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### 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
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/
Associated Topics:
High School Basic Algebra
High School Linear Equations
Middle School Algebra
Middle School Equations
Middle School Word Problems

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