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

\( \frac{1}{9} \) x \( \frac{3}{6} \) = \( \frac{1 x 3}{9 x 6} \) = \( \frac{3}{54} \) = \(\frac{1}{18}\)

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:

\( \frac{6 x 3}{8 x 3} \) + \( \frac{9 x 2}{12 x 2} \)

\( \frac{18}{24} \) + \( \frac{18}{24} \)

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

\( \frac{18 + 18}{24} \) = \( \frac{36}{24} \) = 1\(\frac{1}{2}\)

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

By buying two shirts, Frank will save $14 x \( \frac{45}{100} \) = \( \frac{$14 x 45}{100} \) = \( \frac{$630}{100} \) = $6.30 on the second shirt.

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

\( \frac{8}{100} \) x 10 = \( \frac{8 \times 10}{100} \) = \( \frac{80}{100} \) = 0.8 errors per hour

So, in an average hour, the machine will produce 10 - 0.8 = 9.2 error free parts.

The machine ran for 24 - 2 = 22 hours yesterday so you would expect that 22 x 9.2 = 202.4 error free parts were produced yesterday.