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 area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 42 meters so the equation becomes: 2w + 2h = 42.

Putting these two equations together and solving for width (w):

2w + 2h = 42
w + h = \( \frac{42}{2} \)
w + h = 21
w = 21 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 21 - 2w
3w = 21
w = \( \frac{21}{3} \)
w = 7

Since h = 2w that makes h = (2 x 7) = 14 and the area = h x w = 7 x 14 = 98 m²

52% of the water consumption is industrial use and 20% is personal use so (52% - 20%) = 32% more water is used for industrial purposes. 40,000 gallons are consumed daily so industry consumes \( \frac{32}{100} \) x 40,000 gallons = 12,800 gallons.

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

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

vans = \( \frac{62}{7} \) = 8\(\frac{6}{7}\)

So, it will take 8 full vans and one partially full van to transport the entire team making a total of 9 vans.