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Finding Two Numbers with Known Product and Difference

Date: 08/06/2008 at 02:50:40
From: David
Subject: find two numbers

Two numbers have a product of 6800 and a difference of 5.  Find the
numbers.  How can I solve it?

Date: 08/06/2008 at 22:56:18
From: Doctor Peterson
Subject: Re: find two numbers

Hi, David.

If you knew enough algebra, you could write an equation, which turns
out to be quadratic, and use the quadratic formula to solve it.  I'll
assume that what I just said means nothing to you!

This kind of problem comes up early in the study of algebra, before
you can use that technique, though usually the product is smaller.
There, the usual method taught is trial and error.  You would make a
list of all factor pairs whose product is 6800, and look for a pair
whose difference is 5.  It would start like this:

  1 * 6800: difference is 6799
  2 * 3400: difference is 3398
  4 * 1700: difference is 1696
  5 * 1360: difference is 1355

Notice that the difference started at 6799 and went down very slowly;
the pair we're looking for will be just about the last one we try! 
You wouldn't actually start calculating the difference until you got
much closer.

There is a trick that can make the search a little quicker.

Suppose the difference had been 0 instead of 5. (Since we've seen that
most of the differences are a lot bigger, this will be a lot closer to
our answer!)  Then we would be looking for a number whose square is
6800.  (That is, multiplying it by itself gives 6800.)  If you can use
a calculator, you can find that number; it's the square root of 6800,
and is not a whole number.  If you can't use a calculator (not
available, or not allowed), then you can estimate: 8 squared is 64, so
80 squared is 6400, and that's pretty close; our number must be a bit
more than 80.  Certainly 90 would be too large.

Now, this gives us a better place to start than 1 in my list above.
We'll have one number less than the square root of 6800, and the other
greater.  You might just try dividing 6800 by 80 to start with, since
we know that's not much less than the root; if the quotient is exactly
5 more, then you've got the answer.  If not, try a smaller number (or
maybe larger).

Another thing that might help is to note that since 6800 is a multiple
of 5, at least one of the two numbers must be a multiple of 5.  That
means it must end in 5 or 0!  Actually, that's why I figured that a
round number like 80 would be a good guess to start with; and I would
move by 5's from there.

There are all sorts of bits of information about numbers that can be
useful in solving a puzzle like this.  This is one reason why you are
given lots of experience multiplying and dividing in the early grades:
so you can be familiar enough with how numbers work (such as the
"ending in 5" trick) that you can use them as needed to solve new

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 

Date: 08/07/2008 at 02:27:42
From: David
Subject: Thank you (find two numbers)

Thank you very much for your help.  I was really struggling with that
problem.  Mama solved it doing algebra but without using the formula
she was also blank.  I am home-schooled in Abu Dhabi and do not have
any groups.  For us Doctors like you and sites like these are a great
help.  I will definitely come back with more.  Thank you once again.

Associated Topics:
Middle School Factoring Numbers
Middle School Word Problems

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