To get the smallest possible , subtract the greatest possible  from the smallest possible ; this is .

To get the greatest possible , subtract the smallest possible  from the greatest possible ; this is  .

Either  or 

But without further information, it is impossible to tell which is true. Therefore, the correct choice is that it cannot be determined from the information given.

What we want to do is pick numbers for the number of arrests made during the week and for the number of members in each department.

Let's take the number of arrests first.  Let's say each Trainee arrests 3 criminals.  Then, since the Trainees make 3/5 the number of arrests the Veterans make, we have:

So, each Veteran would arrest 5 criminals.

Next, we know that there are 1/3 as many Veterans as Trainees.  There if we have 3 Trainees, then we have 1 Veteran.

Using this information we can create the following equation for total arrests made by each department:

Where  is the number of Trainee arrests times the number of Trainees and  is the number of Veteran arrests times the number of Veterans

We're almost done.  Since we have 3 Trainees, and each arrests 3 criminals, the total number of Trainee arrests is 9.

Since we have 1 Veteran, and each arrests 5 criminals, the total number of Veteran arrests is 5.

The total number of arrests is

The fraction of the arrests made by the Veterans is:

We start by simplifying the denominators:

 and 

We know that: 

Then we put both fractions to the same denominator, and don't forget to simplify the fraction:

is false because is not displaying the correct way to add two fractions together. When adding fractions we must find a common denominator before adding the numerators.

For example, if , then the above expression would read

 This we know is absurd!

The least common denominator (LCD) is the lowest common multiple of the denominators.

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50

Multiples of 15: 15, 30, 45, 60, 75, 90,105,120, 135, 150

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

The least common denominator is 15.

 is such that:

Add  to each side of the inequality:

Multiply each side of the inequality by :

Multiply each side of the inequality by :

Divide each side of the inequality by :

You can now change the fraction on the right side of the inequality to decimal form.

The correct answer is , since k has to be less than  for the expression to be greater than zero.

In order to add these fractions, we must first find a common denominator for all three. Looking at our three different denominators, we can see that their lowest common denominator is 12. Therefore, we need to rewrite each fraction with a denominator of 12:

Now that the fractions all have a common denominator, we can simply add them together and then simplify the result, recognizing that 27 and 12 have a common factor of 3:

In order to determine which fractions are less than , it's best to convert them to decimals and compare:

Therefore,  is the only choice that is less than .