j_shreyans wrote:100 people are attending a newspaper conference. 45 of them are writers and more than 38 are editors. Of the people at the conference, x are both writers and editors and 2x are neither. What is the largest possible number of people who are both writers and editors?
A)6
B)16
C)17
D)33
E)84
Total = writers + editors - both + neither.
The big idea is to SUBTRACT the overlap.
When we count the total number of writers and the total number of editors, the OVERLAP -- everyone who is BOTH a writer and an editor -- is counted TWICE.
Thus, the people who are in BOTH groups must be subtracted from the total so that they are not double-counted.
In the equation above:
Total = 100.
Writers = 45.
Both = x.
Neither = 2x.
Plugging these values into the equation:
100 = 45 + editors - x + 2x
55 = editors + x.
In order to MAXIMIZE x, we need to MINIMIZE the number of editors.
Since more than 38 of the people are editors, the minimum number of editors = 39:
55 = 39 + x
x = 16.
The correct answer is
B.
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