A number in scientific notation has the format 0.000 x 10^exponent. To convert to scientific notation, move the decimal point to the right or the left until the number is a decimal between 1 and 10. The exponent of the 10 is the number of places you moved the decimal point and is positive if you moved the decimal point to the left and negative if you moved it to the right:

0.0001004 in scientific notation is 1.004 x 10^-4

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:

\( \frac{5}{100} \) x 7 = \( \frac{5 \times 7}{100} \) = \( \frac{35}{100} \) = 0.35 errors per hour

So, in an average hour, the machine will produce 7 - 0.35 = 6.65 error free parts.

The machine ran for 24 - 5 = 19 hours yesterday so you would expect that 19 x 6.65 = 126.4 error free parts were produced yesterday.

To simplify this fraction, first find the greatest common factor between them. The factors of 24 are [1, 2, 3, 4, 6, 8, 12, 24] and the factors of 60 are [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]. They share 6 factors [1, 2, 3, 4, 6, 12] making 12 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{24}{60} \) = \( \frac{\frac{24}{12}}{\frac{60}{12}} \) = \( \frac{2}{5} \)

42% of the water consumption is industrial use and 25% is personal use so (42% - 25%) = 17% more water is used for industrial purposes. 35,000 gallons are consumed daily so industry consumes \( \frac{17}{100} \) x 35,000 gallons = 5,950 gallons.