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, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

The greatest common factor (GCF) is the greatest factor that divides two integers.

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

vans = \( \frac{31}{8} \) = 3\(\frac{7}{8}\)

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

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40]. The first few multiples they share are [4, 8, 12, 16, 20] making 4 the smallest multiple 2 and 4 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{7 x 2}{2 x 2} \) - \( \frac{5 x 1}{4 x 1} \)

\( \frac{14}{4} \) - \( \frac{5}{4} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{14 - 5}{4} \) = \( \frac{9}{4} \) = 2\(\frac{1}{4}\)

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for distance:

distance = \( \text{speed} \times \text{time} \)
distance = \( 45mph \times 8h \)
360 miles