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.

52% of the water consumption is industrial use and 34% is personal use so (52% - 34%) = 18% more water is used for industrial purposes. 10,000 gallons are consumed daily so industry consumes \( \frac{18}{100} \) x 10,000 gallons = 1,800 gallons.

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:

\( \frac{5!}{2!} \)
\( \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} \)
\( \frac{5 \times 4 \times 3}{1} \)
\( 5 \times 4 \times 3 \)
60

First, solve for \( \left|b + 8\right| \):

\( \left|b + 8\right| \) - 4 = 7
\( \left|b + 8\right| \) = 7 + 4
\( \left|b + 8\right| \) = 11

The value inside the absolute value brackets can be either positive or negative so (b + 8) must equal + 11 or -11 for \( \left|b + 8\right| \) to equal 11:

So, b = -19 or b = 3.

To find the average of these 4 numbers add them together then divide by 4:

\( \frac{10 + 4 + 9 + 5}{4} \) = \( \frac{28}{4} \) = 7