To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{6 x 5}{9 x 5} \) + \( \frac{8 x 3}{15 x 3} \)

\( \frac{30}{45} \) + \( \frac{24}{45} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{30 + 24}{45} \) = \( \frac{54}{45} \) = 1\(\frac{1}{5}\)

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{3!}{5!} \)
\( \frac{3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{5 \times 4} \)
\( \frac{1}{20} \)

To raise a term with an exponent to another exponent, retain the base and multiply the exponents:

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 factors of 80 are [1, 2, 4, 5, 8, 10, 16, 20, 40, 80] and the factors of 80 are [1, 2, 4, 5, 8, 10, 16, 20, 40, 80]. They share 10 factors [1, 2, 4, 5, 8, 10, 16, 20, 40, 80] making 80 the greatest factor 80 and 80 have in common.