Understanding rates is fundamental in mathematics and its applications in various fields such as economics, science, and everyday life. A rate<\/strong> 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<\/strong> 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.<\/p>\r\n A rate <\/strong>is the ratio (fraction) of two quantities.<\/p>\r\n <\/p>\r\n A unit rate<\/strong> is a rate with a denominator of one.<\/p>\r\n<\/div>\r\n<\/section>\r\n Ex. [latex]\\dfrac{500}{20}=\\dfrac{25\\cdot 20}{1\\cdot 20}=\\dfrac{25}{1}=25[\/latex]<\/p>\r\n<\/section>\r\n 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.<\/p>\r\n A proportion equation<\/strong> is an equation showing the equivalence of two rates or ratios.<\/p>\r\n<\/div>\r\n<\/section>\r\n <\/p>\r\n\r\nWe 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]. <\/p>\r\n 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<\/em> unit \u201ccancels\u201d and we\u2019re left with a number of miles:<\/p>\r\n <\/p>\r\n Notice if instead we were asked \u201chow many gallons are needed to drive [latex]50[\/latex] miles?\u201d we could answer this question by inverting the [latex]20[\/latex] mile per gallon rate so that the miles<\/em> unit cancels and we\u2019re left with gallons:<\/p>\r\n 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.<\/p>\r\n 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.<\/p>\r\n <\/p>\r\n Now we can multiply by the [latex]15[\/latex] miles\/hr:<\/p>\r\n <\/p>\r\n Now we can convert to feet:<\/p>\r\n <\/p>\r\n We could have also done this entire calculation in one long set of products:<\/p>\r\n <\/p>\r\n View the following video to see this problem worked through.<\/p>\r\n https:\/\/youtu.be\/fyOcLcIVipM<\/p>\r\nrates<\/h3>\r\n
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\r\n[reveal-answer q=\"378596\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"378596\"]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.[\/hidden-answer]<\/section>\r\nProportions<\/h2>\r\n
proportion equation<\/h3>\r\n
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\r\nEx. Solve the proportion:
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\r\n[reveal-answer q=\"526887\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"526887\"]We could certainly answer this question using a proportion:
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\r\n[reveal-answer q=\"946318\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"946318\"]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\u2019t, 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:
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