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Find the Numbers

Date: 08/28/2003 at 22:05:14
From: Jen
Subject: Math Logic?

One number is 61 more than a second number. The sum of the two 
numbers is 127. Find the numbers.

I think you have to have an equation but I don't know how to set it 
up. I think it may be 61 + x = 127, but I am not sure.

Please help!


Date: 08/29/2003 at 09:19:29
From: Doctor Ian
Subject: Re: Math Logic?

Hi Jen,

When I don't know what else to do, I make a guess, and then try to
check it. Often, the process of checking can lead me to an equation.

Here's how that might work. I know I'm looking for a couple of 
numbers. What about 1 and 2?  Let's check that:

  1) One number is 61 more than a second number. 

Okay, so 1 and 2 aren't going to do it.

What this tells me, though, is that once I choose a first number, the
second number is determined: I just add 61 to the number I choose.  

So how about 1 and 62?  That will clearly meet the first test.  What
about the second? 

  2) The sum of the two numbers is 127.

The sum of 1 and 62 is 63, which is way too low.  But note that I can
write my sum this way:

               ?           <---- The '?' reminds me that the 
  1 + (1 + 61) = 127             equation might not be true!

If I choose a different starting number, like 10, I get

                 ?
  10 + (10 + 61) = 127

And at this point, there are a couple of ways we might go.  One would
be to get smarter about our guesses, using each guess to help us
figure out how much larger or smaller our next guess should be. 
Sometimes that's the smartest way to proceed, if checking is easy 
enough. 

But we can also look at our two equations, and notice that only one
thing is changing, which is the number that we choose as a guess. 
That's a clue that we should consider using an uknown to represent it.
Using 'g' to stand for 'guess', we could write

  g + (g + 61) = 127

Before rushing to solve this, we want to check it against the problem
description.  

  1) One number is 61 more than another.  Well, (g+61) is 
     clearly 61 more than g, whatever g turns out to be. 

  2) The sum of the numbers is 127.  We add the two numbers
     and get 127, so we're good to go. 

I find that letting an example or two lead me to an equation, and then
checking the equation against the problem statement, is often easier
than trying to 'translate' the problem directly into an equation. 

Let me note here that checking your equations this way is a good habit
to get into.  Let's look at the equation you came up with:

  61 + x = 127

You have two numbers that add up to 127, so you've nailed the second
condition. But what about the first?  Which of the two numbers is 61
more than the other?  61 is 61 more than x only when x is zero.  And x
is 61 more than 61 only when x is 122.  

It's easy to make mistakes like this when trying to go directly from
words to equations.  But as long as you're careful to check your
equations against the original description, you can catch errors like
this on your own. 

Anyway, so now we have an equation, which seems to represent the
problem statement, and which has a single uknown. Can you take it
from here?

I hope this helps. Write back if you have more questions, about this
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Equations
Middle School Word Problems

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