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.

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.

To simplify this fraction, first find the greatest common factor between them. The factors of 16 are [1, 2, 4, 8, 16] and the factors of 76 are [1, 2, 4, 19, 38, 76]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{16}{76} \) = \( \frac{\frac{16}{4}}{\frac{76}{4}} \) = \( \frac{4}{19} \)

First, solve for \( \left|b - 2\right| \):

\( \left|b - 2\right| \) - 4 = -9
\( \left|b - 2\right| \) = -9 + 4
\( \left|b - 2\right| \) = -5

The value inside the absolute value brackets can be either positive or negative so (b - 2) must equal - 5 or --5 for \( \left|b - 2\right| \) to equal -5:

So, b = 7 or b = -3.

You have 31 out of the total of 350 raffle tickets sold so you have a (\( \frac{31}{350} \)) x 100 = \( \frac{31 \times 100}{350} \) = \( \frac{3100}{350} \) = 9% chance to win the raffle.