A ratio of 3:1 means that there are 3 home fans for every one visiting fan. So, of every 4 fans, 3 are home fans and \( \frac{3}{4} \) of every fan in the stadium is a home fan:

45,000 fans x \( \frac{3}{4} \) = \( \frac{135000}{4} \) = 33,750 fans.

To simplify this fraction, first find the greatest common factor between them. The factors of 20 are [1, 2, 4, 5, 10, 20] and the factors of 52 are [1, 2, 4, 13, 26, 52]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{20}{52} \) = \( \frac{\frac{20}{4}}{\frac{52}{4}} \) = \( \frac{5}{13} \)

If the tiger has consumed 40 pounds of food in 4 days that's \( \frac{40}{4} \) = 10 pounds of food per day. The tiger needs to consume 100 - 40 = 60 more pounds of food to reach 100 pounds total. At 10 pounds of food per day that's \( \frac{60}{10} \) = 6 more days.

The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

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.