The Knewton Blog


To Venn or Not to Venn? Depends on the Question

Posted in Test Prep on August 3, 2011 by

Many students like to ask me the following question when dealing with overlapping sets problems: “Which are better? Venn diagrams or charts?”

I’ve found that students like to ask because every student has a personal preference. Some don’t want to deal with Venn Diagrams because labeling them can get confusing at times. Others stay away from charts, either because they find charts more confusing or because they prefer the compactness of the Venn diagram.

So in essence, what the student really wants me to do is either justify a natural inclination (“I just find charts easier”) or tell them that this inclination is completely wrong and needs to be squelched immediately (“I really like Venn Diagrams, but I always get them wrong, so should I switch to charts?”).

As with so many things in life, the answer lies somewhere in between. Neither a chart nor a Venn diagram is inherently bad. But here’s the rub: When should you use one versus the other?

Let’s take a look at the following Official Guide Data Sufficiency question:

If 75 percent of the guests at a certain banquet ordered dessert, what percent of the guests ordered coffee?

(1) 60 percent of the guests who ordered dessert also ordered coffee.

(2) 90 percent of the guests who ordered coffee also ordered dessert.

Here, we only have two groups (coffee and dessert), and they overlap. In this case, the Venn Diagram is definitely the way to go, because you can very easily see a visual representation of the overlap and separation:

Even if the question dealt with three foods (e.g. coffee, dessert, entree) instead of just two, a Venn diagram would still be the way to go, because everything fits under one basic category: food. Even if there are different overlaps between foods (e.g. coffee and dessert, dessert and entree, entree and dessert), there is the possibility of a single overlap between all three.

On the other hand, let’s take a look at the following Knewton question:

A certain zoo has 288 mammals, 25 percent of which are female. What percent of the mammals in the zoo were born at the zoo?

1. The number of male mammals that were born at the zoo is three times the number of female mammals who were not born at the zoo.

2. The number of male mammals that were not born at the zoo is three times the number of male mammals that were born at the zoo.

Think about what might happen if you go after this question with Venn diagrams. You’d say, “Okay, so, I’ll create a Venn diagram for ‘male’ and ‘born’. But wait… I also need one for ‘male’ and ‘not born’. Wait, so I need two Venn diagrams?? And I haven’t even considered the females yet?! How am I supposed to solve this?? This is too hard!!! I give up.”

Before you are from this question untimely ripped, let’s recall the words of the inimitable George Carlin: “Calm down…have some dip!” If you’re struggling that much to set up a problem, you’re likely going about it in an inefficient way. The big problem here is that if you consider only the split of male/born, you consider none of woman/born. There are several overlaps from two completely separate categories (gender and birthplace), so Venn diagrams don’t make much sense.

But with a chart, things get MUCH more manageable:

Aside from demonstrating my basic online chart-making skills :), this figure works out much better, because it accounts for all the overlaps. And notice that we also leave space for all the totals, which will be just as important.

The big takeaway is: If you have several categories (e.g. male/female, born/not born) and these categories overlap with each other, a chart is definitely the way to go.

So how do we go about solving these two problems? Ah, that’s for you to find out :). Take a stab at them on your own, and feel free to post your solutions in the comments. Bonus points to those who create charts and/or diagrams and post links for us to see!