Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 7: Ratios, Proportions, and Percentages 87

your second ratio is big triangle to small triangle just like the first ratio. If you change the order,
the proportion won’t be true.

In any proportion, the product of the means—the two middle terms—is equal to the product of

the extremes—the first and last terms. For example, in the proportion^5
8

15

24

 , 8 v 15 = 5 v 24.

This multiplying of means and extremes is called cross-multiplying. Whenever you have two equal
ratios you can cross-multiply, and the two product will be equal. Knowing that will often let you
find a number that’s missing from the proportion. You could use a question mark or other symbol
to stand for the missing number, but let’s use an x for now.

When you multiply with a variable, especially when you use the variable x, it’s easy to confuse
the variable with the times sign v, so you may want to use other ways to write multiplication,
like a dot or parentheses. Instead of writing 5 v x, you can write 5™x or 5(x).

Keep in mind that cross-multiplying can only be done in a proportion. You can cross-multiply
when you have two equal ratios, but not in any multiplication with fractions.

DEFINITION

Finding the product of the means and the product of the extremes of a proportion,

and saying that those products are equal, is called cross-multiplying.

Suppose you’re told that two numbers are in ratio 7:4, and the smaller number is 14. But what’s
the larger number? You can use the means-extremes properties of proportions to help you find
out.

If I give you the proportion
x
14

7

4

= , you can use cross-multiplying to help you find the value of

the number I called x. If you multiply the means you get 4 times the unknown number or 4x.

The product of the extremes, 7 v 14, is 98. The product of the means is equal to the product of

the extremes, so 4x must equal 98, or 4x = 98. Dividing 98 by 4 tells you that x^98
4

 24.5.

A

BC

X

Y Z