GMAT Math : Venn alternative

Let’s start with a GMAT Math question to illustrate the concept:

There are 120 employed people in a city. Of the employed people, 60 have skills of a mechanic, 70 have skills of an electrician, 80 have skills of a plumber and 80 have skills of a surveyor. Let M1 denote the number of people with skills of a mechanic, electrician, plumber and surveyor. What is the minimum value of M1?

A. 5

B. 20

C. 15

D. 0

E. None of the above

Readers are encouraged to take a crack at the above problem… 🙂

The traditional approach would suggest drawing four Venn diagrams and figuring out the intersections. However, when the question is asking for ‘minimums’ or ‘maximums’ this approach can get complicated and does not come intuitively enough.

Trick: utilize a straight-line visualization approach for sets when dealing with minima and maxima intersections

Let us see this in action :

We represent the set of 120 people by a straight line. The 60 people with mechanic skills and the 70, 80 and 80 people with electrical, plumbing and surveying skills are represented by additional straight lines below.

It is quite easy to picture the ‘overlaps’ this way

To find the minimum value of M1, we try to shift the straight lines around for the minimum intersection.

..clearly we cannot reduce the intersection below 10 people.

We go with option (E)

Question for readers: What is the maximum intersection possible? 🙂