Calculate the number of vans needed by dividing the number of people that need transported by the capacity of one van:

vans = \( \frac{75}{14} \) = 5\(\frac{5}{14}\)

So, it will take 5 full vans and one partially full van to transport the entire team making a total of 6 vans.

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 add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40] and the first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 4 and 8 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{8 x 2}{4 x 2} \) + \( \frac{7 x 1}{8 x 1} \)

\( \frac{16}{8} \) + \( \frac{7}{8} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{16 + 7}{8} \) = \( \frac{23}{8} \) = 2\(\frac{7}{8}\)

A prime number is an integer greater than 1 that has no factors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, and 11.

To add these radicals together their radicands must be the same:

6\( \sqrt{48} \) + 9\( \sqrt{3} \)
6\( \sqrt{16 \times 3} \) + 9\( \sqrt{3} \)
6\( \sqrt{4^2 \times 3} \) + 9\( \sqrt{3} \)
(6)(4)\( \sqrt{3} \) + 9\( \sqrt{3} \)
24\( \sqrt{3} \) + 9\( \sqrt{3} \)

Now that the radicands are identical, you can add them together: