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/