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5.2 Proportions

A proportion is a statement that two ratios are equal. Thus

40 10
=
20 5
2
is a proportion, because both ratios are equivalent to the ratio 2 : 1 (= the fraction ).
1

Aproportion is a statement of the form

a c
=
b d
where b, d ̸=0.

5.2.1 The cross-product property
There is a very useful fact about proportions: If the proportion a = c is true, then the “cross-products,”
b d
ad and bc, are equal, and, conversely, if the cross-products are equal, then the proportion must be true.
Example 189. Verify that the cross-products are equal in the (true) proportion

40 10
= .
20 5
Solution. 40 × 5= 10 × 20 = 200.

The general cross-product property is stated below for reference:

a c
= if and only if ad = bc.
b d

To prove the cross-product property, we need to say a little bit about equations.An equation is a
mathematical statement of the form X = Y .If the equation X = Y is true, and N is any nonzero
number, then the following equations are also true, and have exactly the same solution(s):

X Y
N × X = N × Y and = .
N N
a c
Now go back to the proportion = , and multiply both sides by the nonzero number bd.We get
b d
a c
bd = bd .
b d
Cancelling b on the left and d on the right yields the cross-product property, ad = bc. Conversely, if
we have four numbers a, b, c, d,(b, d ̸=0), and we know that ad = bc,then,dividing both sides by
c
the nonzero number bd gives us the proportion a = .
b d

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