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Ratios and Proportions
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Ratios and Proportions

Objective Learn the concepts of ratio and proportion and to solve proportion problems.

 

Ratios

Mathematically, a ratio is simply a fraction viewed as a division of two numbers. Its importance is that it is used to compare two numbers (the numerator and the denominator of the fraction). For example, the ratio of x to y is simply the fraction . A ratio can also be written as x to y or x:y.

In the following example you can see how a ratio compares two numbers.

 

Example 1

In Ms. Cunningham's class there are 18 girls and 14 boys. Write the ratio of boys to girls.

Solution

The ratio of boys to girls in the class can be written as 14 to 18 or 14:18 or When simplified, this ratio can also be expressed as 7 to 9 or 7:9 or

 

Proportions

An equation that states that two ratios are equal is called a proportion. The equation is a proportion. This proportion can also be written as 14:18 = 7:9

More generally a proportion will often involve variables. Solving these problems usually involves elementary algebra, because they involve solving for the value of a variable. To solve these kinds of problems we use a process called cross multiplying. The cross multiplying fact should be explained carefully to your students.

Cross Multiplication Fact Suppose 0 y and 0 b are not zero. Then occurs exactly when xb = ya.

Example 1

Solve for a .

Solution

Use the cross multiplication fact and solve the resulting equation.

3(15) = 5( a )

45 = 5a

9 = a

 

Why is the cross multiplication fact true?

Notice that occurs exactly when . Subtract these fractions by finding a common denominator.

occurs exactly when .

Remember that a fraction equals 0 only when its numerator equals 0. So for this equation, when xb - ya = 0 or when xb = ya. This is the result of cross multiplying .

The above discussion about why the cross multiplication fact is true is a very important example of mathematical reasoning. Namely, it uses a computational technique (subtraction of fractions) to derive a general principle (cross multiplication fact). In order to solidify this concept you should do many examples of how this technique is applied.

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