The Difference of x and y...Date: 06/05/2003 at 20:55:25 From: Matt Subject: What does: "The difference of x and y..." mean? What does "the difference of x and y" mean? I pretty sure it means x - y. However, I have a problem with a word problem. It is: "the difference of a number and its square is 42" (actually, it wasn't 42, it was 100 something, but that shouldn't affect anything). It's not difficult or confusing, but I do have a disagreement. A friend says that the equation is x^2 - x = 42, so the number is 7 or -6. I disagree. I think the equation should be x - x^2 = 42, which would be no solution because the equation can be rewritten as x^2 - x = -42, which isn't likely to end as a real number solution. Who's correct? Date: 06/05/2003 at 23:16:21 From: Doctor Peterson Subject: Re: What does: "The difference of x and y..." mean? Hi, Matt. I myself would say "the difference between ..." not "the difference of ..." But that doesn't affect the meaning. The important thing is that a difference is always positive, regardless of which number is larger; the order in which the numbers are given need not be larger to smaller. I would translate the phrase as |x-y|, the absolute value; if I knew which was larger, I could just use x-y or y-x. In your example, the equation would be |x - x^2| = 42 To solve this, we need two cases. First, we suppose that x - x^2 >= 0, and solve x - x^2 = 42 x^2 - x + 42 = 0 which has no (real) solution, since the discriminant is negative. Then we suppose that x - x^2 < 0, and solve x^2 - x = 42 x^2 - x + 42 = 0 (x - 7)(x + 6) = 0 x = 7 or x = -6 Now we have to check that in fact x^2 > x for these cases; not surprisingly, it is. So your friend's solution is correct, though not quite thoroughly supported. To check the answers against the original question (which is the ultimate basis for calling an answer correct), ask yourself, "Is the difference between 7 and 49 equal to 42? How about the difference between -6 and 36?" I think you'll say that they are. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/28/2006 at 23:28:19 From: Mark Subject: Meaning of "difference between" Recently I helped my eighth grade son with his algebra. He was to write out and simplify the following word problem: -54 decreased by the difference between -37 and 15 I wondered about the sign of the difference part of the equation and didn't know if it should be -54 - (-37 - 15) or -54 - (15 - -37) I found your posting here and advised him to solve it as: -54 - |-37 - 15| His teacher insists that it should be -54 - (-37 - 15) and that "difference between" should be translated to math as straight subtraction. All other web sites I have found seem to support her position as this seems to be the common practice. Can you comment on this issue? Date: 08/29/2006 at 09:51:33 From: Doctor Peterson Subject: Re: Meaning of Hi, Mark. I would really call it ambiguous; that happens often in English. In this case, however, I think part of the ambiguity has been introduced by educators who want to make an easy problem out of a tricky one by pretending that English is under their control. In any real situation I can think of, if you were to ask someone for the difference between, say, 3 and 5, the answer would be 2 -- not 3 - 5 = -2! We tend to think of differences as positive numbers, and thus the proper rendition of that expression algebraically would be |3-5| (or |5-3|). This is the same idea as the "distance" between two numbers on the number line, which is always positive. In textbooks, as you've observed, it seems common in "word problems" to make a different convention, that "the difference between a and b" means a - b. My guess is that they do that because an absolute value equation is beyond the students they are usually presenting this to, and they want to promote the illusion that everything you can write in words can be translated to algebra by a simple operation. This convention does make some sense: when we do a subtraction, such as 3 - 5, we call the answer the "difference". So from the perspective of a textbook author, who has mostly been writing math problems rather than real-world problems, this is a natural way to interpret the question. The problem is just that you have to have that context in mind, rather than how the language is really used outside the class; and in some application problems, it's hard to tell which context should be in view! I've seen some texts use "difference of" rather than "difference between", which in my mind carries a little less of the real-world sense. (Oddly, in the page you refer to, the question WAS written that way, yet appears to have had the opposite order in view!) So in the classroom context, and especially if the text has explicitly stated what they mean by difference, you just have to go along with it. But if I were grading a problem and a student chose to use an absolute value, I would not mark it wrong -- I would just point out that, in order to make subsequent problems in the text doable, it would be prudent to adopt their convention for the time being. Here's a related thread from our archives: Interpreting the Difference Between Two Numbers http://mathforum.org/library/drmath/view/69177.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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