MathChat

In the first “Slaying Proportions” post, you learned how to save time by canceling horizontally as well as vertically. In this post you’ll learn how to save even more time with another shortcut. Let’s look at an example to refresh our memory.

Given a proportion such as this:

15   =   5
 a         3

most people would do the traditional “cross-multiplying” step, to get:

5 x a = 15 x 3  (the x here is a times sign)

Using the usual steps, the next step is to ÷ both sides by 5, to get:

a  =  (15 x 3) ÷ 5

But let’s look more closely at this answer expression:   (15 x 3) ÷ 5

We can view this expression more clearly if we see he original proportion:

15   =  5   
 a        3

as containing two DIAGONALS.

One diagonal holds the 15 and the 3; the other holds the ‘a’ and the 5.

Let’s call the diagonal with the ‘a’ the ‘first diagonal.’ And since ‘5’ accompanies ‘a’  in that diagonal, we’ll call 5 the “variable’s partner.”

We’ll call the other diagonal just that, the “other diagonal.”

Now I know you’re getting ‘antsy’ for the shortcut, so just know it’s right around “the bend.”

Using our new terms, we can better understand the expression we got up above:

a = (15 x 3) ÷ 5

The (15 x 3) is the product (result of multiplication) of the “other diagonal,”
and ‘5’ is the “variable’s partner.

So the answer,

                                      (15 x 3)                     ÷              5

is simply (and here’s the shortcut):

         (product of other diagonal) ÷ by  (“variable’s partner.”)

We’ll call this the Diagonal Trick.

The Diagonal Trick saves a BIG STEP. Using it, we no longer need to write out the cross-multiplication product the usual way, as:

5 x a = 15 x 3

Instead, using the Diagonal Trick, we can go straight from the proportion to an expression for ‘a‘:

a  =  (15 x 3) ÷ 5

Let’s see how the Diagonal Trick works in another proportion, such as:

 9    =   45  
13         a

What’s the “variable’s partner”?  9.
What’s in the “other diagonal”? 13 and 45.

So using the Diagonal Trick, the answer is this:

a  =  (13 x 45) ÷ 9

This simplifies to 65, of course. Isn’t it nice not to have to “cross-multiply” any more?

Another nice thing: the Diagonal Trick works no matter where the variable is located in the original proportion. All you need to do is identify the “variable’s partner,” and the “other diagonal,” and then you’re all good go with the Diagonal Trick.

Try a few of these to see how easy and convenient the Diagonal Trick makes it to solve proportions.

PROBLEMS:

1)   a   =      15  
     12          36

2)   18   =    a  
      24         4

3)   21   =   75  
      14          a

ANSWERS (using the Diagonal Trick first):

1)   a  =  (12 x 15) ÷ 36
  a  =  5

2)   a  =  (18 x 4) ÷ 24
      a  =  3

3)   a  =  (14 x 75) ÷ 21
      a  =  50