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 take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{81}{64}} \)
\( \frac{\sqrt{81}}{\sqrt{64}} \)
\( \frac{\sqrt{9^2}}{\sqrt{8^2}} \)
\( \frac{9}{8} \)
1\(\frac{1}{8}\)

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

\( \frac{6}{100} \) x 7 = \( \frac{6 \times 7}{100} \) = \( \frac{42}{100} \) = 0.42 errors per hour

So, in an average hour, the machine will produce 7 - 0.42 = 6.58 error free parts.

The machine ran for 24 - 4 = 20 hours yesterday so you would expect that 20 x 6.58 = 131.6 error free parts were produced yesterday.

The yearly interest charged on this loan is the annual interest rate multiplied by the amount borrowed:

interest = annual interest rate x loan amount

i = (\( \frac{6}{100} \)) x $100
i = 0.02 x $100

No payments were made so the total amount due is the original amount + the accumulated interest:

The number of students taking German or Spanish is 15 + 9 = 24. Of that group of 24, 7 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 24 - 7 = 17 who are taking at least one language. 24 - 17 = 7 students who are not taking either language.