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in solving the means extremes, I get lost where you "divide each side by 2". I don't get what gets divide by what.

example:  1/2 = x/6.    what is x

  1.  multiply the means (2 times x)  and multiply the extremes (1 times 6)---  this I understand

  2.  then the equation is 2x equals 6.---- this I understand

  3.   divide both sides by  2  (2x / 2  = 6 / 2)----Why divide by 2?

  4.   if 2x  is divided by 2,  doesn't that equal one ?

  5.   and if 6 is divided by 2,  that equals 3---this I understand   

  6.  Why isn't the answer 1/3?   I know the answer to x is 3 just by looking at the problem,  but I don't understand the dividing in step 3?   Why do you divide both sides by 2?  Why not 3? Or 4?  Explain.

Just last year at the age of 57 I found out I have ADD.  No wonder math was always SO HARD for me.




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2 Answers

You have the following problem:          1/2 = x/6

This is a proportion equation, in which you are comparing 2 ratios ( 1/2 and x/6 ). Since the ratios you are comparing here are fractions, your goal is to make them equivalent by solving for the unknown variable (x).

1.) You know that the means are 2 and x, and the extremes are 1 and 6. We use these means and extremes to solve for the proportion. Properties of algebra tell us that the product of the means (2*x) is equal to the product of the extremes (1*6).

2.) From the above info, you have found that

            2 * x = 1 * 6       ==>       2x = 6

3.)    2x = 6  :    Since our goal is to solve for the proportion by solving for x, we want to isolate x to one side of the equation (that is, to have the equation in the form of " x =  "). This is why we divide both sides of the equation by 2....dividing the left side of the equation (2x) by 2 we get:

          2x / 2 = 1x = x

...dividing the right side of the equation (6) by 2 we get:

          6 / 2 = 3

So,      2x / 2 = 6 / 2    ==>    1x = 3    ==>    x = 3

4.) No, 2x divided by 2 is not 1 ==>    (2*x) / 2 = (2/2)*x = 1x   .... don't know the value of x yet, which is why you are trying to solve for it. You are dividing both sides of the equation by 2 to get rid of the 2 on the left hand side of the equation so as to solve for x.

5.) Yes, 6 divided by 3 is equal to 2.

Let's summarize what we did:

       1/2 = x/6

       2*x = 1*6    ==>    2x = 6

       2x / 2 = 6 / 2    ==>    1x = 3    ==>    x = 3

Thus, the answer, to solve for the proportion by making the fractions equivalent, is x = 3.

Check to see that the answer is correct:

        1/2 = x/6          plug in 3 for x in the equation 

        1/2 = 3/6

        1/2 = 1/2         CORRECT!

In step 3, we we can do the same thing to both sides of the equation and the equation will stay true. So we can divide both sides by 2, which we choose because we want to cancel the 2 in 2x.

In step 4, 2x/2 = x, not 1. Think about it, the 2's cancel but the x does not. More rigorously, multiplication is commutative, so 2x/2 can be written as x*2/2 = x*1 = x.


I do not understand what "commutative" means.  The equation following that word does not make sense to me.  I am a college grad and an LPN but when dealing with math it is better to communicate with me in terms a 4th grader would understand

Commutativity is a property of an operation where the answer never changes if you swap the operands. Examples:

Addition is commutative: a+b = b+a

Multiplication is commutative: ab = ba

Division and Subtraction are not:

a/b and b/a are not always equal, and neither are a-b and b-a.