In order to find how many additional workers are needed to staff the extra crews you first need to calculate how many workers are on a crew. There are 4 workers at the company now and that's enough to staff 2 crews so there are \( \frac{4}{2} \) = 2 workers on a crew. 5 crews are needed for the busy season which, at 2 workers per crew, means that the roofing company will need 5 x 2 = 10 total workers to staff the crews during the busy season. The company already employs 4 workers so they need to add 10 - 4 = 6 new staff for the busy season.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{4!}{5!} \)
\( \frac{4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{5} \)
\( \frac{1}{5} \)

The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 50% the radius (and, consequently, the total area) increases by \( \frac{50\text{%}}{2} \) = 25%

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{2}{5} \) x \( \frac{4}{6} \) = \( \frac{2 x 4}{5 x 6} \) = \( \frac{8}{30} \) = \(\frac{4}{15}\)

A ratio of 3:1 means that there are 3 home fans for every one visiting fan. So, of every 4 fans, 3 are home fans and \( \frac{3}{4} \) of every fan in the stadium is a home fan:

50,000 fans x \( \frac{3}{4} \) = \( \frac{150000}{4} \) = 37,500 fans.