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.

To fill a 10 gallon tank exactly halfway you'll need 5 gallons of fuel. Each fuel can holds 2\(\frac{1}{2}\) gallons so:

cans = \( \frac{5 \text{ gallons}}{2\frac{1}{2} \text{ gallons}} \) = 2

To simplify this fraction, first find the greatest common factor between them. The factors of 20 are [1, 2, 4, 5, 10, 20] and the factors of 56 are [1, 2, 4, 7, 8, 14, 28, 56]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{20}{56} \) = \( \frac{\frac{20}{4}}{\frac{56}{4}} \) = \( \frac{5}{14} \)

The commutative property states that, when adding or multiplying numbers, the order in which they're added or multiplied does not matter. For example, 3 + 4 and 4 + 3 give the same result, as do 3 x 4 and 4 x 3.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{4!}{3!} \)
\( \frac{4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} \)
\( \frac{4}{1} \)
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