To fill a 9 gallon tank exactly halfway you'll need 4\(\frac{1}{2}\) gallons of fuel. Each fuel can holds 1\(\frac{1}{2}\) gallons so:

cans = \( \frac{4\frac{1}{2} \text{ gallons}}{1\frac{1}{2} \text{ gallons}} \) = 3

You have 21 out of the total of 300 raffle tickets sold so you have a (\( \frac{21}{300} \)) x 100 = \( \frac{21 \times 100}{300} \) = \( \frac{2100}{300} \) = 7% chance to win the raffle.

To multiply terms with radicals, multiply the coefficients and radicands separately:

9\( \sqrt{7} \) x 5\( \sqrt{9} \)
(9 x 5)\( \sqrt{7 \times 9} \)
45\( \sqrt{63} \)

Now we need to simplify the radical:

45\( \sqrt{63} \)
45\( \sqrt{7 \times 9} \)
45\( \sqrt{7 \times 3^2} \)
(45)(3)\( \sqrt{7} \)
135\( \sqrt{7} \)

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

\( \frac{7}{100} \) x 8 = \( \frac{7 \times 8}{100} \) = \( \frac{56}{100} \) = 0.56 errors per hour

So, in an average hour, the machine will produce 8 - 0.56 = 7.4399999999999995 error free parts.

The machine ran for 24 - 8 = 16 hours yesterday so you would expect that 16 x 7.4399999999999995 = 119 error free parts were produced yesterday.