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:

48,000 fans x \( \frac{3}{4} \) = \( \frac{144000}{4} \) = 36,000 fans.

To simplify this fraction, first find the greatest common factor between them. The factors of 32 are [1, 2, 4, 8, 16, 32] and the factors of 68 are [1, 2, 4, 17, 34, 68]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{32}{68} \) = \( \frac{\frac{32}{4}}{\frac{68}{4}} \) = \( \frac{8}{17} \)

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

Solving for time:

time = \( \frac{\text{distance}}{\text{speed}} \)
time = \( \frac{30mi}{30mph} \)
1 hour

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{-a^7}{a^2} \)
\( \frac{-1}{1} \) a^{(7 - 2)}
-a⁵

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

\( \frac{40\sqrt{28}}{8\sqrt{4}} \)
\( \frac{40}{8} \) \( \sqrt{\frac{28}{4}} \)
5 \( \sqrt{7} \)