Kiss those Math Headaches GOODBYE!

Posts tagged ‘Solving Proportions’

Conquering Proportions, Part 2

In my first “Conquering Proportions” post, I showed how to save time by canceling terms 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 true times sign; that’s why I’m using ‘a‘ as the variable, not ‘x.’)

If you follow the usual steps, the next thing would be 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 conceptualize this expression better if we think of the original proportion:

15   =  5   
 a        3

as containing two DIAGONALS.

One diagonal holds the 15 and the 3; the other diagonal 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 Proportion Shortcut Formula, or the PSF, for short.

The PSF 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 PSF, we can go straight from the proportion to an expression for ‘a‘:

a  =  (15 x 3) ÷ 5

Let’s see how the PSF 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 PSF, 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 PSF 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 PSF.

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


1)   a   =      15  
     12          36

2)   18   =    a  
      24         4

3)   21   =   75  
      14          a

ANSWERS (using the PSF 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


Conquering Proportions, Part 1

Proportions can seem intimidating, but actually they are one of the easiest kinds of word problems to solve. In this series I’ll offer a number of tips that help you conquer this type of algebra word problem. Search “Conquering Proportions” to see the whole series.

First, a cool shortcut that you can use whenever you have to solve an algebraic proportion …

In working with algebraic proportions, I’m amazed to see that few students know how to do a certain kind of canceling that allows them to find the solution more quickly and efficiently.

So I thought I’d share the trick, for anyone who has never seen it.

Of course, given a problem like:  6/x  =  24/32

we all know that we can cancel vertically with the two numbers in the fraction on the right, to get:

6/x  =  3/4

Then you can cross-multiply to get:

3x  =  24, and then x  =  8

In other words, everyone knows that you can cancel vertically when you have a proportion, just as you can cancel vertically for any fraction.

What not everyone realizes is that there’s another way you can cancel when solving proportions — horizontally.

— What? you say.

— Horizontally, I say. And no, I’m not joshing.

For example,  in the problem:  7/4  =  21/x

you can cancel horizontally with the two numbers in the numerator: the 7 and the 21. These reduce to 1 and 3.

The problem then becomes:

1/4  =  3/x

Cross-multiplying, you get the answer in one quick step:   x = 12.

What’s really convenient is that you can also cancel both vertically and horizontally in the same problem. For example, in

6/x  =  42/28

you could first cancel horizontally, to get:

1/x  = 7/28

Then you can cancel vertically, to get:

1/x  =  1/4

Cross-multiplying, you get the answer in just a step:  x = 4

I find that when students cancel before cross-multiplying, they are more apt to get the right answer, and to become less frustrated, for the numbers they deal with remain small.

For example, in the last problem, if the student had not canceled at all, he would have a cross-multiplication mess of:

6 x 28 = 42x

That sort of problem just opens up the door to arithmetic mistakes. But canceling before cross-multiplying closes that door since it makes the numbers smaller and easier to manage.

So now you get a chance to practice.

First use horizontal cancelling to get the answer to these
proportions. Those who’d like an added challenge might like to try them in their head:

a)   x/12  =  3/4

b)  3/7  =  x/35

c)   z/48  =  7/12

d)  y/56  =  7/8

Now go really wild. Use both horizontal and vertical canceling to make quick work of these proportions:

e)  x/9  =  16/36

f)   x/22  =  30/66

g)  32/56  =  y/14

h)  13/q  =  65/35

And here are the answers to these problems:

a)  x  =  9

b)  x  =  15

c)   z  =  28

d)   y  =  49

e)   x  =  4

f)   x  =  10

g)   y  =  8

h)  q  =  7