## The Halls and the Oateses

Try your hand at this question:

Two family reunions are happening at the same hotel, the Oates reunion and the Hall reunion. All 100 guests at the hotel attend at least one of the reunions. If 40 people attend the Oates reunion and 62 people attend the Hall reunion, how many people attend both reunions?

(A) 2

(B) 5

(C) 10

(D) 16

(E) 22

Answer after the jump.

Explanation:

The question asks us to determine the number of people who attend both the Hall and Oates reunions. Since this is a question about overlapping sets, we will find it useful to draw a Venn diagram.

We will call the number of people who attend only the Hall reunion *h*, the number of people who attend only the Oates reunion *o*, and the number of people who attend both *b*. The information provided in the prompt allows us to set up three equations:

*o* + *h* + *b* = 100

40 – *b* = *h*

62 – *b* = *o*

If we add the second and third equations together, we get: 102 – 2*b* = *h* + *o*.

Rearranging the first equation, we get: 100 – *b* = *h* + *o*.

Now subtracting this equation from the equation above it, we find: 2 – *b* = 0, or *b* = 2.

As expected, 2 people attend both the Hall and Oates reunions. They play all of their hits and a few new songs.

Answer **choice A** is correct.

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The Halls and the Oateses | B-School Admissions FormulaNovember 11, 2009 at 6:02 PM