A) $4000 B.) $5000 C). $6000
A complete answer to the general question about abstract reasoning would require many pages of discussion, so I will focus only on one tiny example. In the US election of 2024 it became apparent that one of the biggest challenges facing the average American was financial–paying for groceries and meeting financial obligations. With escalating costs, the purchasing power of the dollar drops, savings are depleted and the prospect of owning a home recedes into a more distant future. Understanding how to emerge from this conundrum requires an understanding of some abstract concepts such as exponential growth and decay, the rule of 72 and the distinction between expenditures and investments.
Exponential Growth
Einstein is reputed to have said, “Compound interest is the 8th wonder of the world,” 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.
The Distinction between Expenditures and Investments
The discussion above shows that the banks are on the winning side of these transactions because you are spending while they are lending you money (investing). People who understand this abstract concept of the difference between expenditures and investments often postpone the short term pleasure of updating their car or kitchen and invest their money in an asset with growth potential. Of course, we all have to spend, but prudent monitoring of expenditures pays huge long-term benefits.
Perhaps, The Greatest Benefit of Abstraction
Abstraction is the jewel in the crown of human intelligence. It has enabled us to move to the top of the food chain and live relatively comfortable lives. But its greatest benefit may be its ability to enable us to view this exciting universe through the lens of general conceptual understanding whereby daily, minuscule observations can be gathered together and generalized into meaningful concepts that enrich our understanding of who we are.
In a later post, I will suggest methods for increasing your ability to think in the abstract.
