100 Day Challenge 2020

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:

  • 1818 are beautiful,
  • 2222 are loving, and
  • 1717 are talented.

More specifically,

  • 77 are both beautiful and loving,
  • 33 are both beautiful and talented, and
  • 1111 are both loving and talented.

Also, there is 11 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=79,18+22+17+7+3+11+1=79, but there are not 7979 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 1818 beautiful cats, that includes the 77 cats that are both beautiful and loving, the 33 that are both beautiful and talented, and that 11 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=57.18+22+17=57. The number of beautiful cats, 18,18, includes the cats who are beautiful and loving. The number of loving cats, 22,22, also includes the cats who are beautiful and loving. So, by adding 18+22,18+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=36.18+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. 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 1818 cats).). In today’s challenge, can you apply this kind of thinking to calculating an area?

Today's Challenge

Three rectangles have areas of 72,80,72, 80, and 130.130.

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

What is the total shaded area?