To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

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

\( \frac{8 x 3}{8 x 3} \) + \( \frac{8 x 2}{12 x 2} \)

\( \frac{24}{24} \) + \( \frac{16}{24} \)

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

\( \frac{24 + 16}{24} \) = \( \frac{40}{24} \) = 1\(\frac{2}{3}\)

If 77% of the town's 14,000 voters cast ballots the number of votes cast is:

(\( \frac{77}{100} \)) x 14,000 = \( \frac{1,078,000}{100} \) = 10,780

The mayor got 71% of the votes cast which is:

(\( \frac{71}{100} \)) x 10,780 = \( \frac{765,380}{100} \) = 7,654 votes.

First, solve for \( \left|a + 1\right| \):

\( \left|a + 1\right| \) - 9 = -3
\( \left|a + 1\right| \) = -3 + 9
\( \left|a + 1\right| \) = 6

The value inside the absolute value brackets can be either positive or negative so (a + 1) must equal + 6 or -6 for \( \left|a + 1\right| \) to equal 6:

So, a = -7 or a = 5.

The number of students taking German or Spanish is 6 + 13 = 19. Of that group of 19, 4 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 19 - 4 = 15 who are taking at least one language. 20 - 15 = 5 students who are not taking either language.

The equation for this sequence is:

a_n = a_n-1 + 4(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 4(6 - 1)
a₆ = 41 + 4(5)
a₆ = 61