A university library surveyed 240 graduate students regarding the research methodologies u… — GMAT

A university library surveyed $240$ graduate students regarding the research methodologies used in their dissertations: Quantitative Analysis ( $Q$ ), Qualitative Interviews ( $I$ ), and Archival Research ( $A$ ). Every student surveyed used at least one of these three methodologies. A total of $130$ students used Quantitative Analysis, $110$ used Qualitative Interviews, and $90$ used Archival Research. Additionally, $40$ students used both Quantitative Analysis and Qualitative Interviews, $30$ used both Qualitative Interviews and Archival Research, and $35$ used both Quantitative Analysis and Archival Research. How many students used all three research methodologies?

$15$ Answer
B
$25$
C
$40$
D
$90$
E
$105$

Answer

15 students used all three research methodologies.

According to the Principle of Inclusion-Exclusion for three sets,

\text{Total} = |Q| + |I| + |A| - (|Q \cap I| + |I \cap A| + |Q \cap A|) + |Q \cap I \cap A| + \text{Neither}

. Substituting the given values gives

240 = 130 + 110 + 90 - (40 + 30 + 35) + x + 0

, which simplifies to

240 = 225 + x

, yielding

x = 15

. Thus, the option stating 15 is correct.

Step-by-Step Solution

State the three-set inclusion-exclusion principle formula for total population.

Total =

|Q| + |I| + |A| - (|Q \cap I| + |I \cap A| + |Q \cap A|) + |Q \cap I \cap A| + \text{Neither}

To set up an algebraic equation relating the total number of students to their set intersections.

Substitute the known numerical values into the formula.

240 = 130 + 110 + 90 - (40 + 30 + 35) + |Q \cap I \cap A| + 0

Every student used at least one methodology, so

\text{Neither} = 0

Simplify the equation and solve for the target variable

|Q \cap I \cap A|

240 = 330 - 105 + |Q \cap I \cap A| \implies 240 = 225 + |Q \cap I \cap A| \implies |Q \cap I \cap A| = 15

Subtracting 225 from 240 yields the number of students who used all three methodologies.

Key Concept

Three-Set Inclusion-Exclusion Principle

Estimated Time:1m 30s

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