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 50% the radius (and, consequently, the total area) increases by \( \frac{50\text{%}}{2} \) = 25%

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

The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 5 and 9 have in common.

If 47% of the town's 41,000 voters cast ballots the number of votes cast is:

(\( \frac{47}{100} \)) x 41,000 = \( \frac{1,927,000}{100} \) = 19,270

The mayor got 59% of the votes cast which is:

(\( \frac{59}{100} \)) x 19,270 = \( \frac{1,136,930}{100} \) = 11,369 votes.

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 64 are [1, 2, 4, 8, 16, 32, 64]. They share 5 factors [1, 2, 4, 8, 16] making 16 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{16}{64} \) = \( \frac{\frac{16}{16}}{\frac{64}{16}} \) = \( \frac{1}{4} \)