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:

\( \sqrt{\frac{81}{4}} \)
\( \frac{\sqrt{81}}{\sqrt{4}} \)
\( \frac{\sqrt{9^2}}{\sqrt{2^2}} \)
\( \frac{9}{2} \)
4\(\frac{1}{2}\)

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{2!}{6!} \)
\( \frac{2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{6 \times 5 \times 4 \times 3} \)
\( \frac{1}{360} \)

The equation for this sequence is:

a_n = a_n-1 + 2(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 2(6 - 1)
a₆ = 21 + 2(5)
a₆ = 31

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

\( \frac{3}{100} \) x 5 = \( \frac{3 \times 5}{100} \) = \( \frac{15}{100} \) = 0.15 errors per hour

So, in an average hour, the machine will produce 5 - 0.15 = 4.85 error free parts.

The machine ran for 24 - 5 = 19 hours yesterday so you would expect that 19 x 4.85 = 92.1 error free parts were produced yesterday.

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for distance:

distance = \( \text{speed} \times \text{time} \)
distance = \( 40mph \times 1h \)
40 miles