Ratio, Rate, and Proportion

If there are seven boys and twelve girls in a class, then the ratio of boys to girls can be expressed as 7 to 12, , or 7:12. A ratio compares the size, or magnitude, of two numbers. Two other related concepts, rate and proportion, together with ratio, are used for solving many real-world problems that involve comparing different quantities.

Calculating Ratios

Suppose a parking garage contains six blue cars and two green cars. The ratio of blue cars to green cars can be expressed as a fraction . If the two green cars leave the garage, then there are zero green cars and the ratio becomes . Division by zero, however, is not defined, so this form of the ratio is meaningless. Expressing a ratio as a fraction, , is valid as long as b is not equal to zero. However, the ratio of blue to green cars can still be written as 6 to 0 or 6:0.

Ratios can be used to compare quantities of the same type of objects and of different types. There are two types of ratios that compare quantities of the same type. When the comparison is to part of the whole to the whole, then the ratio is a part-whole ratio. When the comparison is to part of the whole to another part of the whole, then the ratio is a part-part ratio.

For example, suppose there is a wall made up of twelve blocks, five white blocks and seven red blocks. The ratio of white blocks to the total number of blocks is , which is a part-whole ratio. The ratio of white blocks to red block is , which is a part-part ratio.

Figuring Rates

A ratio that compares quantities of different types is called a rate. A phone company charges $0.84 for 7 minutes of long distance, and a student reads 10 pages in 8 minutes. The first rate is minutes, which is equal to minute (obtained by dividing both terms by 7). The second rate is minutes, which is equal to minutes.

The rate in the first example is called a unit rate. In a unit rate, the denominator quantity is 1. A unit rate is often used for comparing the cost of two similar items. If a 12-ounce box of cereal sells for $2.40, and a 16-ounce box sells for $2.88, which is the better buy? The unit rate of the first box is $0.20/ounce ( ounces), and the unit rate of the second box is $0.18/ounce ( ounces). Therefore, the second box is a better buy.

Understanding Proportions

When two ratios are equal, the mathematical statement of that equality is called a proportion. The statement that is a proportion. If is equal to , then is called a proportion. To find out if two ratios form a proportion, one can evaluate the cross product. If and are ratios, then the two ratios form a proportion if ad = bc.

Proportions are used when three quantities are given, and the fourth quantity is an unknown. Suppose a person drives 126 miles in 3 hours. At the same speed, how many miles would the drivertravel in 4 hours? Because the rate of travel remains the same, a proportion can be written.

The unknown quantity, the distance traveled by the car in 4 hours, can be indicated by x. Therefore, the two ratios and form a proportion.



Multiplying both sides by 4, or using cross multiplication, yields x = 168 miles.

Summarizing the Concepts

A ratio compares the magnitude of two quantities. When the quantities have different units, then a ratio is called a rate. A proportion is a statement of equality between two ratios.