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 30% the radius (and, consequently, the total area) increases by \( \frac{30\text{%}}{2} \) = 15%

A factor is a positive integer that divides evenly into a given number. For example, the factors of 8 are 1, 2, 4, and 8.

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

6\( \sqrt{5} \) x 8\( \sqrt{5} \)
(6 x 8)\( \sqrt{5 \times 5} \)
48\( \sqrt{25} \)

Now we need to simplify the radical:

48\( \sqrt{25} \)
48\( \sqrt{5^2} \)
(48)(5)
240

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{4}{8} \) x \( \frac{3}{9} \) = \( \frac{4 x 3}{8 x 9} \) = \( \frac{12}{72} \) = \(\frac{1}{6}\)

You have 32 out of the total of 400 raffle tickets sold so you have a (\( \frac{32}{400} \)) x 100 = \( \frac{32 \times 100}{400} \) = \( \frac{3200}{400} \) = 8% chance to win the raffle.