On a company-sponsored cruise, 2/3 of the passengers were

Another approach is to use the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it.
Here, we have a population of passengers, and the two characteristics are:
- company employee or not company employee
- a manager or not a manager

Okay, onto the question...
Target question: What was the number of company-employee passengers who were NOT managers?

Given: 2/3 of the passengers were company employees and the remaining passengers were guests. 3/4 of the company-employee passengers were managers.

Let's begin by setting up our diagram to show the two sets of characteristics:


Since we don't know the total number of passengers, let's let x = total number of passengers:


If "2/3 of the passengers were company employees and the remaining passengers were guests," then 1/3 are guests. This means the sum of the boxes in the left-hand column is (2/3)x and the sum of the boxes in the right-hand column is (1/3)x:


Then we're told that "3/4 of the company-employee passengers were managers"
So 3/4 of the (2/3)x passenger were managers.
3/4 of (2/3)x = (1/2)x, so (1/2)x passengers were managers:


Since the sum of the two boxes in the left-column must add to (2/3)x, we can conclude that the other box must be (1/6)x, so we can add that here.


IMPORTANT: Since none of the guests were managers, we know that the top-right box contains zero passengers as follows:


Finally, the sum of the two boxes in the right-column must add to (1/3)x, we can conclude that the other box must be (1/3)x, so we can add that here.


Okay, our goal is to determine the number of company-employee passengers who were NOT managers. In other words, we want to know the number of passengers in the bottom-left box. So, let's place a STAR in this box to remind us of this:


We're now ready to check the statements...

Statement 1: There were 690 passengers on the cruise
In other words, x = 690
From this, we can take our diagram and plug in 690 for x:


As we can see, we can determine the number of passengers in every box, which means we can definitely determine the number of company-employee passengers who were NOT managers
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: There were 230 passengers who were guests of the company employees.
In other words, the boxes in the left-hand column have a sum of 230


If (1/3)x = 230, we can determine the value of x.
Once we know the value of x, we can we can definitely determine the number of company-employee passengers who were NOT managers
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer =

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Two practice questions