You have 24 out of the total of 400 raffle tickets sold so you have a (\( \frac{24}{400} \)) x 100 = \( \frac{24 \times 100}{400} \) = \( \frac{2400}{400} \) = 6% chance to win the raffle.

To multiply terms with radicals, multiply the coefficients and radicands separately:

3\( \sqrt{4} \) x 8\( \sqrt{5} \)
(3 x 8)\( \sqrt{4 \times 5} \)
24\( \sqrt{20} \)

Now we need to simplify the radical:

24\( \sqrt{20} \)
24\( \sqrt{5 \times 4} \)
24\( \sqrt{5 \times 2^2} \)
(24)(2)\( \sqrt{5} \)
48\( \sqrt{5} \)

The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 65% the radius (and, consequently, the total area) increases by \( \frac{65\text{%}}{2} \) = 32\(\frac{1}{2}\)%

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