The commutative property states that, when adding or multiplying numbers, the order in which they're added or multiplied does not matter. For example, 3 + 4 and 4 + 3 give the same result, as do 3 x 4 and 4 x 3.

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%

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

To find the average of these 4 numbers add them together then divide by 4:

\( \frac{16 + 14 + 16 + 14}{4} \) = \( \frac{60}{4} \) = 15

To divide fractions, invert the second fraction and then multiply:

\( \frac{2}{6} \) ÷ \( \frac{4}{5} \) = \( \frac{2}{6} \) x \( \frac{5}{4} \)

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

\( \frac{2}{6} \) x \( \frac{5}{4} \) = \( \frac{2 x 5}{6 x 4} \) = \( \frac{10}{24} \) = \(\frac{5}{12}\)