First, solve for \( \left|a + 4\right| \):

\( \left|a + 4\right| \) - 5 = 7
\( \left|a + 4\right| \) = 7 + 5
\( \left|a + 4\right| \) = 12

The value inside the absolute value brackets can be either positive or negative so (a + 4) must equal + 12 or -12 for \( \left|a + 4\right| \) to equal 12:

So, a = -16 or a = 8.

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

vans = \( \frac{61}{15} \) = 4\(\frac{1}{15}\)

So, it will take 4 full vans and one partially full van to transport the entire team making a total of 5 vans.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

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

\( \frac{5 x 5}{9 x 5} \) + \( \frac{6 x 3}{15 x 3} \)

\( \frac{25}{45} \) + \( \frac{18}{45} \)

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

\( \frac{25 + 18}{45} \) = \( \frac{43}{45} \) = \(\frac{43}{45}\)

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

The factors of a number are all positive integers that divide evenly into the number. The factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56.