# How To Tackle Problems Involving Venn Diagrams

Problems involving Venn Diagrams do make a compulsory appearance on the Quantitative section of the GMAT® . But oftentimes it makes a lot of sense to treat these problems like logical reasoning problems rather Venn Diagrams. In fact in some cases you might not need to draw any diagram at all! The GMAT® problem below is the best illustration of the same.

DIRECTIONS: Each data sufficiency problem below consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you are to option

A if statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
B if statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
C if BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient;
D if EACH statement ALONE is sufficient to answer the question asked;
E if statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

At least 100 students in a school study Japanese. If 4% of students who study French also study Japanese, do more students in the school study French than Japanese?
(1) 16 students study both French and Japanese
(2) 10% of students at school who study Japanese also study French

One of the first instincts of test-takers on encountering such a problem is to draw two intersecting circles or a two by two matrix to represent the data.

On the GMAT the standard method might result in lost time as the test always leaves the door open for you to think logically about the information at hand rather than follow set patterns.

So it makes a lot of sense to first actively process the information in the question before you draw the Venn diagram, if necessary, rather than passively transfer the information into a Venn Diagram.

Statement (1) says that 16 students study both French and Japanese.
The question says 4% of the students studying French also study Japanese.

Therefore if 4% of the number of French-studying (F) students is equal to 16 or .4F=16 and F = 400. The question only says that the number of students studying Japanese is at least 100, which can be greater than or less than 400. Hence, (1) alone is insufficient to answer the question.

Statement (2) says that 10% of the students who study Japanese also study French.
Therefore, students who study both French and Japanese are equal to
4% of French-studying (F) students and
10% of Japanese-studying (J) students
In other words, .04F = .1J, this means F has to be greater than J.
Hence the question can be answered using (2) alone.

So, it can be seen that there is no need to draw Venn Diagrams at all for this problem!

A few guidelines to solve Data Sufficiency problems involving Venn Diagrams

• Refrain from drawing anything while reading the question
• Draw the first diagram, if necessary, after reading the first statement
• Draw a separate diagram for Statement (2) so that information from Statement (1) is not carried over; this results in test-takers choosing option (C) assuming that both statements are required

Remember, Venn Diagrams are just a way of representation and logic precedes representation. The common theme running throughout the GMAT Quant will be logical reasoning since aptitude cannot be plugging values into formulas!