To see what I mean, check out this basic sets problem:
In a photography exhibition, some photographs were taken by Octavia the photographer, and some photographs were framed by Jack the framer. Jack had framed 24 photographs taken by Octavia, and 12 photographs taken by other photographers. If 36 of the photographs in the exhibition were taken by Octavia, how many photographs were either framed by Jack or taken by Octavia?
How can we organize all this information in a systematic manner?
There are many approaches, but few are satisfactory. Some instructors insist on approaching sets problems using formal set theory. For students, however, using abstract mathematical symbols can often make such a confusing problem even more difficult to grasp. Venn diagrams are not much better; although they are familiar tools, they do not display information efficiently.
To solve these deficiencies, Master GMAT uses the technique of the sets table. Based on our teaching experience, a simple 3×3 table has proven itself as the most organized and effective approach for solving sets problems. Once you learn this technique, you will not want to go back.
The first step in creating a sets table is to identify the two pairs of complementary sets in the problem. (Every pair of complementary sets will include the entire sample space. For example, a pair of complementary sets might be “Smokers/Non-smokers” or “Male/Female”) In this problem, the first pair of complementary sets consists of those photos that Jack framed and did not frame; let’s call this “Jack” and “Not Jack.” The second pair consists of those photos that Octavia took and those she didn’t; we’ll call this “Octavia” and “Not Octavia.”
Once you identify the two pairs, draw up a 3×3 table. Label the headers of the rows with the first pair of complementary sets. In this case, the headers will be “Octavia,” “Not Octavia,” and “Total.” We then label the headers of the columns with the remaining two sets, “Jack,” “Not Jack,” and “Total.”
One of the key properties of sets tables is that within a row, the cells will add up to the right-most cell within that row (under the header “Total”). Within a column, also, each cell will add up to the bottom-most cell within that column (also under the header “Total”). This property will help us glean hidden information in the question stem.
Once we’ve drawn out the sets table, the next step is to fill in each cell with the proper data:
Each cell represents the intersection of the row and column. For example, the cell with the number 24 represents the number of photos that were both framed by Jack and taken by Octavia.
At this point, we use the special property of the sets table to quickly fill in the missing value:
We’re almost done. Returning to the question stem, we are asked to find the number of photographs that were either taken by Octavia or framed by Jack. It seems as if the question stem is merely asking you to add the two values: 36 + 36 = 72.
Before you do so, pay attention to an interesting fact: the 24 photographs which were both taken by Octavia and framed by Jack (those in the Octavia / Jack box) are actually counted twice: Once as part of the top row, and again as part of the left column. These photographs, however, should be counted only ONCE.
Therefore, the correct calculation must discount them by subtracting them from the total: 36 + 36 – 24 = 48, which is the final answer.
This technique of the sets table will now allow you to tackle all of the sets problems on the GMAT. Whenever you see a sets problem, draw up a 3×3 table to organize your information.
Now that you’ve learned the powerful technique of the sets table, it’s time to try your hand at a challenging problem:
According to a certain estimation, the total number of black cats is 25% greater than the number of black male cats, and the number of all female cats is 5 times the number of black female cats. If the male cats are 50% of all cats, then what percent of male cats are black?