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:

\( \frac{5!}{2!} \)
\( \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} \)
\( \frac{5 \times 4 \times 3}{1} \)
\( 5 \times 4 \times 3 \)
60

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 - 4 = 20 hours yesterday so you would expect that 20 x 6.65 = 133 error free parts were produced yesterday.

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:

\( \sqrt{\frac{16}{49}} \)
\( \frac{\sqrt{16}}{\sqrt{49}} \)
\( \frac{\sqrt{4^2}}{\sqrt{7^2}} \)
\(\frac{4}{7}\)

First, solve for \( \left|a + 1\right| \):

\( \left|a + 1\right| \) - 2 = -9
\( \left|a + 1\right| \) = -9 + 2
\( \left|a + 1\right| \) = -7

The value inside the absolute value brackets can be either positive or negative so (a + 1) must equal - 7 or --7 for \( \left|a + 1\right| \) to equal -7:

So, a = 6 or a = -8.

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.