Constructing Proportions to Solve Real World Problems

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will concentrate on proportions. Through definition and example, we will learn how to construct proportions and how to use proportions to solve real world problems.


Suppose you recently got a job, and you just received your first paycheck. You worked 22 hours, and your paycheck is $223. Next week, you are scheduled to work 31 hours, and you are wondering how much your paycheck for that week will be. The answer to this question can be found using proportions.

A proportion is an equation that sets two ratios equal to each other, where a ratio is a fraction comparing two different values.


Let's take a look at how to find an unknown in a proportion.

Cross Multiplication

In a proportion, if one of the numbers is unknown, we can use a process called cross multiplication to solve for that unknown. To perform cross multiplication, we multiply the numerator of the left hand ratio by the denominator of the right hand ratio, and we multiply the denominator of the left hand ratio by the numerator of the right hand ratio. Then we set the two products equal to each other and solve for the unknown.


For example, suppose we have the following proportion.


Let's use cross multiplication to solve for the unknown quantity in the proportion.


Constructing Proportions to Solve Real World Problems

Let's look back at our paycheck example. We can use proportions to solve this problem, but first we have to construct the proportion that represents this problem. To construct a proportion, we just need to set up two ratios comparing the same quantities and then set them equal. In our example, we have the number of hours you work and the amount of your paycheck as our quantities.

We know that when you work 22 hours, you make $223. This information is enough to set up one ratio comparing the number of hours worked to the amount of money made.


The rest of the information that we have is that next week you will be working 31 hours, and you want to know how much money you'll make for that many hours. The unknown here is the amount of money you'll make for 31 hours of work. Let's call the unknown x, and set up another ratio comparing these two quantities.


We now have two ratios comparing number of hours worked to the amount of money made. All we have to do is set them equal, and we have our proportion.


It is important to note that you want your quantities in the numerator and denominator to be consistent in both ratios of the proportion. We see that we did this with our example, since hours is in the numerator in both ratios and dollars is in the denominator in both ratios. Lastly, we can use the proportion to solve for the unknown.


We see that you will get a paycheck of $314.22 for your 31 hours of work next week. Aren't these proportions handy when it comes to real world applications? Let's look at one more example.

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