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 
convenient, but the answer will not be exact.  (Your answer will only 
be as accurate as you can read your graph.)
 
-Doctor Barney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/