Number Puzzle: 26, 70

Date: 09/12/2001 at 01:45:44
From: Bryce  Aspell
Subject: Word problem

The sum of two numbers is 26; 4 times the first number plus 2 times 
the second number equals 70. What are the two numbers?

Date: 09/12/2001 at 14:54:52
From: Doctor Ian
Subject: Re: Word problem

Hi Bryce,

Just for fun, let's try guessing that the two numbers are 12 and 14.  

Is this the answer?  Well, 

  12 + 14 = 26              (good)

  4*12 + 2*14 = 48 + 28 

              = 76          (too big)

We overshot on the second criterion. How can we bring that down a 
little?  

Well, notice that the first number gets multiplied by 4, while the 
second number gets multiplied by only 2. So if we can make the first 
number a little smaller, maybe we can get the second number down to 
70:

How about 10 and 16?

  10 + 16 = 26             (good)

  4*10 + 2*16 = 40 + 32

              = 72         (still too big)

We still overshot by a little. But we're clearly going in the right 
direction. What about 8 and 18?

  8 + 18 = 26              (good)

  4*8 + 2*18 = 32 + 36

             = 68          (too small)

So the first number has to be a little bigger. That's good! It means 
that the answer has to be somewhere in between the numbers we've 
tried. 

You can keep going this way, or you can try to solve the problem 
directly, by translating it into equations.  Let's try that:

  1.  The sum of two numbers is 26:

      If one of the numbers is N, then the other one has to be 26-N.
      Do you see why? 

  2. Four times the first number (4 * N) plus two times the
     second number (2*(26-N) equals 70:

         4N + 2(26-N) = 70

Now you can try various values for N:

      N=5  ->   4(5) + 2(26 - 5)   = 20 + 42 = 62       (too low)

      N=6  ->   4(6) + 2(26 - 6)   = 24 + 40 = 64       (too low)

         [the answer has to be in here somewhere]

      N=10 ->   4(10) + 2(26 - 10) = 40 + 32 = 72       (too big)

Or you can use algebra to solve the equation directly:

         4N + 2(26 - N) = 70

      4N + 2(26) - 2(N) = 70

           4N - 2N + 52 = 70

                     2N = 70 - 52

and so on. 

There are often lots of different ways that you can solve any given 
problem, so it's good to try to be aware of your options, and to keep 
in mind that _any_ method that helps you solve the problem quickly is 
the 'right' method... even if it's guessing, or making a graph, or 
counting on your fingers, or moving pennies around on a desk. 

As W.W. Sawyer wrote in his excellent book, _Vision in Elementary 
Mathematics_, 

  Pupils ask 'Am I allowed to do this?' as if we were playing a game
  with certain rules. A pupil is allowed to write anything that
  is true, and not allowed to write anything untrue! These are 
  the only rules of mathematics. 

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions.                               

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/