The ratio of football cards to baseball cards is 7:2 and the ratio of baseball cards to basketball cards is 7:1. To solve this problem, we need the baseball card side of each ratio to be equal so we need to rewrite the ratios in terms of a common number of baseball cards. (Think of this like finding the common denominator when adding fractions.) The ratio of football to baseball cards can also be written as 49:14 and the ratio of baseball cards to basketball cards as 14:2. So, the ratio of football cards to basketball cards is football:baseball, baseball:basketball or 49:14, 14:2 which reduces to 49:2.

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 8 = \( \frac{6 \times 8}{100} \) = \( \frac{48}{100} \) = 0.48 errors per hour

So, in an average hour, the machine will produce 8 - 0.48 = 7.52 error free parts.

The machine ran for 24 - 8 = 16 hours yesterday so you would expect that 16 x 7.52 = 120.3 error free parts were produced yesterday.

A ratio of 3:1 means that there are 3 home fans for every one visiting fan. So, of every 4 fans, 3 are home fans and \( \frac{3}{4} \) of every fan in the stadium is a home fan:

33,000 fans x \( \frac{3}{4} \) = \( \frac{99000}{4} \) = 24,750 fans.

To divide fractions, invert the second fraction and then multiply:

\( \frac{1}{8} \) ÷ \( \frac{1}{5} \) = \( \frac{1}{8} \) x \( \frac{5}{1} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{1}{8} \) x \( \frac{5}{1} \) = \( \frac{1 x 5}{8 x 1} \) = \( \frac{5}{8} \) = \(\frac{5}{8}\)

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:

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