## Venn Rectangles

We can all agree: every cat is beautiful, or loving, or talented, or a combination of the three. At the Brilliant Cat Shelter, every cat up for adoption has been categorized as follows:

• $18$ are beautiful,
• $22$ are loving, and
• $17$ are talented.

More specifically,

• $7$ are both beautiful and loving,
• $3$ are both beautiful and talented, and
• $11$ are both loving and talented.

Also, there is $1$ special cat who is beautiful and loving and talented.

So how many cats are there in total at the Brilliant Cat Shelter? It would be a mistake to simply add up the numbers above. To see why, keep reading. Otherwise, jump right to today’s challenge which appears to be completely different. (It’s not!)

The sum of all the numbers above is $18+22+17+7+3+11+1=79,$ but there are not $79$ cats at the shelter. Consider the Venn diagram below. We don't actually know the values of the three important regions in this figure that are highlighted here:

When we count the $18$ beautiful cats, that includes the $7$ cats that are both beautiful and loving, the $3$ that are both beautiful and talented, and that $1$ special cat who is beautiful and loving and talented.

By including all of these cats in the sum, some cats get double- or triple-counted, so the calculation is inaccurate. Now let’s calculate the actual number of cats at the shelter.

Suppose we start with the sum $18+22+17=57.$ The number of beautiful cats, $18,$ includes the cats who are beautiful and loving. The number of loving cats, $22,$ also includes the cats who are beautiful and loving. So, by adding $18+22,$ we have counted those beautiful and loving cats twice.

Now, however, we need to subtract the number of beautiful and loving cats once so that the result is only counting them once.

By the same logic, we need to subtract the number of beautiful and talented cats, as well as the number of loving and talented cats. Then we get $18+22+17-7-3-11=36.$

But we can’t forget about that one special cat who is beautiful and loving and talented! The number of beautiful cats, the number of loving cats, and the number of talented cats all include this special cat. Also, the number of beautiful and loving cats, the number of beautiful and talented cats, and the number of loving and talented cats all include this special cat. So we included this cat in the sum three times, and then we removed it from the sum three times. We need to include this special cat exactly one time, so the total number of cats is $18+22+17-7-3-11+1=37.$

Can you take this problem one step further? How many cats at the shelter only have one attribute — only beautiful, or only loving, or only talented?

Try it for yourself $($the answer is $18$ cats$).$ In today’s challenge, can you apply this kind of thinking to calculating an area?

# Today's Challenge

Three rectangles have areas of $72, 80,$ and $130.$

When we arrange them so that they overlap, this is what we get:

What is the total shaded area?