Understanding rates is fundamental in mathematics and its applications in various fields such as economics, science, and everyday life. A rate is essentially a comparison between two different quantities with different units, which allows us to measure one quantity relative to another. When we talk about speed, for example, we are referring to the rate at which distance changes over time. A unit rate is a type of rate with a denominator of one, which means it compares the quantity to one unit of another quantity, making it easier to understand and compare different rates.
rates
A rate is the ratio (fraction) of two quantities.
A unit rate is a rate with a denominator of one.
When calculating rates, it is important to remember how to reduce a fraction. The Equivalent Fractions Property states thatIf [latex]a,b,c[/latex] are numbers where [latex]b\ne 0,c\ne 0[/latex], then [latex]{\dfrac{a\cdot c}{b\cdot c}}={\dfrac{a}{b}}[/latex].
Your car can drive [latex]300[/latex] miles on a tank of [latex]15[/latex] gallons. Express this as a rate.
Expressed as a rate, [latex]\displaystyle\frac{300\text{ miles}}{15\text{ gallons}}[/latex]. We can divide to find a unit rate:[latex]\displaystyle\frac{20\text{ miles}}{1\text{ gallon}}[/latex], which we could also write as [latex]\displaystyle{20}\frac{\text{miles}}{\text{gallon}}[/latex], or just [latex]20[/latex] miles per gallon.
Proportions
Proportions are a cornerstone concept in mathematics, particularly when we seek to understand the relationship between quantities. They provide a way to represent the equality of two ratios or fractions and are immensely useful in various fields, from creating recipes in culinary arts to representing scales in maps or models. Understanding proportions allows us to maintain consistency, predict outcomes, and scale different quantities up or down while keeping the same ratio.
proportion equation
A proportion equation is an equation showing the equivalence of two rates or ratios.
Recall that we can use letters we call variables to “stand in” for unknown quantities. Then we can use the properties of equality to isolate the variable on one side of the equation. Once we have accomplished that, we say that we have “solved the equation for the variable. “In the example below, you are asked to solve the proportion (an equality given between two fractions) for the unknown value [latex]x[/latex].
Ex. Solve the proportion:
[latex]\dfrac{7}{3}=\dfrac{x}{15}[/latex]
We see that the variable we wish to isolate is being divided by [latex]15[/latex]. We can reverse that by multiplying on both sides by [latex]15[/latex].
However, we earlier found that [latex]300[/latex] miles on [latex]15[/latex] gallons gives a rate of [latex]20[/latex] miles per gallon. If we multiply the given [latex]40[/latex] gallon quantity by this rate, the gallons unit “cancels” and we’re left with a number of miles:
Notice if instead we were asked “how many gallons are needed to drive [latex]50[/latex] miles?” we could answer this question by inverting the [latex]20[/latex] mile per gallon rate so that the miles unit cancels and we’re left with gallons:
Notice that with the miles per gallon example, if we double the miles driven, we double the gas used. Likewise, with the map distance example, if the map distance doubles, the real-life distance doubles. This is a key feature of proportional relationships, and one we must confirm before assuming two things are related proportionally.
You have likely encountered distance, rate, and time problems in the past. This is likely because they are easy to visualize and most of us have experienced them first hand. In our next example, we will solve distance, rate and time problems that will require us to change the units that the distance or time is measured in.
A bicycle is traveling at [latex]15[/latex] miles per hour. How many feet will it cover in [latex]20[/latex] seconds?
To answer this question, we need to convert [latex]20[/latex] seconds into feet. If we know the speed of the bicycle in feet per second, this question would be simpler. Since we don’t, we will need to do additional unit conversions. We will need to know that [latex]5280 \text{ft} = 1[/latex] mile. We might start by converting the [latex]20[/latex] seconds into hours:
Sometimes when working with rates, proportions, and percents, the process can be made more challenging by the magnitude of the numbers involved. Sometimes, large numbers are just difficult to comprehend.
The 2022 U.S. military budget was [latex]$773[/latex] billion. To gain perspective on how much money this is, answer the following questions.
What would the salary of each of the [latex]2.3[/latex] million Walmart employees in the US be if the military budget were distributed evenly amongst them?
If you distributed the military budget of 2022 evenly amongst the [latex]332[/latex] million people who live in the US, how much money would you give to each person?
If you converted the US budget into [latex]$100[/latex] bills, how long would it take you to count it out – assume it takes one second to count one [latex]$100[/latex] bill.
Here we have a very large number, about [latex]$773,000,000,000[/latex] written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities.
If that amount of money was used to pay the salaries of the [latex]2.3[/latex] million Walmart employees in the U.S., each would earn over [latex]$336,000[/latex].
There are about [latex]332[/latex] million people in the U.S. The military budget is about [latex]$2,328[/latex] per person.
If you were to put [latex]$773[/latex] billion in [latex]$100[/latex] bills, and count out [latex]1[/latex] per second, it would take [latex]245[/latex] years to finish counting it.