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Posts tagged ‘Solving Proportions’

“Hacks” for Slaying Proportions, Part 2: the Diagonal Trick


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

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“Hacks” for Slaying Proportions, Part 1: the Amazing Horizontal Canceling Trick


Proportions can seem intimidating, but they’re actually one of the easiest types of word problems to master. In this series I’ll offer a number of tips that help you conquer algebraic proportion problems. 

But first, a cool shortcut you can use whenever you’re facing down an algebraic proportion …

High-Octane Boost for MathIn working with proportions, I’m amazed that few students know how a canceling process that would help them find the solution more quickly and efficiently.

So I want to share the trick, for all who’ve never seen it.

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

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

6/x  =  3/4

Then we just cross-multiply to get:

3x  =  24, and see that x  =  8.

In other words, we know we can cancel vertically given a proportion, just as we can cancel vertically with any fraction.

What many people don’t know though, is that there’s another way we can cancel when solving proportions — horizontally!

— What? you say.

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

For example,  given the proportion:  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 proportion then becomes:

1/4  =  3/x  [I’m really not kidding.]

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’re more apt to get the right answer, and to get 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 shuts that door since it makes the numbers smaller and easier to manage.

So now you get a chance to practice horizontal cancelling!

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 all of 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