There are many great mathematical concepts that are relevant to life, but it’s often difficult to help people understand why they are relevant. However, I have found that most people agree on the relevance of money, so I will provide one example that gets the attention of most people, and that is exponential growth.
Einstein is reputed to have said, “Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn’t, pays it,” and while this quote may be falsely attributed to him, it captures the fact that compounding small investments over a period of decades is a powerful method for building wealth. Years ago, I traveled across the country working with students of all ages to introduce them to financial and mathematical literacy. To a class of students at the junior high level, I would propose the following:
Would you sign a no-cut contract to cut lawns for 28 days that pays you 1¢ the first day, 2¢ the second day, 4¢ the third day, and so on, doubling each day the amount paid on the previous day?
The students in overwhelming numbers would give a resounding “NO!” Then we would make a list showing how much they earn in the first 7 days: $0.01 + $0.02 + $0.04 + $0.08 + $0.16 + $0.32 + $0.64 = $1.27. The students would unanimously agree that the contract is a “rip-off.” We would then continue for another week and find that the payment for the 14th day is $81.92. At this point, I have the students’ attention and many of them are beginning to revise their opinions. By the 21st day, the payment is now $10,485.76– just for that day. The students cheer as they watch us list the payment for the 28th day at $1,342,177.28. The total payment for 28 days at $2,684,354.55. We then abstract to show how we can calculate the payment on the nth day as 2^n cents and the total payment after n days as 2^(n + 1) – 1 cents.
So what good is this abstraction? An understanding of compound interest enables people to see the importance of investing money early and letting it grow. It also paves the way for the next concept.
The Rule of 72
The rule of 72 is a way of estimating how long it takes money to double when invested at various interest rates. You merely divide 72 by the interest rate to get the doubling time. So, $1000 invested at 8% per annum takes about 72 ÷ 8 or 9 years to double in value to $2000. Money invested at 24% per annum doubles in about 72 ÷ 24 or 3 years. This means that if you carry a balance of $1000 on your credit card at 24% per annum, you will accrue almost $1000 in interest (exactly $1000(1.24)^3 – $1000 = $906.62) in addition to the amount of your purchases. That’s why banks encourage people to carry balances on their credit cards and why bank stocks are considered by many investment managers as a necessary part of an investment portfolio.
If we could help people to understand these concepts when they are young, we could save a lot of people from economic hardship during middle age. It’s one of the many concepts in mathematics that is invaluable to living a productive and comfortable life.
