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.

**PROBLEMS:**

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**