Of the 40 researchers at a certain company, 60 percent work on project P and 30 percent work on project...

Understanding the Question

We need to find how many researchers work on project P but NOT on project Q.

Given Information

Total researchers: 40
60% work on project P → \(0.60 \times 40 = 24\) researchers
30% work on project Q → \(0.30 \times 40 = 12\) researchers


What We Need to Determine
To find "P but not Q," we need to know how many researchers work on BOTH projects. Once we know the overlap, we can subtract it from the 24 who work on P.

In other words: P only = Total P - Overlap

We know Total P = 24, but we don't know the overlap yet. If we can determine this overlap, we can answer the question definitively.

Analyzing Statement 1

Statement 1 tells us: 4 researchers work on project Q but not on project P.

Let's think about what this means:

12 people total work on Q
4 of them work ONLY on Q (not on P)
Therefore, the remaining \(12 - 4 = 8\) who work on Q must also work on P


This gives us the overlap! If 8 people work on both P and Q, then:

P but not Q = \(24 - 8 = 16\) researchers


Since we can determine a unique answer, Statement 1 is SUFFICIENT.

[STOP - Statement 1 is Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: 12 researchers work on neither project P nor project Q.

Here's where logical reasoning really shines:

40 total researchers
12 work on neither project
So \(40 - 12 = 28\) researchers work on at least one project


But wait - if we add up all the project assignments:

24 work on P
12 work on Q
Total "slots" = \(24 + 12 = 36\)


We have 36 project assignments but only 28 people actually working on projects. The difference (\(36 - 28 = 8\)) represents people being counted twice - these are the researchers working on BOTH projects!

To visualize: When we count "24 on P plus 12 on Q," we're double-counting the 8 people who work on both.

Therefore:

P but not Q = \(24 - 8 = 16\) researchers


Statement 2 is SUFFICIENT.

[STOP - Statement 2 is also Sufficient!]

The Answer: D

Both statements independently allow us to determine the overlap between projects P and Q, which is all we need to answer the question. Each gives us a different piece of information that leads to the same overlap value of 8 researchers.

Answer Choice D: "Each statement alone is sufficient."