To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [10, 20, 30, 40, 50] making 10 the smallest multiple 2 and 10 share.

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

\( \frac{5 x 5}{2 x 5} \) + \( \frac{2 x 1}{10 x 1} \)

\( \frac{25}{10} \) + \( \frac{2}{10} \)

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

\( \frac{25 + 2}{10} \) = \( \frac{27}{10} \) = 2\(\frac{7}{10}\)

To fill a 12 gallon tank exactly halfway you'll need 6 gallons of fuel. Each fuel can holds 1\(\frac{1}{2}\) gallons so:

cans = \( \frac{6 \text{ gallons}}{1\frac{1}{2} \text{ gallons}} \) = 4

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

First, solve for \( \left|c - 8\right| \):

\( \left|c - 8\right| \) + 4 = 2
\( \left|c - 8\right| \) = 2 - 4
\( \left|c - 8\right| \) = -2

The value inside the absolute value brackets can be either positive or negative so (c - 8) must equal - 2 or --2 for \( \left|c - 8\right| \) to equal -2:

So, c = 10 or c = 6.

The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.