Arithmetic and Facebook

One of my Facebook friends commented on this simple, apparently long-circulating, arithmetic problem and so it prompted many other of my Facebook acquaintances to also weigh in. The statistics of this FB post, as a whole, captured  my attention. With 516K votes (“likes” I suppose) and 5.5M comments that is almost too impressive.

I don’t think the interest in this question would have been so high if everyone had arrived at the same answer. Since some did not, from the few answers I’ve seen, it seems many were prompted not only to correct the errors, but to also provide detailed explanations on the order of precedence rules in arithmetic.

Now, I’m naturally skeptical and suspicious (not always a good trait) and so I could not help suspecting that some of the wrong answers presented with conviction were nothing more than “click bait” and perhaps led to the phenomenal response to this simple post.

As a tutor in chemistry and physics this discussion provoked some interesting thoughts …

1. In any math problem that involves a mathematical expression, the expression is a language that connects the person who set up and solved a math problem and someone else who uses the solution to find the correct answer to a similar problem. The conventions around the rules of precedence, that is to say: “multiplication and division must be done before addition and subtraction” are established so that the users of the equation understand how they are to perform the various operations correctly to get the right answer. If they are not followed, then using hyperbole, bridges will fall down, planes won’t leave the runway, and patients will received incorrect dosages. The rules involve shorthand (default rules that everyone is supposed to know) that make the expression as compact as possible.
   • since multiplication is done first, as most respondents noted, the expression simplifies to:
   • 50+50+0+2+2=? and so the answer is 104
   • If the person who set up the expression wanted a different outcome, then brackets would be used to change the order of operations … (50+50-25)x0+2+2=? … this answer would be 4
   • The conventions communicate from the person who set up the equation to the user. Like all conventions of this sort, they are only effective if we all agree to the same ones.
2. The second interesting point has to do with calculators. Depending on the calculator, a person who has not bothered to learn the order of precedence conventions can easily get the wrong answer. Using a calculator is no guarantee of accuracy.
   • For an older, simpler calculator that forces you to enter a number and an operation and another number to complete the operation, going from left to right will give the incorrect answer. By beginning at the left the user imposes an incorrect order of operation on the whole equation. You will in effect solve … (50+50-25)x0+2+2=?
   • A more sophisticated calculator that lets you enter the whole expression will follow the rules of precedence

Bottom line: having a calculator does not necessarily keep you from making mistakes if you don’t learn the rules.