Calculate the number of vans needed by dividing the number of people that need transported by the capacity of one van:

vans = \( \frac{53}{16} \) = 3\(\frac{5}{16}\)

So, it will take 3 full vans and one partially full van to transport the entire team making a total of 4 vans.

The number of students taking German or Spanish is 6 + 6 = 12. Of that group of 12, 4 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 12 - 4 = 8 who are taking at least one language. 22 - 8 = 14 students who are not taking either language.

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

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:

\( \frac{2!}{6!} \)
\( \frac{2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{6 \times 5 \times 4 \times 3} \)
\( \frac{1}{360} \)

To divide fractions, invert the second fraction and then multiply:

\( \frac{4}{9} \) ÷ \( \frac{3}{9} \) = \( \frac{4}{9} \) x \( \frac{9}{3} \)

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

\( \frac{4}{9} \) x \( \frac{9}{3} \) = \( \frac{4 x 9}{9 x 3} \) = \( \frac{36}{27} \) = 1\(\frac{1}{3}\)