A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

To simplify this fraction, first find the greatest common factor between them. The factors of 36 are [1, 2, 3, 4, 6, 9, 12, 18, 36] and the factors of 72 are [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72]. They share 9 factors [1, 2, 3, 4, 6, 9, 12, 18, 36] making 36 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{36}{72} \) = \( \frac{\frac{36}{36}}{\frac{72}{36}} \) = \( \frac{1}{2} \)

The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

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

vans = \( \frac{30}{12} \) = 2\(\frac{1}{2}\)

So, it will take 2 full vans and one partially full van to transport the entire team making a total of 3 vans.

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

\( \frac{3}{9} \) ÷ \( \frac{3}{5} \) = \( \frac{3}{9} \) x \( \frac{5}{3} \)

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

\( \frac{3}{9} \) x \( \frac{5}{3} \) = \( \frac{3 x 5}{9 x 3} \) = \( \frac{15}{27} \) = \(\frac{5}{9}\)