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 …
In 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