Dimensional Analysis of Rates: Fresh Take

  • Write a proportion to express a rate or ratio
  • Use units to determine which conversion factors are needed for dimensional analysis
  • Use a conversion factor to convert a rate

Proportions and Rates

The Main Idea 

Ratio: A ratio is a way of comparing or relating one quantity to another. It is a mathematical expression that presents the relationship between two numbers indicating how many times the first number contains the second.

Rate: A rate is a special kind of ratio, used to compare quantities of different kinds. Rates are often used in everyday life, such as when we talk about speed (miles per hour), cost (dollars per item), or pay (dollars per hour). For example, if a car travels [latex]60[/latex] miles in [latex]2[/latex] hours, the rate is [latex]60[/latex] miles per [latex]2[/latex] hours, or [latex]30[/latex] miles per hour.

Unit Rate: A unit rate is a rate in which the second quantity in the comparison is one unit. In other words, it’s the rate for just one of something. For example, if a car travels [latex]60[/latex] miles in [latex]2[/latex] hours, the unit rate is [latex]30[/latex] miles per [latex]1[/latex] hour, which is often simply expressed as 30 miles per hour. Unit rates are useful in understanding the relationship between two units and are commonly used in everyday life, such as when comparing prices at the grocery store (e.g., dollars per pound).

A proportion is a statement that two ratios or rates are equivalent. It is an equation that equates two ratios.

Solve the proportion [latex]\displaystyle\frac{5}{3}=\frac{x}{6}[/latex] for the unknown value [latex]x[/latex].

A map scale indicates that [latex]\frac{1}{2}[/latex] inch on the map corresponds with [latex]3[/latex] real miles. How many miles apart are two cities that are [latex]\displaystyle{2}\frac{1}{4}[/latex] inches apart on the map?

Suppose you’re tiling the floor of a [latex]10[/latex] ft by [latex]10[/latex] ft room, and find that [latex]100[/latex] tiles will be needed. How many tiles will be needed to tile the floor of a [latex]20[/latex] ft by [latex]20[/latex] ft room?

You can view the transcript for “An Intro to Ratios | What is a Ratio? | Understanding Ratios | Math with Mr. J” here (opens in new window).

You can view the transcript for “How to Solve Proportions Using Cross Multiplication | Solving Proportions | Math with Mr. J” here (opens in new window).

Dimensional Analysis of Rates

The Main Idea Converting rates involves transforming a given rate into a different unit of rate. This is achieved by applying the appropriate conversion factors in a process known as dimensional analysis.

Dimensional Analysis: A mathematical technique used to convert one unit of measurement to another by using conversion factors.

Conversion Factors: These are ratios that express the relationship between two different units of measurement.

You can view the transcript for “1.2.3 – Converting Rates Using Dimensional Analysis” here (opens in new window).

You can view the transcript for “Dimensional Analysis and Rates” here (opens in new window).