To discuss either type of interest, we need to define some variables:<\/p>\n
A=the end amount: \u00a0the total value at time t<\/em><\/p>\n t=the elapsed time in years<\/em><\/p>\n r=the interest rate as a decimal (not a percent)<\/em><\/p>\n P=the principal amount (the amount of money deposited at the beginning)<\/em><\/p>\n I=the value of the interest earned<\/em><\/p>\n The simple interest on an investment is calculated as follows:<\/p>\n I=Prt<\/em><\/p>\n Of course, the total value includes the principal as well as the interest:<\/p>\n A=P + Prt<\/em><\/p>\n You can factor out P and get the other form:<\/p>\n A=P(1+rt)<\/em><\/p>\n Example 1: \u00a0calculate the value of a $5000 investment kept in the bank for 6 years at 3.2% simple interest.<\/strong><\/p>\n Solution: \u00a0First, we note the value of each variable given:<\/p>\n A=what we have to find<\/p>\n t=6 years<\/p>\n r=0.032 (to get the decimal, divide the percent by 100).<\/p>\n P=$5000, which is the amount invested.<\/p>\n Plugging into the formula gives us<\/p>\n A=5000(1+0.032(6))<\/p>\n We simplify to arrive at<\/p>\n A=5960<\/p>\n So, if we put $5000 in an account that pays simple interest of 3.2% and leave it in there for six years, the balance will be $5960.<\/p>\n To explain compound interest, we need to define compounding. \u00a0In financial math, compounding means taking the interest earned and adding it to the principal. \u00a0Once that interest is added to the principal, it can earn interest as well.<\/p>\n Hence the difference between simple interest and compound interest: \u00a0with simple interest, only the original deposit can earn interest. \u00a0With compound interest, the interest itself can earn interest.<\/p>\n With compound interest, people usually find the total value at the end, A, rather than the interest itself. \u00a0Of course,<\/p>\n I=A-P<\/p>\n To calculate the end amount, A, using compound interest, you need to know how many times per year the interest is compounded. \u00a0For today’s post, we’ll start with the easiest case: \u00a0annual compounding. \u00a0Then our formula for the end amount, A, after time t is<\/p>\n A=P(1+r)t<\/span><\/sup><\/p>\n Example 2: \u00a0Find the value of a $5000 investment kept in the bank for 6 years at 3.2% compounded annually.<\/strong><\/p>\n Solution:<\/p>\n A=5000(1+0.032)6<\/span><\/sup><\/p>\n From the calculator, we get<\/p>\n A=6040.16<\/p>\n So, if we leave $5000 in an account paying 3.2% compounded annually for six years, the balance at the end will be $6040.16.<\/p>\n Comparing Example 2 with Example 1, you see that with all else equal, the account paying compound interest grows faster than the one paying simple interest. \u00a0As time goes on, the difference gets more pronounced. \u00a0We’ll have more to say about that in a future post.<\/p>\n