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:

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

40% of the water consumption is industrial use and 24% is personal use so (40% - 24%) = 16% more water is used for industrial purposes. 10,000 gallons are consumed daily so industry consumes \( \frac{16}{100} \) x 10,000 gallons = 1,600 gallons.

A prime number is an integer greater than 1 that has no factors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, and 11.

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 56 are [1, 2, 4, 7, 8, 14, 28, 56]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{36}{56} \) = \( \frac{\frac{36}{4}}{\frac{56}{4}} \) = \( \frac{9}{14} \)

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

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

\( \frac{2 x 5}{9 x 5} \) - \( \frac{7 x 3}{15 x 3} \)

\( \frac{10}{45} \) - \( \frac{21}{45} \)

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

\( \frac{10 - 21}{45} \) = \( \frac{-11}{45} \) = -\(\frac{11}{45}\)