Order of Operations
Date: 05/19/99 at 13:54:24 From: Stephanie Wu and Meghan Heil Subject: Algebraic expressions and order of operation The problem was presented like this: a = 1.56 b = 1.2 x = 7.2 y = 0.2 ax/by = ? Here are two ways that I solved it: 1) I first rewrote the problem as [1.56(7.2)/ 1.2](0.2). Second, a was multiplied by x. The product was 11.232. Then, since no parentheses were present, I followed the order of operations and divided 11.232 by b, which was 1.2. The quotient was 9.36. Then I multiplied 9.36 by y, which was 0.2. The final answer was 1.872. 2) The other way, the first thing I did was multiply a by x. The product, which was 11.232, was set aside for the time being. Then b was multiplied by y, which gave the product of 0.24. The problem was now solved by dividing 11.232 (or ax) by 0.24 (or by) to reach a final answer of 46.8. Can you please tell us which answer is correct and why?
Date: 05/19/99 at 17:03:49 From: Doctor Peterson Subject: Re: Algebraic expressions and order of operation Hi, Stephanie and Meghan. You are not alone in wondering about this. We have had several other questions about expressions similar to yours, from confused teachers and students who have found that different books or teachers have different answers, and even calculators disagree. As written, your expression ax/by should be evaluated left to right: a times x, divided by b, times y. The multiplication is not done before the division, but both are done in the order they appear. Your first solution is right. Some texts make a rule, as in your second solution, that multiplication without a symbol ("implied multiplication") should be done before any other operations in an expression, including "explicit multiplication" using a symbol. Following this rule, you would multiply a by x, then multiply b and y, then divide one by the other. Some (probably most) texts don't mention such a rule - but some of those may use it without saying so, which is far worse. I don't know of a general rule among mathematicians that implied multiplication should be done before explicit multiplication. As far as I'm concerned, all multiplications fit in the same place in the order of operations. It's not an unreasonable rule, though, since it does seem that implied multiplication ties the operands together more tightly, at least visually; but the idea of Order of Operations (or precedence, as it is called in the computer world) is supposed to be to ensure that everyone will interpret an otherwise ambiguous expression the same way - so if some texts change the rules, or if people do what feels natural, the purpose has been lost. The problem here is that the expression looks as if it were meant to be ax ---- by In the Dr. Math FAQ about writing math in e-mail, one of our recommendations is to use parentheses wherever possible to avoid ambiguity, even where the rules should make it clear, because it can be easy to forget them in some situations: http://mathforum.org/dr.math/faq/faq.typing.math.html (click on the Fractions link). So in e-mail we would write it like this: ax/(by) or (ax/b)*y depending on what is intended. In my research for another Dr. Math "patient," I found that some calculators have experimented with this rule. Calculators have somewhat different needs than mathematicians, since they have to take input linearly, one character after another, so they are forced to make a decision about it. On the TI Web site I learned that they deliberately put this "feature" into the TI 82, and then took it out of the TI 83, probably because they decided it was not a standard rule and would confuse people. Take a look at their explanation: http://www.ti.com/calc/docs/faq/83faq039.htm They also talk about a similar issue for exponentiation of the form a^b^c, and give the same conclusion we give: always use parentheses where a statement is ambiguous without special rules: http://www.ti.com/calc/docs/faq/83faq058.htm So to answer your question, I think both answers can be considered right - which means, of course, that the question itself is wrong. I prefer the standard way (your first answer) when talking to students, unless their own text gives the "implicit multiplication first" rule; but in practice if I came across that expression, I would probably first check where it came from to see if I could tell what was intended. The main lesson to learn is not which rule to follow, but how to avoid ambiguity in what you write yourself. Don't give other people this kind of trouble. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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