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ZingPath: Ratios, Rates, and Proportions

Ratio and Rate

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Ratios, Rates, and Proportions

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Ratio and Rate

Math Foundations

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Students learn the definitions of ratio and rate, and apply their properties to solve problems.

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Now You Know

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

  • Define ratio and rate.
  • Write and simplify ratios and rates.
  • Apply the properties of ratios and rates to solve problems.

Everything You'll Have Covered

A ratio is a type of relationship between two quantities. The precise definition of the concept can vary between sources, but usually involves the quotient between two quantities. For example, the ratio between male to female ducks in a pond is the number of male ducks divided by the number of female ducks. Some definitions of ratio stipulate that they be dimensionless, or that the quantities in a ratio have like kind. Some mathematicians might object to such definitions, given the ambiguity of like kind and that restricting to dimensionless quantities could preclude a definition of rate in terms of ratio. Technical considerations aside, ratios provide a measure of how two quantities are related; for example, the ratio of miles to inches in a map, the ratio of flour to water in a recipe, or the ratio of deer to trees in a forest.

Ratios are most commonly expressed in three different ways. The first of these uses fractions. For example, in a recipe calling for 2 cups of flour and 1 cup of water, the ratio of water to flour is . The second expression of this ratio is 1:2. The third method expresses the ratio as a statement: 1 to 2. The last method is most naturally suited to language: The ratio of water to flour is 1 to 2. Apart from these three methods, ratios are sometimes expressed as percentages or decimal numbers.

Rates are a type of ratio that are frequently, but not always, used to measure changing quantities. For example, the rate of change in distance with respect to time is the ratio of change in distance to the change in time. One should not attempt to stringently define the concept of rate or to strictly distinguish ratios from rates. The precise distinction between ratio and rate frequently depends on the context. One might, for example, consider the ratio of male to female ducks in a pond as a rate if wondering how the attractive powers of female ducks affects the pond's male duck population. Other examples of rates involve quantities such as distance and time, pollution and temperature, or dollars and ounces of gold.

This picture is further complicated by the concept of unit rates, which luckily have a commonsense interpretation. For example, unit price is a type of unit rate, namely the price per one unit of some good. In general, unit rates measure an amount of some quantity per one unit of another quantity.

This Activity Object introduces the concepts of ratio, rate, and unit price without resorting to arcane technical restrictions. Ratios are defined as the quotient between quantities, rates are defined as a relationship between quantities, and a commonsense approach is taken to the concept of unit price. Students have the opportunity to practice finding the ratio between various quantities from ordinary life.

Tutorial Details

Approximate Time 20 Minutes
Pre-requisite Concepts apply the concept of division, and evaluate expressions using operations with fractions.
Course Math Foundations
Type of Tutorial Concept Development
Key Vocabulary equivalent ratio, rate, ratio