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.

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.