It can be solved by calculating all five answers:

\(\displaystyle \dpi{100} \small x+y = \frac{5}{6}\)

\(\displaystyle \dpi{100} \small x-y=\frac{1}{6}\)

\(\displaystyle \dpi{100} \small (xy)^{2}=\frac{1}{36}\)

\(\displaystyle \dpi{100} \small x^{2}+y^{2}=\frac{13}{36}\)

\(\displaystyle \dpi{100} \small (x+y)^{2}=\frac{25}{36}\)

The smallest is \(\displaystyle \dpi{100} \small \frac{1}{36}\).

We need to convert this percentage into a fraction.  This is one of the conversions you should remember.

\(\displaystyle \dpi{100} \small 66\frac{2}{3}\)% = 0.666666 = \(\displaystyle \dpi{100} \small \frac{2}{3}\)

\(\displaystyle 18 \ast \frac{2}{3} = 6 \ast 2 = 12\)

It's easiest to convert the fractions into decimals.

\(\displaystyle \small \frac{3}{8}\ =\ 0.375\)

\(\displaystyle \small \frac{2}{4}\ =\ 0.50\)

\(\displaystyle \small \frac{2}{5}\ =\ 0.40\)

Therefore, the correct answer is 0.25.

To get the smallest possible \(\displaystyle A - B\), subtract the greatest possible \(\displaystyle B\) from the smallest possible \(\displaystyle A\); this is \(\displaystyle 4 - 3 = 1\).

To get the greatest possible \(\displaystyle A - B\), subtract the smallest possible \(\displaystyle B\) from the greatest possible \(\displaystyle A\); this is  \(\displaystyle 5 - 2 = 3\).

\(\displaystyle \small 1 \leq A -B \leq 3\)

\(\displaystyle \small x + y = 10\)

\(\displaystyle \small (x + y )^{2} = 10^{2}\)

\(\displaystyle \small x^{2} + 2xy + y^{2} = 100\)

\(\displaystyle \small \small x^{2} + 2xy + y^{2} - 4xy= 100 - 4xy\)

\(\displaystyle \small \small x^{2} - 2xy + y^{2} = 100 -4xy\)

\(\displaystyle \small (x - y)^{2} = 100 - 4 (20)\)

\(\displaystyle \small \small (x - y)^{2} = 20\)

Either \(\displaystyle \small \small \small x - y = \sqrt{20} = 2 \sqrt{5}\) or \(\displaystyle \small \small x - y = - \sqrt{20} = - 2 \sqrt{5}\)

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:

\(\displaystyle \frac{3}{5}v_a=3\)

\(\displaystyle v_a=3\bigg(\frac{5}{3}\bigg)=5\)

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:

\(\displaystyle \frac{t_a(t)}{v_a(v)}\)

Where \(\displaystyle t_a(t)\) is the number of Trainee arrests times the number of Trainees and \(\displaystyle v_a(v)\) is the number of Veteran arrests times the number of Veterans

\(\displaystyle \frac{t_a(t)}{v_a(v)}=\frac{3(3)}{5(1)}=\frac{9}{5}\)

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 \(\displaystyle 9+5=14\)

The fraction of the arrests made by the Veterans is:

\(\displaystyle \frac{5}{14}\)

We start by simplifying the denominators:

\(\displaystyle \frac{1}{1-\frac{2}{5}}=\frac{1}{\frac{5}{5}-\frac{2}{5}} = \frac{1}{\frac{3}{5}}\) and \(\displaystyle \frac{1}{1+\frac{1}{5}} = \frac{1}{\frac{5}{5}+\frac{1}{5}}=\frac{1}{\frac{6}{5}}\)

We know that: \(\displaystyle \frac{1}{\frac{3}{5}}+\frac{1}{\frac{6}{5}} = \frac{5}{3}+\frac{5}{6}\)

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

\(\displaystyle \frac{5}{3}+\frac{5}{6}=\frac{10}{6}+\frac{5}{6}=\frac{15}{6}=\frac{5}{2}\)

\(\displaystyle \frac{a}{b} +\frac{c}{d} = \frac{a+c}{bd}\) 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 \(\displaystyle a=1, b=2, c=1, d=2\), then the above expression would read

\(\displaystyle \frac{1}{2} +\frac{1}{2} = \frac{1}{2}\) 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.

\(\displaystyle k\) is such that:

\(\displaystyle \frac{3}{k}-\frac{5}{7}>0\)

Add \(\displaystyle \frac{5}{7}\) to each side of the inequality:

\(\displaystyle \frac{3}{k}>\frac{5}{7}\)

Multiply each side of the inequality by \(\displaystyle 3\):

\(\displaystyle \frac{21}{k}>5\)

Multiply each side of the inequality by \(\displaystyle k\):

\(\displaystyle 21>5k\)

Divide each side of the inequality by \(\displaystyle 5\):

\(\displaystyle k< \frac{21}{5}\)

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

\(\displaystyle k< \frac{21}{5}=\frac{20}{5}+\frac{1}{5}= 4.2\)

The correct answer is \(\displaystyle 4\), since k has to be less than \(\displaystyle 4.2\) for the expression to be greater than zero.