To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 14 are [14, 28, 42, 56, 70, 84, 98]. The first few multiples they share are [42, 84] making 42 the smallest multiple 6 and 14 share.

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

\( \frac{4 x 7}{6 x 7} \) - \( \frac{7 x 3}{14 x 3} \)

\( \frac{28}{42} \) - \( \frac{21}{42} \)

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

\( \frac{28 - 21}{42} \) = \( \frac{7}{42} \) = \(\frac{1}{6}\)

The hourly error rate for this machine is the error rate in parts per 100 multiplied by the number of parts produced per hour:

\( \frac{5}{100} \) x 9 = \( \frac{5 \times 9}{100} \) = \( \frac{45}{100} \) = 0.45 errors per hour

So, in an average hour, the machine will produce 9 - 0.45 = 8.55 error free parts.

The machine ran for 24 - 8 = 16 hours yesterday so you would expect that 16 x 8.55 = 136.8 error free parts were produced yesterday.

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 divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{-5b^6}{8b^3} \)
\( \frac{-5}{8} \) b^{(6 - 3)}
-\(\frac{5}{8}\)b³

If 30% of the town's 47,000 voters cast ballots the number of votes cast is:

(\( \frac{30}{100} \)) x 47,000 = \( \frac{1,410,000}{100} \) = 14,100

The mayor got 61% of the votes cast which is:

(\( \frac{61}{100} \)) x 14,100 = \( \frac{860,100}{100} \) = 8,601 votes.