. We first want to know what equation can be used to determine the account balance after two years, and then what the account balance would be after four years. The compounded interest formula is given here below, where P is the principal or starting amount, R is the annual interest rate expressed as a decimal, N is the number of compounds per year. Notice it occurs here and here. So in this case, since the interest is compounded quarterly, and there are four quarters in one year, N would be equal to four. If it was monthly, N would be twelve. T is time in years, and A is the amount after time T. So for this situation, we'd have the equation A equals the starting amount, which is two thousand dollars, times the quantity one plus the annual interest rate expressed as a decimal. Three point four percent as a decimal would be zero point zero three four. We drop the percent symbol and divide by one hundred, or put the decimal point to the left two places, divided by N, the number of compounds per year. It's compounded quarterly, and there are four quarters per year, so N is four. The exponent is going to be N times T. Again, N is four, so we have four T as our exponent. So notice by dividing the annual interest rate by four, this would be the quarterly interest rate. Then multiplying four times the number of years, this exponent would be the number of quarters. Now let's go ahead and simplify this. We'd have A equals two thousand, times one plus zero point zero three four, divided by four. This is going to be one plus zero point zero zero eight five, so we have one point zero zero eight five, raised to the power of four T. So this would be the equation that would give us the account balance after T years. So if we want to know the account balance after four years, we're just going to substitute four for T. So the amount of money in the account would be two thousand, times one plus zero zero eight five, raised to the four times four power, that'd be the sixteenth power. Let's go ahead and determine this value. We'll round to the nearest cent. So we'll have two thousand, times one point zero zero eight five, raised to the power of four times four, or sixteen. So the account balance would be two thousand, two hundred ninety dollars, and five cents.