A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

50% of the water consumption is industrial use and 34% is personal use so (50% - 34%) = 16% more water is used for industrial purposes. 30,000 gallons are consumed daily so industry consumes \( \frac{16}{100} \) x 30,000 gallons = 4,800 gallons.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [40, 80] making 40 the smallest multiple 8 and 10 share.

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

\( \frac{3 x 5}{8 x 5} \) - \( \frac{2 x 4}{10 x 4} \)

\( \frac{15}{40} \) - \( \frac{8}{40} \)

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

\( \frac{15 - 8}{40} \) = \( \frac{7}{40} \) = \(\frac{7}{40}\)

If the tiger has consumed 70 pounds of food in 7 days that's \( \frac{70}{7} \) = 10 pounds of food per day. The tiger needs to consume 140 - 70 = 70 more pounds of food to reach 140 pounds total. At 10 pounds of food per day that's \( \frac{70}{10} \) = 7 more days.

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