Solving Proportions
Objective Learn the concept of proportion,
and solve proportion problems.
In this lesson, you will be introduced to the concept of
proportion. This topic is related to the concept of ratio, and
studying proportions will help to solidify your understanding of
ratios. You will also be asked to solve proportion problems using
the technique of cross multiplication. This will help develop
your skills with fractions. In this lesson, you will also see the
use of several fundamental algebra techniques.
Proportions
An equation that shows two ratios are equal is called a
proportion. The equation is a proportion. Neither y nor b can be
zero. This proportion is sometimes also written as x : y = a : b
(read as “x is to y as a is to b”.). Solving proportion
problems usually involves the use of algebra, because one of the
four values in the proportion is unknown and is represented by a
variable. To solve these kinds of problems we use the following
fact.
Key Idea
Suppose y 0 and b
0. If , then xb = ya .
In the Key Idea above, the values xb and ya are called the cross
products of the proportion. The process of finding the
cross products of a proportion is called cross
multiplication.
Example 1
Solve .
Solution
Step 1 Use cross multiplication.
Step 2 Solve the equation from Step 1.
3 Â· 15 
= 5 Â· a 
Write the cross products. 
45 
= 5a 
Divide each side by 5. 
9 
= a 

Step 3 Verify the solution. When a is
replaced by 9, the ratio on the right side of the given
proportion becomes . Dividing numerator and denominator by
their GCF, 3, gives the other ratio, .
Why Does Cross Multiplication Work?
Why do you think that cross multiplication works? A
justification can be derived by using your knowledge about
subtraction of fractions. The proportion is an equation. Show students that if the
fraction is subtracted from both
sides of the equation , the result is . The left side of this equation involves
the subtraction of two fractions with unlike denominators. Remind
students that the first step when subtracting two fractions with
unlike denominators is to rewrite the fractions so they have the
same denominator. Multiplying by and multiplying by will result in a like denominator of yb.
Now we need only subtract the numerators and retain the like
denominator.
But a fraction equals zero only when its numerator equals
zero. So, the value of the expression xb  ya must equal zero.
This only occurs when xb = ya . Therefore, if a proportion is
true, its cross products must be equal.
The above discussion about why the cross multiplication works
is a very important example of mathematical reasoning. Namely it
uses a computational technique (subtraction of fractions) to
derive a general principle (the cross products of a proportion
are equal). In order to solidify your understanding of these
concepts, you should do several examples and try to see how this
principle is applied.
