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How to Use Venn Diagrams to Solve Problems

Learn how to solve SAT math problems using Venn diagrams.

By
Jason Marshall, PhD
August 12, 2011
Episode #073

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Now that we know how to make and interpret the meaning of Venn diagrams, it’s time to put our knowledge to the test and use them to solve problems that you might encounter in the real world. And when I say put our knowledge to the test, I mean that literally since problems like the one we’ll look at today frequently show up on standardized tests like the SAT…and lots of other places too.

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Today’s SAT-Inspired Question

Imagine you’re at a local city council meeting at which an important vote about where to build a new dog park is taking place. The options are to build the dog park at either Washington Park or Waterfront Park, or to build a new dog park at each location. When people who support building at Washington Park are asked to raise their hands, 47 votes are registered. When people who support building at Waterfront Park are asked to raise their hands, 36 votes are registered. As you’re watching the votes come in, you notice that 24 of the voters have raised their hands in support of building parks in both locations.

So the question is: How many people voted? In other words, how can you use the knowledge that 47 people raised their hands for Washington Park and 36 raised their hands for Waterfront Park, while also knowing that 24 people raised their hands for both parks, to figure out how many people cast votes?

The Wrong Answer

Some of you might immediately jump to the conclusion that since 47 and 36 people cast votes for Washington and Waterfront parks, the total number of people who voted must be 47 + 36 = 83. If you made that leap, let me emphasize that each of the voters had the option of voting for more than one park…and we know that 24 of them did exactly that. In other words, some of the 47 people who voted for Washington Park also voted for Waterfront Park. So although there were 47 + 36 = 83 votes cast, there were actually fewer voters. How can we figure out how many?

How to Set Up the Venn Diagram

Well, fortunately we now have a great tool at our disposal that’s ready to help us tackle problems like this. Of course, the tool I’m referring to is the venerable old Venn diagram. Here’s how it works in this case. Let’s start by drawing a rectangle that represents the set of everyone who cast a vote about the dog parks. Next, let’s draw a circle within that rectangle to represent the set of people who raised their hands in support of a dog park at Washington Park. Finally, let’s draw a second circle that represents the set of voters who raised their hands in support of a dog park at Waterfront Park.

As with all Venn diagrams, these two circles must overlap to show all the possible relationships between the various sets. In this case, the overlapping region between the two circles represents the set of people who raised their hands in support of both dog parks. In other words, this overlapping region is the intersection of these two sets.

That’s all well and good, but how does it help us answer our original question: How many people voted?

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