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Searching for ## Ratio, Rate, and Proportion

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Pre-Algebra

Students determine if two ratios form a proportion, and solve for the unknown value of a proportion.

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After completing this tutorial, you will be able to complete the following:

- Determine if two ratios form a proportion.
- Determine if a given relationship is proportional.
- Use the properties of proportions to solve problems involving proportional relationships.
- Solve for the unknown value of a proportion in mathematical statements and real-life situations.

Proportionality is a type of relation between quantities. Two quantities are proportional if they are constant multiples of each other. More specifically, two variables x and y vary directly if there is a nonzero constant k such that y = k . x. The constant k is called the constant of proportionality.

There is a constant ratio between proportional quantities, which can be seen by manipulating the equation y = k . x.

This fact is the basis for proportional reasoning questions, such as the one below.

The cost of purchasing coffee is proportional to the amount purchased. If five pounds of coffee cost $7.00, how much can be purchased with $21.00?

The quantities are proportional, which implies that the following ratio is constant for any amount of coffee purchased.

In particular, the constant ratio can be determined by using the information given in the problem: five pounds of coffee cost $7.00.

Therefore, if x is the number of pounds of coffee that can be purchased for $21.00, then the ratio must equal the ratio above. This fact can be expressed in an equation, from which it follows that x equals 15 pounds.

In order to facilitate discussions and justify solution methods, there are a handful of terms referring to the parts and properties of a proportional relationship. Suppose and that and d are nonzero. The quantities b and c are called the means, whereas the quantities a and d are called the extremes. Simple arithmetic yields the means-extremes, or cross-product, property of proportions.

Cross-product property

The majority of proportional reasoning problems can be solved by using some variation of this property.

Approximate Time | 10 Minutes |

Pre-requisite Concepts | Students should know the concept of division and be able to evaluate expressions using operations with fractions. |

Course | Pre-Algebra |

Type of Tutorial | Concept Development |

Key Vocabulary | cross-product, equivalent ratios, extremes |