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### Stick Problem

```
Date: 05/13/97 at 07:19:38
From: The Khoos
Subject: Sticks

Dear Dr. Math,

How do you solve this problem?

Stick A is 8 cm longer than Stick B. Two thirds the length of Stick B
is equal to half the length of Stick A. What is the length of Stick B?
```

```
Date: 07/03/97 at 10:46:52
From: Doctor Barney
Subject: Re: Sticks

Sometimes the trick to solving word problems like this is to select
the right variables to solve for. Since the question asks for the
length of stick B, let's let that be one of our variables: I'll call
it simply B.

Let B = the length of stick B.

Since the problem gives us some information about the length of
stick A, let's use the letter A in our equations to represent the
length of stick A:

Let A = the length of stick A.

Now, the first sentence tells us that stick A is 8 cm longer than
stick B, and if we write the first sentence in equation form using the
two variables we have defined, it looks like this:

A = 8 + B     equation (1)

I am calling this equation (1) so that I can refer to it later.  Next,
to use the information that two thirds the length of Stick B is equal
to half the length of Stick A. We will write this in equation form
using the same two variables:

2/3 x B = 1/2 x A     equation (2)

Can you see how these equations relate to the first two sentences in
the word problem? Since we have two equations and only two unknowns
(A and B), we can find numbers for A and B. The method is to
eliminate one of the variables from one of the equations.  In this
case, since equation (1) already has an expression for A, we can
substitute the expression on the right side of equation (1) into
equation (2) every place where there is an A (in this case there is
only one place).

In other words, since A is equal to 8 + B,  8 + B is another way of
expressing A.  Knowing this, we can rewrite equation (2) as:

2/3 x B = 1/2 x (8 + B)    equation (3)

Next we will manipulate equation (3) until we get B all by itself on
one side of the equation, as follows:

2/3 x B = (1/2 x 8) + (1/2 x B)
2/3 x B = 4 + (1/2 x B)
(2/3 x B) - (1/2 x B) = 4
(2/3 - 1/2) x B = 4
1/6 x B = 4
B = 24 cm

Since B is the length of stick B, that is the answer to the question.
Since the length given in the problem was in cm., the answer is also
in cm. If you want to find the length of stick A, you can put 24 cm
into equation (1) in place of B, since we now know what B is. Go
ahead and try it, and then read the word problem again and see if the
words are true for the numbers we have found.

Another way to solve this problem is to plot equations (1) and (2) on
a graph (A on one axis and B on the other). The answer will be the
point where the two lines intersect.  Sometimes this is more

-Doctor Barney,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Word Problems

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