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:

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

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:

\( \sqrt{\frac{49}{9}} \)
\( \frac{\sqrt{49}}{\sqrt{9}} \)
\( \frac{\sqrt{7^2}}{\sqrt{3^2}} \)
\( \frac{7}{3} \)
2\(\frac{1}{3}\)

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.

A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

40,000 fans x \( \frac{4}{5} \) = \( \frac{160000}{5} \) = 32,000 fans.

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.