Two Quantities, Two Relations

Date: 04/04/2000 at 14:55:26
From: Danni*J
Subject: Find...?

Hi,

A few days ago my maths teacher set a homework called 'Find....' and 
it had 25 questions using a calculator, e.g. Find two consecutive 
numbers whose squares, when subtracted, make 42. Or something like 
that. 

I only managed 9 out of 25. Everyone else seemed to know what they 
were doing when they handed it in, which surprised me because I'm 
normally quite good at maths. Do you know why I didn't understand 
this?
Danni J.

Date: 04/04/2000 at 16:36:09
From: Doctor Ian
Subject: Re: Find...?

Hi Danni,

I would have a better chance of helping you if you would send me (1) 
some of the questions that you missed, along with (2) the steps you 
went through trying to answer them. 

For now, let's look at a question like the one you recall:

  Find two consecutive numbers whose squares, when subtracted, 
  equal 43. 

If you know that the numbers are consecutive, then if one of them is 
N, the other must be N+1. Do you see why this is true? 

So, now the question reads

  Find N such that (N+1)^2 - N^2 = 43.  

We can simplify that:

         (N+1)^2 - N^2 = 43
  (N^2 + 2N + 1) - N^2 = 43
    N^2 - N^2 + 2N + 1 = 43
                2N + 1 = 43
                    2N = 42
                     N = 21

To check this, we note that 22^2 = 484, and 21^2 = 441.  The 
difference between them is 43.  

So, what's going on here? In general, these questions will always want 
you to find the values of two quantities, and as clues they will give 
you two relations between the quantities. For example:

  Bill is two years older than Tom.  The sum of their ages
  is 18. How old are they? 

   - The first relation is that if Bill's age is B, and Tom's
     age is T, then T = B+2. The second relation is that 
     B + T = 18.  

  Sandy and Randy went bowling.  Randy's score was 10 pins higher
  than Sandy's.  The sum of their scores was 215.  What were their
  scores? 

    - The first relation is that if Sandy's score is S, and
      Randy's score is R, then R = S + 10. The second relation
      is that R + S = 216. 

  Spot's doghouse is two feet longer than it is wide. The area
  of the doghouse is 8 square feet. What are the dimensions of the
  doghouse? 

    - The first relation is that if the width is W and the 
      length is L, then L = W + 2. The second relation is that 
      W * L = 8.  

These are basically all the same problem, and they all have basically 
the same solution:

1) One of the relations will let you write the value of one quantity 
   in terms of the other, e.g., 

   T = B + 2
   R = S + 10
   L = W + 2

2) This means that you can substitute for one of the quantities in the 
   other relation, e.g., 

   B + T = 18       -->   B + (B + 2) = 18
   R + S = 215      -->  (S + 10) + S = 216
   W * L = 8        -->   W * (W + 2) = 8

3) Now this other relation is expressed in terms of one quantity, and 
   some numbers, so you should be able to solve for the value of 
   quantity, e.g.

   B + (B + 2) = 18
        2B + 2 = 18
            2B = 16
             B = 8

4) Now that you know the value of one quantity, you can go back to the 
   relation you identified in step (1) and use it to find the value 
   of the second quantity, e.g., 

   T = B + 2
     = 8 + 2
     = 10

I'll leave the other problems for you to finish.

The important thing to understand here is that on a test like the one 
you were given, you were really being asked to solve the same problem 
25 times. And except for the particular relations involved, they all 
have the same solution.  

I hope this helps.  Be sure to write back if you're still confused, or 
if you have any other questions. 

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