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### Two Quantities, Two Relations

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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?

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/
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Associated Topics:
Middle School Puzzles
Middle School Word Problems

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