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:

31,000 fans x \( \frac{3}{4} \) = \( \frac{93000}{4} \) = 23,250 fans.

To divide terms with radicals, divide the coefficients and radicands separately:

\( \frac{10\sqrt{42}}{2\sqrt{6}} \)
\( \frac{10}{2} \) \( \sqrt{\frac{42}{6}} \)
5 \( \sqrt{7} \)

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so add the coefficients and retain the base and exponent:

8a² + 3a²
(8 + 3)a²
11a²

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.

To multiply terms with radicals, multiply the coefficients and radicands separately:

4\( \sqrt{6} \) x 9\( \sqrt{7} \)
(4 x 9)\( \sqrt{6 \times 7} \)
36\( \sqrt{42} \)